point (t, x) in the boundary corresponds to an anyon emerging or sinking into the ...... the Fibonacci anyonic model, cannot be obtained from the quantum double ...
Aspects of Topological Quantum Field Theory Rajath Krishna Radhakrishnan Supervisor: Dr. Matthew Buican
Submitted in partial fulfillment of the requirements for the degree of Master of Science of Imperial College London Theoretical Physics Group Department of Physics September, 2018
Abstract In this dissertation, we will define a Topological Quantum Field Theory (TQFT) and discuss some of its properties. We will emphasise on anyonic models, and explore how the algebraic data in an anyonic model is contained in a Modular Tensor Category (MTC). We will define Hopf algebras, and discuss two series of anyonic models obtained from Hopf algebras. Namely, those obtained from the quantum double of a finite group, and quantum groups. We will study how 3D TQFTs and 2D Conformal Field Theories are related to each other, and explore the relationships between Chern-Simons-Witten theory and Wess-Zumino-Witten Models. We will motivate how TQFTs are ‘simpler’ compared to general quantum field theories, and how they could be used to do explicit computation to learn about the subtle properties of QFTs in general. Keeping this goal in mind, we discuss a recent development in defining topological entanglement entropy in Chern-Simons-Witten theories.
Acknowledgments First of all, I would like to express my gratitude to Dr. Matthew Buican for accepting my request to be my supervisor for the project even though I am not a student in his university. I would like to thank him for his guidance, and for his patience while explaining numerous topics to me in detail. I would like to thank my family for their constant support and encouragement. I thank Imperial College for providing workspace in the MSc computer room in which most of this work was done and my friends who made my year in London fun and enjoyable.
Dedicated to my parents.
Contents 1 Introduction
1
2 Anyons and their Statistics
8
3 Algebraic Structure of Anyonic Models 3.1
18
Building up to Modular Tensor Category
18
3.1.1
Category
18
3.1.2
Monoidal Category
20
3.1.3
Braided Monoidal Category
21
3.1.4
Rigid Braided Monoidal Category
22
3.1.5
Ribbon Category
23
3.1.6
Semisimple Ribbon Category
23
3.1.7
Modular Tensor Category
26
3.1.8
Fusion Rules
27
4 Modular Tensor Categories from Hopf Algebras
33
4.1
Hopf Algebra
33
4.2
The Category Z(G) of Representations of Quantum Double of a finite group
38
4.2.1
Quantum Double of a finite group D(G)
38
4.2.2
Representations of D(G) and Category Z(G)
39
4.2.3
Quantum Double of Z2
40
4.2.4
Quantum Double of S3
43
4.3
Quantum Groups and their Representations
45
4.3.1
Universal Enveloping algebra U(g) of Lie algebra g
45
4.3.2
Quantum Group Uq (g)
47
4.3.3
Quantum Group Uq (sl(2))
48
5 Fusion and Braiding in 2D Conformal Field Theories 5.1
50
The Virasoro Algebra and its Representations
50
5.1.1
The Witt algebra and its central extension
51
5.1.2
Highest Weight Representation of Virasoro Algebra
52
5.2
CFTs based on Kac-Moody algebras
53
5.3
Null states and constraints on correlation functions
54
5.4
Fusion Algebra
55
5.5
Verlinde Formula and Modular Invariance
55
5.6
Fusion and Braiding Matrices
56
6 3D Topological Field Theory and 2D Conformal Field Theory 6.1
6.2
6.3
59
Topological Quantum Field Theory
59
6.1.1
59
Atiyah Axioms
Modular Functor of Conformal Field Theory
62
6.2.1
Vertex Operators in CFT and the geometric picture
62
6.2.2
Segal’s Definition
65
Relationships between Topological and Conformal Field Theories
66
6.3.1
Wess-Zumino-Witten Model
66
6.3.2
Chern-Simons-Witten Theory
69
6.3.3
2D Duality Identities and 3D General Covariance
71
7 Entanglement Entropy in Topological Quantum Field Theories
73
7.1
Manifold with torus boundaries and link states
73
7.2
Entanglement entropy for the abelian case G=U(1)
76
7.3
Entanglement entropy for non-abelian case G=SU(2)
77
8 Conclusion
79
1
Introduction
In modern physics, the basic laws of nature are described by quantum field theories. An exception is gravity, and one expects that a crucial part of trying to construct a consistent theory of quantum gravity involves understanding the subtle details of quantum field theories (QFTs). However, QFTs are complicated, and it is only recently that we have been able to find answers to questions that we knew already in the context of quantum mechanics. Most of quantum field theory revolves around scattering amplitudes as it is an observable quantity. However, being a quantum theory, one wonders about the Hilbert space structure of an interacting QFT, the entanglement properties of the states etc. These questions, as important as they are, are difficult to answer within the framework of QFTs. To give a concrete idea of the difficulties involved, let us look into how we can associate a Hilbert space with a spacetime. For this, one has to choose a time foliation of the spacetime. Then, we can associate a Hilbert space with a spacelike hypersurface in this foliation. Let M be the spacetime and let Σ be a spacelike hypersurface in its time foliation. We know that for two local observables Oi (x) and Oj (y)
[Oi (x), Oj (y)] = 0
(1.1)
if x and y are spacelike separated. Any two points on Σ are spacelike separated. Hence, for any x ∈ Σ, a local observable O(x) will commute with every observable inserted at a 0
point x 6= x. If we have a complete set of local observables at x, then we can construct a Hilbert space Hx associated with it. The full Hilbert space H is given by
H=
O
Hx
(1.2)
x∈Σ
Such a decomposition is not always possible. Subtleties arise when there are gauge fields or gravitational anomalies in the theory [1] [2]. We can construct a Hilbert space associated
–1–
Σ
A A
Figure 1. Region A in the spacelike hypersurface Σ and its compliment.
to a region A in Σ. HA =
O
Hx
(1.3)
x∈A
We can also obtain the Hilbert space corresponding to the compliment of the region A, A.
HA =
O
Hx
(1.4)
x∈A
The total Hilbert space H can thus, be written as
H = HA ⊗ H A
(1.5)
Since we now decomposed the total Hilbert space into Hilbert spaces of a region and its compliment, for a given |Ψ i ∈ H, we can define the reduced density matrix
ρA = trA (ρ)
(1.6)
where ρ = |Ψ i hΨ |. As an attempt to understand the entanglement structure of the theory one can formally define the Von-Neumann entanglement entropy of this state.
S(ρ) = trρA ln(ρA )
–2–
(1.7)
Usually, the above quantity is calculated by finding the eigenvalues of ρA . However, since in QFTs, the Hilbert spaces are infinite dimensional, one has to find alternate methods to do it. In general, obtaining an exact solution is difficult and even then, due to ultraviolet effects, without a UV cut-off, the answer would be infinite. This difficulty motivates us to try and answer these questions in a more straightforward setting. Two important class of theories which help in this long-term goal of understanding the details of QFTs are Conformal Field Theories (CFTs) and Topological Quantum Field Theories (TQFTs). A CFT is a quantum field theory in which the Lorentz symmetry is promoted to a bigger symmetry called the conformal symmetry. CFT: Lorentz Symmetry → Conformal Symmetry 0
A conformal transformation is a coordinate transformation f , such that under x → x = f (x), the metric tensor transforms as
gµν (x) → Λ(x)gµν (x)
(1.8)
for some positive scalar function Λ(x). CFTs are theories which are invariant under conformal tranformations. Lorentz transformations are indeed conformal tranformation for λ(x) = 1, and hence the Lorentz group forms a subgroup of the conformal group. But, the conformal group has two additional symmetries, known as the scaling symmetry and special conformal tranformations. These correspond to the coordinate tranformations
x → λx and x →
xµ − (x · x)bµ 1 − 2(b · x) + (b · b)(x · x)
(1.9)
respectively, for a given scalar λ and vector bµ . Thus, the conformal group is a bigger symmetry group than the Lorentz group, and hence it constrains the QFT even further. For example, in QFTs, we require the correlation functions in the theory to be Lorentz invariant. In the case of CFTs, we require them to be conformally invariant which significantly restricts its functional form.
–3–
In 2 dimensions, the set of conformal transformations is the set of holomorphic functions [3]. For a 2D CFT defined on the complex plane, z → f (z) is a conformal transformation, if f (z) is a holomorphic function. Thus, the conformal algebra is infinite dimensional in 2D, and it is so restrictive that in many cases, one can find the correlation functions in the theory from the symmetry alone without writing down a Lagrangian. 2D CFTs are crucial in explaining critical phenomena. Conformal field theories are scale invariant and hence live in the fixed points of the RG flow of a general QFT. Problems in CFTs are often much easier to tackle than their QFT counterparts, and an in-depth understanding of CFTs will give a better understanding of QFTs in general. CFTs have also gathered the interests of mathematicians, and within mathematical physics, they are described by various objects like Vertex Operator Algebras, Modular Functor etc. One can add further constraints on a QFT by demanding that the theory is independent of the metric of the manifold on which it is defined. Such a theory is called a Topological Quantum Field Theory (TQFT). A TQFT has an action which is independent of the metric, i.e., δS = 0 = Tµν δgµν
(1.10)
Thus, the stress-energy tensor of the theory is zero which implies that there is no local dynamics in the theory. Also, since the metric does not play a role, a TQFT is a trivial CFT. Since the input data in a TQFT only contains information about the topology of the manifold, the observables of TQFT should be topological invariants. A TQFT is crucial for explaining experimental results like the Fractional Quantum Hall effect [4]. TQFTs are also known as anyonic models within the context of condensed matter physics, and they play a crucial role in a method of quantum computation known as topological quantum computation [5]. A TQFT can be characterised entirely using an algebraic object called a Modular Tensor Category (MTC). They can also be defined using a more abstract set of axioms and conditions originally given by Atiyah [6]. Even though TQFTs are naturally thought of as ‘topological’ in nature, it has a robust algebraic aspect too. We will explore this aspect by studying the properties of anyons in 3 spacetime dimensions. Unlike in 4
–4–
spacetime dimensions, the nature of wordlines of particles in 3 space-time dimensions allows for a richer statistics than the Fermi-Dirac and Bose-Einstein statistics that we are familiar with. Particles which follow these general statistics are called anyons, and their kinematics can be encompassed entirely in an MTC. After going through the details of how a modular tensor category defines an anyonic model, we will look at specific realisations of MTCs which correspond to some of the most important examples of TQFTs. We will look at the following two families of MTCs 1. MTC formed from the representations of quantum double of a finite group. These are the simplest anyonic models in the sense that it depends on finite groups and many computations can be done explicitly. These form the series of TQFTs called Dijkgraaf-Witten theories [7]. 2. MTC formed from the representations of quantum groups. These theories depend on a choice of a Lie group and correspond to Chern-Simons-Witten theories [8]. Note that a TQFT often refers to a broader class of theories. One can construct theories in which the action does depend on the metric, but the theory has observables which do not depend on the metric. The former theories are called Schwartz type TQFTs while the later ones are called Witten type TQFTs. In general, TQFTs are theories in which the observables are topological invariants. We will be concerned only with Schwartz type TQFTs. TQFTs and CFTs are deeply related to each other. This relationship can be explored from a physical perspective. This is mainly done using the relationship between a particular class of TQFTs called Chern-Simons-Witten theories and a particular class of CFTs called Wess-Zumino-Witten models. One can also look at this relationship from a mathematical perspective, as the relation between modular tensor categories and modular functors. In this dissertation, we will be mainly concerned with 3D TQFTs and 2D CFTs. For a physical picture of the correspondence between 3D TQFTs and 2D CFTs, one can consider the TQFT to be defined on a 3-manifold with boundary and the CFT to be defined on that boundary. It turns out that the algebraic properties of anyons moving in the bulk corresponds to the
–5–
symmetries of the correlation functions in the CFT. For example, consider the anyons to be living in a solid cylinder and let the CFT be defined on the surface of the cylinder. We consider particle worldlines A and B of anyons in the bulk. The insertion of a field at a point (t, x) in the boundary corresponds to an anyon emerging or sinking into the bulk at that point. As we see in figure 2, these anyon worldlines can twist and braid around each other, and it turns out that these braiding properties are exactly same as the symmetries of correlation functions of fields on the surface of the cylinder. For example, a counterclockwise exchange of anyons in the bulk corresponds to moving the fields counter-clockwise on the surface.
B τ A
x Figure 2. Braiding anyon wordlines in the bulk of the cylinder
Recently, there has been an increase in research work on studying entanglement structures in quantum field theories. Even though people had developed ways to calculate the entanglement entropy in field theories, one of the significant advancements which enabled easier calculations is the AdS-CFT correspondence [9]. AdS-CFT and the Ryu-Takayanagi entanglement entropy formula made the entanglement entropy calculations in CFTs easier [10]. Since TQFTs are much more constrained than general QFTs, one could expect that entanglement entropy calculations in TQFTs are simpler. This is indeed the case, and we will be looking at how to define an entanglement entropy for links in Cherns-Simons-Witten
–6–
theory and what information it could give us about the link. Even though one can compute entanglement entropies in a CFT, the infinite dimensional degrees of freedom in the theory make such a computation hard. On the other hand, the Hilbert space of a CSW theory is finite dimensional [8], and hence the entanglement entropy calculations become relatively straightforward. This dissertation provides an introduction to several aspects of TQFTs and its relationship with CFTs. It is organised in the following manner. In Chapter 2, we will start by exploring the relationship between particle statistics and the topology of its worldlines, and define an anyonic model. We will describe the kinematics of particles in 3 spacetime dimensions. These define TQFTs which have been of vital importance in topological quantum computation. We will realise that anyonic theories have a very strong algebraic nature, and in Chapter 3, we will define an algebraic object called a modular tensor category which contains the essential ingredients to define an anyonic model. In Chapter 4, we will study Hopf algebras, and how one can obtain a modular tensor category from them. We will also study two important series of TQFTs, one obtained from the quantum double of a finite group and the other from the representation category of quantum groups. In Chapter 5, we will look at how certain aspects of TQFTs naturally arise in CFTs. In Chapter 6, we will define TQFTs and CFTs more rigorously and study the relationship between them. We will argue the details of this relationship using the correspondence between CSW theories and WZW models, and then we will comment on the general picture. In Chapter 7, we will define a state corresponding to a link in a CSW theory called the link state and define its entanglement entropy. We will see that calculating entanglement entropy and obtaining general analytic formulas for it is considerably simpler in TQFTs than in general quantum field theories. We will conclude this dissertation by mentioning some conjectured relationships between TQFTs and CFTs.
–7–
2
Anyons and their Statistics
The statistics of a group of particles is an exciting physical problem. The periodic table exists because of a crucial property of electrons (and of other fermions) called the Pauli exclusion principle. On the other hand, Bosons are friendly and like to be with each other. However, are these the only possible types of particles? It turns out that the quantum statistics of particles depend crucially on the topology of their worldlines. We can show that in 4 or higher dimensions, the topology of worldlines only allows for two types of particles. More interestingly, in 3 dimensions we have a vast variety of particles called anyons as particle worldlines in 3 dimensions is more complicated. Let us consider the worldlines of two indistinguishable particles with initial positions A1 and B1 and final positions A2 and B2 . The second diagram corresponds to the exchange of the two particles while the first diagram does not involve any exchange. What about more complicated paths like that in diagram 3?
A2
B2
B2
A2
B1
A1
A1
B1
A1
B1
A1
B1
1
2
3
Figure 3. 1. Two particle worldline without exchange 2. Two particle worldline with exchange 3. An example of a complicated two particle wordline
It can be shown that in 4 dimensions, any such worldlines can be reduced to the direct or exchange diagrams. These worldlines with fixed ends are generally called as braids in mathematics, and the above statement essentially means that any complicated braiding (as in diagram 3 of figure 3) can be reduced to that in diagram 1 or 2 of figure 3. We will demonstrate this property with Knots, which are just braids with the two ends connected
–8–
to each other. A mathematical knot embedded in Rd is defined by the map S 1 → Rd . Let us focus on the case d = 3. A trivial knot (unknot) in R3 is the circle itself. The simplest non-trivial knot is the Trefoil knot.
Figure 4. Trefoil knot
Note that given a knot, it is considered trivial if it can be deformed into the unknot. Knots are characterised by crossings like the one shown below. The Trefoil knot is non-trivial, we cannot unknot a Trefoil knot as the knot cannot intersect itself, which means that any two points on the knot cannot have the same 3 coordinate.
Figure 5. Crossings which characterise a Knot
Now, let us look at the case d = 4. It is much harder to imagine knots embedded in 4 dimensions and hence for visual representation; we use the colour of a point on the knot as its fourth coordinate. Again, two points on the knot are not allowed to have the same 4 coordinates. However, since we can always move two parts of the knot through each other as long as they do not have the same colour, we can ‘uncross’ every crossing which renders the knot trivial. Hence, colour provides an extra degree of freedom using which we can unknot every crossing. Using the same argument, we can show that braids are trivial in 4
–9–
dimensions. Anyons live in 3 spacetime dimension, and thus the braiding formed by their worldlines are non-trivial compared to fundamental particles, whose worldlines form trivial braiding as they live in 4 spacetime dimensions. Hence, we see that braiding plays a crucial role in the kinematics of anyons. Suppose we formulate the exchange of two indistinguishable particles in 4 dimensions as an operator E acting on the wavefunction of the two particles. That is, E Ψ (x1 , x2 ) = Ψ (x2 , x1 )
(2.1)
If we exchange the particles twice, we get the following worldline.
Figure 6. Worldlines with double exchange
However, we know that the crossings in the middle can be unknotted, and hence the worldlines reduce to that of particles without an exchange. Thus, we require the operator to satisfy the condition that E 2 =1 which implies that the eigenvalues of E are ±1. If the exchange of particles leaves the wavefunction invariant, the particles are called bosons, and if the same leaves the wavefunction invariant except for an overall minus sign, the particles are called fermions. The topology of worldlines reproduces the two types of particles that we already know exists in 4 dimensions. In quantum theories, the representation of the group corresponding to the symmetry acts on the Hilbert space of the system. In 4 dimensions, the group corresponding to indistinguishable particle exchanges is just the permutation group on n objects Sn . In three dimensions, the anyon exchanges are described
– 10 –
using representations of the Braid Group Bn acting on the Hilbert space. Anyons are not just a mathematical construct. They have been realised experimentally. A major example is the Fractional Quantum Hall Effect [11]. We will briefly describe this experiment and understand different properties of anyons. This will help us to identify the mathematical structure required to develop a theory of anyons. When we expose electrons flowing in a conductor to a perpendicular magnetic field, we can measure a voltage across the conductor as the Lorentz force acting on the electrons forces them to move towards one side of the conductor. The resistance corresponding to that voltage is called the Hall resistance RH , and since the Lorentz force is linear in the applied magnetic field B, we expect the Hall resistance to be linear in the applied magnetic field. Something astonishing happens if the same experiment is done by confining the electrons to a 2D plane using semiconductors at low temperatures. The linear relationship between the Hall resistance and B is broken. The graph for Hall resistance with B has plateau with discontinuous jumps. At those jumps RH changes by
RH =
h ne
(2.2)
where h is Planck’s constant, e is the electric charge and n is a fractional number for which Fractional Quantum Hall Effect gets its name. It is observed that electrons form extended composite particles with fractional charge! Also, the perpendicular magnetic field forms ’flux tubes’ on the 2D semiconductor. These flux tubes and fractionally charged electrons can move and braid around each other. Thus, anyons have several crucial properties. First of all, anyons have charge. The set of possible charges in the theory will be denoted by a, b, c etc. Two anyons with specific charges can be combined, and this process is called fusion. We would like to describe the possible charges of the system after fusing two anyons. Fusion rules of the theory contain information about the possible charges of the system after fusing two anyons. These are
– 11 –
c . Fusion of anyons a and b gives specified by the non-negative integers Nab
a×b=
X
c Nab c
(2.3)
c c = 0 for a particular c, then a and b cannot fuse to give a total charge c. If N c is If Nab ab
non-zero, then it gives the number of possible ways in which a and b can fuse to give a total charge c. Note that 2.3 is symmetric in a and b. The total charge of the system after the fusion of a and b does not depend on whether a is on the left or right. Note that in a general fusion algebra, the fusion rules need not be commutative, though that is the case in c can also be interpreted as anyonic models. The relation 2.3 can be read backwards and Nab
the number of ways in which an anyon with charge c can split into two anyons with charges a and b. c called the fusion space with dimension For every triple a, b, c, we associate a vector space Vab c . V c can be viewed as the space of operators which ‘fuses’ two anyons. Similarly, the Nab ab
vector space Vcab is the splitting space. It is the space of operators which ‘splits’ an anyon c, c into a and b. If ψ ∈ Vcab and ψ † ∈ Vab
c
a
b
ψ† a
ψ b
c
Figure 7. Fusion and Splitting c can be written as |ab; c, µi where µ = 1, 2, ..., N c . The Hermitian inner A basis for Vab ab
product is given by operator multiplication, η † ξ = hη|ξi idc which can be represented diagrammatically as given in figure 8. Also, using completeness, the identity operator acting on two labels can be expanded as in figure 9. When two anyons a and b undergoes a counterclockwise exchange, their total charge remains unchanged. Since fusion is commutative, we have an isomorphism R between the vector c and V c . R is a unitary matrix called the braiding matrix. The components of spaces Vab ba
– 12 –
c
c η†
a
hη|ξi
b ξ c
c
Figure 8. Inner product of operators
a
b
a
b µ
XX c c
a
µ
µ a
b
b
Figure 9. Completeness relation for identity over two elements. The c summation is over the possible intermediate states and the µ summation is over a basis of the fusion space.
the R matrix are defined by a
b
a
X µ µ
b
0
c )µ (Rab µ
0
c
µ
0
c
Figure 10. Action of the R-matrix braids the incoming particles in a fusion
Fusion is also associative. For three anyons a, b and c
(a × b) × c = a × (b × c)
(2.4)
This is because we do not expect the total charge of a system of three anyons to depend on
– 13 –
d , which corresponds to the fusion the order in which they fuse. Thus, the fusion space Vabc
of anyons a, b and c to give d can be decomposed in the following ways, d ∼ Vabc =
M
e Vab ⊗ Vecd ∼ =
M
e
d Var ⊗ Vbcr
(2.5)
r
d , and there exists a These decompositions correspond to different choices of bases for Vabc
unitary matrix F which relates these bases. The components of the F matrix are defined as
a
b
c
a
c
b
µ e
X ν
0
d )e µ ν (Fabc eµν
µ0 ν 0 e0
µ
0 0
0
0
ν
e
0
d
d
Figure 11. Action of the F-matrix in a fusion process
We can consider a sequence of F and R moves and obtain isomorphism between certain Hilbert spaces. These constrain the F and R matrices. There are five different ways to fuse 4 particles, and they all are related by F moves as given in figure 12. This gives a set of consistency conditions on the F matrices. Let the basis in the left diagram be |left; a, bi. Let the basis in the right most diagram be |right; c, di. These two basis vectors can be related in two different ways, by applying the F moves across the top of figure 12,
|left; a, bi =
X
5 d 5 |right; c, di (F12c )a (Fa34 )cb
(2.6)
c,d
and by applying the F moves across the bottom of the figure 12,
|left; a, bi =
X
d c 5 d b |right; c, di (F234 )e (F1e4 )b (F123 )ea
c,d,e
1
Picture taken from [12]
– 14 –
(2.7)
Figure 12. Pentagon Constraint1
Equating the two expressions, we get,
5 F12c
a d
5 Fa34
c b
=
X
d F234
c e
5 F1e4
d b
b F123
e a
(2.8)
e
These constitute a series of equations for every possible choice of {1, 2, 3, 4, 5, a, b, c, d} which is consistent with the fusion rules. These are called the Pentagon equations. We can also consider the 6 ways in which 3 anyons can fuse which are related by F and R matrices. (ref. figure 13). This gives another series of constraints on the F and R matrices. Again, the left and right bases are related by the series of moves across the top of figure 13,
|left; ai =
X
4 c 4 4 b |right; ci (F231 )b R1b (F123 )a
(2.9)
b,c
Also, they are related by the series of moves across the bottom of figure 13,
|left; ai =
X
c 4 c a |right; ci (R13 )(F213 )a R12
c
– 15 –
(2.10)
Figure 13. Hexagon Constraint2
Equating these two expressions, we get X c a c 4 b c 4 4 4 R13 F213 R = F F R 12 231 123 1b a b a
(2.11)
b
These constitute a series of equations for every possible choice of {1, 2, 3, 4, a, c} which is consistent with the fusion rules. These are called the Hexagon equations. One could imagine much more constraints on these two matrices. Given a set of labels and fusion rules, MacLane Coherence Theorem ensures that the Pentagon and Hexagon equations are the only conditions on F and R matrices and the solutions to those give viable anyonic models [13]. It guarantees that if the F and R matrices satisfy the Pentagon and Hexagon equations, then it automatically satisfies all the other more complicated constraints that one could construct. Also, it has been shown that these equations allow only for a finite number of inequivalent solutions, a property known as Oceanu Rigidity. Two solutions to these equations are physically equivalent if they correspond to a simultaneous change of basis in the spaces Vcab . Onceanu Rigidity means that once we quotient over such trivial transformations corresponding to simultaneous basis changes, we are left with only a discrete set of inequivalent solutions. This, in turn, means that the algebraic structure is 2
Picture taken from [12]
– 16 –
rigid. Oceanu Rigidity is proved by considering deformations of F and R matrices [14]. A deformation of F and R matrices is permissible if they satisfy the Pentagon and Hexagon equations up to first order in Taylor expansion. A deformation is called trivial if it can be obtained from a basis change up to first order. Proving rigidity involves taking the quotient, permissible modulo trivial deformations and showing that it vanishes. For an anyonic model defined using a finite group G, with group elements as the labels, group multiplication as the fusion rules and trivial associativity relations, possible deformations using phase factors are classified using the third cohomology space H3 (G, U (1)) [14]. For an arbitrary fusion category C, the non-trivial deformations are again given by a similar object H3 (C) and thus, proving Oceanu Rigidity reduces to proving that H3 (C) is trivial for arbitrary unitary fusion category C. Hence, we start with a set of labels and fusion rules and solve the Pentagon and Hexagon equations. If a solution does not exist, an anyonic model with the given labels and fusion rules is not consistent. If several solutions exist, then it suggests that we have more than one inequivalent anyonic model for the given set of labels and fusion rules. We need a mathematical structure in which we can label anyons by its charge. We need the mathematical structure to be able to deal with composite systems of anyons. It should also have a ‘semi-simple’ nature using which we can decompose the product of two labels as a sum of a finite number of labels. Moreover, anyons braid around each other, and as they are extended objects, rotations of a single anyon have a non-trivial effect on its kinematics. Hence, the worldines of anyons are more of a ‘ribbon’ than a line which can be twisted. It turns out that the mathematical structure called Modular Tensor Categories can be used to describe all these properties and hence, it is apt to describe anyonic models. In the next chapter, we will give a brief introduction to Category Theory. We will describe the structures that we have to add to a category to represent the above properties of anyons, and we will slowly build our way to Modular Tensor Categories.
– 17 –
3
Algebraic Structure of Anyonic Models
Using the Fractional Quantum Hall effect experiment, we recognised the properties of anyons. These properties can be encoded in a beautiful algebraic formalism called the Modular Tensor Category. In this chapter, we will go through the definition of a category and add additional structures to it which will eventually make it suitable for describing an anyonic model. Such a mathematical treatment will enable us to realise important properties of anyonic theories, and then we will describe how to construct a general anyonic model by constructing the so-called Quantum Double of a finite group as well as Quantum Groups. The motivation is similar to that of a second course in General Relativity in which one initially defines a topological manifold and then adds structures like a metric, defines a covariant derivative, various curvature tensors, time orientation etc. in order to eventually encode the physics of General Relativity in that mathematical formalism. 3.1 3.1.1
Building up to Modular Tensor Category Category
In anyonic models, anyons are labelled by their charges. We will now define a category whose elements label the charges of the anyons. Definition: A category C consists of (i) a class of objects ob(C) (ii) for any a, b ∈ ob(C), a class of morphisms (maps) C(a, b) (iii) For any a, b, c ∈ ob(C) and f ∈ C(a, b) and g ∈ C(b, c), there is a composition operator
◦ : C(a, b) × C(b, c) → C(a, c)
(3.1)
and the composition of f and g is denoted by f ◦ g. The composition operator is associative,
(f ◦ g) ◦ h = f ◦ (g ◦ h)
– 18 –
(3.2)
and for every a ∈ ob(C) there exists a morphism 1a ∈ C(a, a), such that for any f ∈ C(a, b)
f = f ◦ 1a = 1a ◦ f
(3.3)
Let us introduce some more concepts in introductory category theory which will be crucial in adding additional structures to a category. Let A and B be two categories. A map F between them denoted as F:A→B
(3.4)
is called a functor. Note that since a category contains objects and morphisms, F maps objects in A to that in B as well as morphisms in A to that in B. Given two categories A and B, and functors F and G, we can define a family of maps ηa between F and G ηa : F(a) → G(a)
(3.5)
for each a ∈ A. To be compatible with the already existing composition operation in the category, we require ηa to satisfy
ηb ◦ F (f ) = G(f ) ◦ ηb
(3.6)
for every morphism f ∈ C(a, b). The above equality can be depicted using the following diagram. F(a)
F(f )
ηa
G(a)
F(b) ηb
G(f )
G(b)
This is an example of a commutative diagram. In a commutative diagram, all directed paths from the same start and end points (colour coded in the diagrams) give the same result (are equal). We say that η is a natural transformation and ηa are its components. If ηa is an isomorphism for each a ∈ ob(C), then η is called a natural isomorphism.
– 19 –
3.1.2
Monoidal Category
We want to consider compound systems of more than one anyon. A compound system of anyons labelled by a and b is given by the tensor product a ⊗ b. Hence, we have to define a tensor product structure on the category. A category with a tensor product structure is called a monoidal category. Definition: A monoidal category is a category C equipped with a functor
⊗:C ×C →C
(3.7)
and three natural isomorphisms α, λ and ρ with components
αa,b;c : (a ⊗ b) ⊗ c → a ⊗ (b ⊗ c)
(3.8)
λa : 1 ⊗ a → a
(3.9)
ρa : a ⊗ 1 → a
(3.10)
such that it satisfies the (i) Pentagon Axiom
((a ⊗ b) ⊗ c ⊗ d) αa⊗b,c;d αa,b;c ⊗1d
(a ⊗ (b ⊗ c)) ⊗ d
(a ⊗ b) ⊗ (c ⊗ d)
αa,b⊗c;d
a ⊗ ((b ⊗ c) ⊗ d)
αa,b;c⊗d 1a ⊗αb,c;d
(ii) Triangle Axiom
– 20 –
a ⊗ (b ⊗ (c ⊗ d))
αa,1;b
(a ⊗ 1) ⊗ b
a ⊗ (1 ⊗ b)
ρa ⊗1b 1a ⊗λb
a⊗b for all a, b, c, d ∈ ob(C). As mentioned earlier, the objects in the category are labels for the charge of the anyons. For a compound system of anyons a1 , a2 , ..., an , the total charge of the system is denoted by a1 ⊗ a2 ⊗ ... ⊗ an . The unit object 1 ∈ ob(C) denotes the trivial charge. The fusion of two anyons with charges a1 and a2 is denoted by a1 ⊗ a2 . Consider the fusion of three anyons a, b, c. Then we would like to translate from fusing a and b first and then fusing the result with c to fusing b and c first and then fusing a with the result. This is exactly what the natural isomorphism α allows us to do. Note that the Pentagon axiom is none other than the Pentagon constraint in figure 12 written in the language of category theory. Hence, the Pentagon equations come up as an axiom in the definition of a monoidal category. Also, the natural isomorphisms λ and ρ tells us that the total charge after fusing an anyon a with the trivial charge is again a. 3.1.3
Braided Monoidal Category
One of the most striking properties of anyons is the way their worldines form braids. We introduced the monoidal structure above which enables us to consider a configuration of several anyons. We need a braid structure which can act on a tensor product of anyons. Definition: A braided monoidal category C is a monoidal category equipped with a family off isomorphisms σa,b : a ⊗ b → b ⊗ a
(3.11)
where a, b ∈ ob(C) such that λa
a⊗1 σ1,a
1⊗a
– 21 –
ρa
a
and both
a ⊗ (b ⊗ c)
σa,b⊗c
(b ⊗ c) ⊗ a
σa,b ;c
αb,c;a
(a ⊗ b) ⊗ c
b ⊗ (c ⊗ a) σa,b ⊗1c
1b ⊗σa,c
(b ⊗ a) ⊗ a
αa,b;c
b ⊗ (a ⊗ c)
and the same diagram with σ replaced by σ −1 commute for all a, b, c ∈ ob(C). In the above structure, it is easy to see that the σa,b depicts the exchange of anyons a and b. Since we have both associativity as well as braiding in a braided monoidal category, we expect the Hexagon equations 2.11 to come into the picture and the above commutative diagram is indeed same as the Hexagon constraint in figure 13. 3.1.4
Rigid Braided Monoidal Category
We would like to incorporate a structure into the category in order to capture charge conjugation. This is done by introducing the concept of duals of objects in a category. Definition: Consider an object a ∈ ob(C), where C is a braided monoidal category. Dual of a is an object a∗ ∈ ob(C) along with two morphisms ia : 1 → a ⊗ a∗ and ea : a∗ ⊗ a → 1 such that
a∗
1a∗ ⊗i
a∗ ⊗ a ⊗ a∗ 1a∗
a
i⊗1a
a ⊗ a∗ ⊗ a 1a
e⊗1a∗
a∗
1a ⊗e
a
commute. If each object in C has a dual, then the category is called rigid. The physical interpretation of ia and ea are clear from their definition. The former refers to creating a particle anti-particle pair of anyons, and the latter refers to the annihilation of a particle anti-particle pair of anyons. These can be denoted diagrammatically as
– 22 –
Figure 14. Birth and death processes: The left diagram correspond to the creation of an anyon and its anti-particle from the vacuum while the right diagram corresponds to their annihilation.
3.1.5
Ribbon Category
The algebraic structure of rigid braided monoidal categories already allows us to depict the braiding of anyons worldlines as well as the concept of charge conjugation. However, as we saw in the specific case of Fractional Quantum Hall Effect, anyons are extended objects, and hence they can rotate about a point which can have non-trivial effects on its kinematics. Hence, the worldlines of anyons should be seen as ribbons which can be twisted rather than just lines. We would like to introduce this structure into the rigid braided monoidal category to make the aptly named Ribbon Categories. Definition: A rigid braided monoidal category C is said to be a ribbon category if it comes equipped with a family θa : a → a of isomorphisms called twists (also known as topological spins) indexed by a ∈ ob(C) that satisfy
θa⊗b = (θa ⊗ θb ) σb,a ◦ σa,b θa∗ = (θa )∗
3.1.6
and
(3.12) (3.13)
Semisimple Ribbon Category
The anyons in an anyonic model are irreducible in the sense that they cannot be decomposed into more basic entities. Hence, we need a notion of irreducible objects in a category to capture this idea. Irreducible objects in a category are called simple. Also, following the fusion rules, we would like to write the fusion of two objects as a finite direct sum of simple objects. A category in which such a decomposition is possible is called a semi-simple category. Now, we will make these notions precise. Definition: An object i ∈ ob(C) is an initial object if for every x ∈ ob(C) there exists
– 23 –
precisely one morphism i → x. Definition: An object t ∈ ob(C) is a terminal object if for every x ∈ ob(C) there exists precisely one morphism x → t. Definition: An object 0 ∈ ob(C), is a zero object if it is both initial and terminal. Definition: A morphism m: a → b is a monomorphism if for any two f, g : c → a, we have m◦f =m◦g ⇒f =g
(3.14)
Definition: A morphism h: a → b is an epimorphism if for any two f, g : b → c, we have
f ◦h=g◦h⇒f =g
(3.15)
Definition: The kernel of a morphism f : a → b in C, a category with a zero object, is an arrow Ker(f ) := k : s → a
(3.16)
such that if f ◦ k = 0, then for every h: c → a such that h ◦ f = 0, h factors uniquely 0
through k as h = h ◦ k.
s
0 k
0
a
h
f
b
h 0
c
Definition: The cokernel of a morphism f : a → b in C, a category with a zero object, is an arrow CoKer(f ) := u : b → s
– 24 –
(3.17)
such that if u ◦ f = 0, then for every h: b → c such that h ◦ f = 0, then h factors uniquely 0
through u as h = u ◦ h .
s
0 u
a
f
0
b
h h
0
c
Definition: (i) A category is preadditive if C(a, b) for all a, b ∈ ob(C) and the composition of morphisms is bilinear. (ii) A preadditive category is additive if every finite set of objects has a biproduct. This essentially means that we can form finite direct sums and finite direct products. (iii) An additive category is preabelian if every morphism has both a kernel and a cokernel. (iv) A preabelian category is abelian if every monomorphism and epimorphism is normal. (Note that a monomorphism is normal if it is the kernel of some morphism, and a normal epimorphism is one that occurs as the cokernel of some monomorphism.) Definition: In an abelian category C, an object x ∈ ob(C), x 6' 0, is a simple object if for every b ∈ ob(C), f : b → x is either the zero morphism or an isomorphism. As discussed before, the charges in an anyonic model are irreducible, and hence they will be labelled by the simple objects in a category. Definition: An abelian category C is semisimple if any a ∈ ob(C) is such that
a'
M
Nj xj
(3.18)
j∈J
where xj are simple objects and J is the set of isomorphism classes of simple objects. Nj are integers and such that only a finite number of them are non-zero.
– 25 –
Definition: A semisimple ribbon category C is a semisimple category endowed with a ribbon structure where the tensor unit 1 is simple, the tensor product is bilinear and where for each simple object x ∈ ob(C), C(x, x) ' K, a field of characteristic zero. 3.1.7
Modular Tensor Category
A modular tensor category (MTC) is a special type of semisimple ribbon category in which there are only a finite number of charges. Moreover, in the definition of a modular tensor category, we will define the s-matrix which has several interesting properties. Also, we will find that one can derive the fusion coefficients from the s-matrix. Definition: A modular tensor category is a semisimple ribbon category C such that (i) There are only a finite number of isomorphism classes of simple objects. (ii) Modularity condition: The matrix s defined by (s)ij = [λ1 ◦ (ei ⊗ ej ) ◦ (1i ⊗ σij ⊗ 1j ) ◦ (1i ⊗ σji ⊗ 1j ) ◦ (ii ⊗ ij ) ◦ λ−1 1 ]ij
(3.19)
is invertible. The components of this matrix give the amplitude for a process in which two particleantiparticle pairs are created, particle from one pair is wound around that with the other, the pairs are annihilated. Diagrammatically, the definition of the s-matrix is
s = a∗
b∗
a
b
Figure 15. s-matrix
Topologically, this diagram is just two circles knotted together and is known as the Hopf link. The s-matrix can be written in terms of the familiar R-matrix and a trace
f b∗ a Rab∗ ) s = Tr(R
– 26 –
(3.20)
f is defined as Tr(R f ab ) = P dc Tr(Rc ). Note that taking traces correspond to where Tr c ab closing the open ends of a diagram. 3.1.8
Fusion Rules
When two anyons are combined, the possible total charge of the combined system is given by the fusion rules of the anyonic model. Definition: Let xi and xj be simple objects in ob(C), where C is a modular tensor category. J is the index set of isomorphism classes of simple objects. The fusion rule of xi and xj is given by xi ⊗ xj '
M
Nijk xk
(3.21)
k∈J
where Nijk are called the fusion coefficients. Note that as already mentioned, the anyonic charges correspond to the simple objects in the category, and hence the above definition does give a rule for fusing anyons. Note that the semisimplicity of the category was crucial in defining the fusion rules.
We went through the construction of an MTC which has all the crucial elements to describe an anyonic model. We can also add unitarity into the picture which requires additional constraints. Also, it is not necessary that we have to consider only a finite number of anyons. We can even consider fusion categories in which a braiding is not necessarily defined. A fusion category has simple objects and fusion rules defined in it. To summarise, there are different algebraic objects that we can construct based on the properties of the physical system that we are trying to explain. MTCs are special in the sense that they have the right amount of structure to open up relationships between different theories, about which we will explore in the following chapters. We will now define quantum dimensions of anyons and the Verlinde formula. Quantum Dimension A quantum dimension da of label a is a real number. Consider the creation of two particleantiparticle pairs with trivial total charge. Then, if the particle from one pair fuses with
– 27 –
the antiparticle of the other,
1 d2a
is the probability that they both annihilate. The quantum
dimension of an anyon a gives a measure of how the Hilbert space of n-anyons of type a increases with n. In the special case in which a labels irreducible representations Ra of a group, da = |Ra |, the dimension of the representation. It is useful to renormalize the birth √ and death functions for an anyon a by multiplying them with da making their norm da . Using the diagrammatic representation of the functions given in figure 14, we get
a
da
Figure 16. Diagrammatic representation of quantum dimension
We define the renormalized inner product in figure 17, and the renormalized completeness relations is given in figure 18.
c
c η†
hhη|ξii =
q
dc da db
a
b
q
dc da db
hη|ξi
ξ c
c
Figure 17. Renormalized inner product of operators
c for each label a in the MTC using the fusion coefficients. Let us define the matrices Na = Nab
Let us also define the vector d~ whose components are the quantum dimensions. It turns out that d~ is an eigenvector of Na with eigenvalue da . This property is obvious from the following identity that the quantum dimensions satisfy.
– 28 –
Claim: The quantum dimensions satisfy the fusion rules.
da db =
X
c Nab dc
(3.22)
c
Proof:
a
da db
b
a b
ab for V ab where j = 1, ..., N c and using the completeness Choosing a set of basis vectors ψc,j c ab
relation 18,
b s X c,j
dc da db
a
ab ψc,j
a
ab )† (ψc,j
b
Now, using the diagram for the inner product 17, we get
da db =
X
=
X
dc
DD EE ab ab ψc,j ψc,j
c,j c dc Nab
(3.23)
X
(3.24)
c
Hence, we have da db =
c Nab dc
c
– 29 –
a
b
a
b µ
q
P
cµ
dc da db
c µ
a
a
b
b
Figure 18. Completeness relation after renormalization
Definition: Let D be
qP
j∈J
d2j where dj are the quantum dimensions. The modular S
matrix is defined to be S=
1 s D
(3.25)
Note that the following identity holds (figure 19). It can be proved by taking the trace both sides which correspond to closing the open ends of the diagram, and comparing the diagram obtained with figure 16 and 15. We will use it in the subsequent proof for the Verlinde formula. x∗
x∗
Sax S1x
a x∗
x∗ Figure 19.
Claim: (Verlinde Formula) The fusion coefficients are given in terms of the modular S matrix by the formula c Nab =
X Sax Sbx Sc∗ x x
– 30 –
S1x
(3.26)
where 1 denotes the trivial label. Proof: The proof for the Verlinde formula is obtained by inserting a x∗ line within the diagrams of the proof of the algebraic relation between quantum dimensions.
x∗
x∗ a
Sax Sbx idx∗ = 2 S1x
a
b
b
x∗
x∗
Using the completeness relation 18, we get, x∗
b s X c,j
dc da db
a
ab ψc,j
a
ab )† (ψc,j
b x∗
Using the diagram for inner product 17, x∗
X
c Nab
c
c
X c
x∗
– 31 –
=
c Nab
Scx idx∗ S1x
where the final equality was obtained using the identity in figure 19. Hence, we get Sax Sbx X c Nab Scx = S1x c
(3.27)
Multiplying both sides by Sd∗ x ans summing over x, we get, X Sax Sbx Sd∗ x x
S1x
= = =
X
c Nab Scx Sd∗ x
c,x c ∗ Nab Scx Sxd
X
∗ ∗ (using Sd∗ x = Sdx = Sxd )
c Nab δdc (here we used the assumption that the S matrix is unitary)
c d = Nab
(3.28)
Thus, we get c Nab =
X Sax Sbx Sc∗ x x
S1x
(3.29)
Within the mathematical physics literature, anyonic theories are broadly called as topological quantum field theories (TQFTs). It is remarkable that even though the theory is topological, we could give a purely algebraic formulation for it in terms of modular tensor categories. The information that the kinematics is in 3 spacetime dimensions is encoded in the algebraic data contained in braiding and fusion matrices. But, is there an alternative geometric description of such theories? The answer is yes, and we will briefly study this description when we look at the relationships between conformal and topological field theories in Chapter 6.
– 32 –
4
Modular Tensor Categories from Hopf Algebras
Towards the end of Chapter 2, we saw that, once we assume a set of labels and fusion rules, the question of whether one or more consistent anyonic models corresponding to it exists or not is answered by solving the Pentagon and Hexagon equations. But, because we know that a modular tensor category carries the algebraic data of an anyonic model, we can search for consistent anyonic models by looking at possible realisations of MTCs. An important class of MTCs is obtained from representations of a particular kind of algebra know as quasi-triangular Hopf algebra. We will focus on two special cases of quasi-triangular Hopf algebras, namely, quantum double of finite groups and quantum groups. Representation category of quantum double of a finite group is an MTC; hence, it gives a series of anyonic models corresponding to every finite group. Quantum groups are constructed from the Lie algebra of Lie groups. Hence, the representation category of quantum groups gives a series of anyonic models corresponding to Lie groups. 4.1
Hopf Algebra
Before defining a Hopf algebra, let us start with the definition of an algebra, then we will define a coalgebra, bialgebra, and antipode. These notions are crucial for defining a Hopf algebra, and then we will add extra structure to it and define a quasi-triangular Hopf algebra. Definition: A k-algebra is a k-vector space, A, with two linear maps, the multiplication map µ:A⊗A→A
(4.1)
u:k→A
(4.2)
and the unit map
such that the diagrams
– 33 –
µ⊗id
A⊗A⊗A
A⊗A µ
id⊗µ µ
A⊗A
A
and
A⊗A u⊗id id⊗u
k⊗A
A⊗k
µ s s
A where s denotes scalar multiplication, commute.
Definition: A coalgebra is a k-vector space, A, with two linear maps, the comultiplication
∆:A→A⊗A
(4.3)
:A→k
(4.4)
and counit
such that the above diagrams hold with reversed arrows, multiplication and unit replaced by comultiplication and counit, and s replaced by a map which takes A to A ⊗ k.
Definition: A tuple (A, µ, u, ∆, ) is called a bialgebra if (A, µ, u) is an algebra, (A, ∆, ) is a coalgebra such that ∆ and are algebra homomorphisms. For the compatibility of the algebra and coalgebra structures, we require the following diagrams to commute. µ
A⊗A
A ∆
∆⊗∆
A⊗A⊗A⊗A
τ
A⊗A⊗A⊗A
– 34 –
µ⊗µ
A⊗A
∆
A
A⊗A
u
u⊗u
A
⊗
k⊗k
k
µ
A⊗A
k⊗k
A
k
k
=
u
k where τ : A⊗4 → A⊗4 is given by τ (a ⊗ b ⊗ c ⊗ d) = a ⊗ c ⊗ b ⊗ d and the unmarked arrows are given by the canonical operations in the field.
Definition:
Consider a bialgebra A, a linear homomorphism S : A → A is called an
antipode if the following diagram commutes. S⊗idA
A⊗A
A⊗A µ
∆
A
k
u
A µ
∆ idA ⊗S
A⊗A
A⊗A
A bialgebra A together with an antipode S is called a Hopf algebra.
We will now go through two particular examples of Hopf algebras, namely a group ring and its dual. These will help us to understand the definition of the quantum double. Examples: (i) A group algebra K[G], where K is a field and G is a group with operation *, is the set of all linear combinations of finitely many elements in G with coefficients in K. A general element in K[G] can be denoted as:
g=
X
ag g
g∈G
– 35 –
(4.5)
where ag ∈ K. We will be interested in the case where K = C. C[G] is an algebra with multiplication defined by µ : C[G] × C[G] → C[G];
X
ag g ⊗
g∈G
X
ah h =
h∈G
X
ag ah gh
(4.6)
g,h∈G
and unit is e ∈ G. It also has a coalgebra structure with comultiplication
∆ : C[G] → C[G] × C[G];
X
ag g →
g∈G
X
ag (g ⊗ g)
(4.7)
g∈G
and counit : C[G] → C;
X
ag g →
g∈G
X
ag
(4.8)
g∈G
Moreover, one can also introduce an antipode using the fact the elements in G are invertible, making K[G] a Hopf algebra. The antipode is defined by
S : C[G] → C[G];
X
ag g →
g∈G
X
ag g −1
(4.9)
g∈G
(ii) It can be proved that the dual of C[G], C∗ [G]=Hom(C[G], K) is also a Hopf algebra. It is useful to go through the Hopf algebra structure of C∗ [G] to define the Quantum Double of a Group. A general element in C∗ [G] will be denoted by g∗ . A basis for C[G] is given by the elements g ∈ G. Similarly, a basis for C∗ [G] is given by δg ∈ Hom(C[G], K) with
δg (h) = δg,h
(4.10)
The multiplication defined on the basis elements is given by 0
µ : C∗ [G] × C∗ [G] → C∗ [G]; (δg · δh )(x) := δg,x δh,x = δg,h δg (x)
– 36 –
(4.11)
The unit is given by 0
0
0
u : C → C∗ [G]; u (1)(x) = 1 ∀x ∈ G ⇔ u (1) =
X
δg
(4.12)
g∈G
For the comultiplication of a basis element δg , we get for x,y ∈ G: 0
0
∆ : C → C∗ [G] × C∗ [G]; ∆ (δg )(x ⊗ y) :=
X
(δgh−1 ⊗ δh )(x ⊗ h)
(4.13)
h∈G
The counit is given by 0
0
: C∗ [G] → C : (δg ) := δg,e
(4.14)
The antipode is 0
0
S : C∗ [G] → C∗ [G] : S (δg )(x) := δg−1
(4.15)
Definition: Let A be a bialgebra. An element R ∈ A ⊗ A is called a universal R-matrix if it is invertible and satisfies the following axioms a) (τ ◦ ∆)(a)R = R∆(a) b) (idA ⊗ ∆)R = R13 R12 c) (∆ ⊗ idA )R = R13 R23 P Here we used the following notation: If R= i ai ⊗ bi ∈ A ⊗ A, then
R13 =
X
ai ⊗ 1 ⊗ bi and R23 =
i
X
1 ⊗ ai ⊗ bi
(4.16)
i
etc. A bialgebra A together with a universal R-matrix is called quasi-triangular. If A is a quasi-triangular bialgebra, then one can show that
σV,W : V ⊗ W → V ⊗ W → W ⊗ V
(4.17)
is a braiding on the monoidal category Rep(A). One can also show that if A is a quasitriangular Hopf algebra with an invertible antipode S. Then, Rep(A) is a rigid braided monoidal category [15].
– 37 –
Definition: Let (A, µ, u, ∆, , R, S) be a quasi-triangular Hopf algebra with invertible antipode. An invertible element θ ∈ Z(A), where Z(A) is the center of A, is called a twist, if it satisfies, a) ∆(θ) = (τ (R)R)−1 (θθ) b) S(θ) = θ A quasi-triangular Hopf algebra A with invertible antipode along with a twist θ is called a ribbon Hopf algebra, then one can show that Rep(A) is a ribbon category [15]. 4.2
The Category Z(G) of Representations of Quantum Double of a finite group
Certain anyonic theories based on a finite group G have particles called dyons which have both an electric charge and a magnetic flux. We know that in quantum field theories, the possible charges of particles are given by the representations of the gauge group. Similarly, we would like to find a mathematical object whose representations give the possible electric and magnetic flux charge pairs in the theory. The object we are looking for is the Quantum Double D(G) of G. Furthermore, it can be proved that the category of representations of D(G) is a modular tensor category. Hence, based on our discussion in the previous section, this defines a consistent anyonic model.
4.2.1
Quantum Double of a finite group D(G)
Definition: The quantum double D(G) of a finite group G is the tensor product C[G] ⊗ C∗ [G] endowed with the structure of a Hopf algebra. The multiplication is given by
(g1 δh1 )(g2 δh2 ) = g1 g2 δg−1 h1 g2 δh2 2
(4.18)
If 1 and 1∗ are the units in C[G] and C∗ [G] respectively, then 1 ⊗ 1∗ is the unit in D(G). The coproduct in D(G) is defined as
∆(gδh ) :=
X
δx ⊗ δhx−1
x∈G
– 38 –
(4.19)
The counit is given by (gδh ) := δe,h
(4.20)
S(gδh ) = g −1 δgh−1 g−1
(4.21)
and the antipode is defined as
4.2.2
Representations of D(G) and Category Z(G)
It can be shown that D(G) is a ribbon Hopf algebra and hence, the representation category, denoted Z(G), of D(G) is a ribbon category. It can be further shown that Z(G) is a semisimple category with a finite number of simple objects [16]. Let us look into how one can construct the representations of D(G). Consider an element a ∈ G. Let a be the conjugacy class of a and Z(a), the centralizer of a. Let π the representation of Z(a) over a vector space W with basis elements {w1 , ..., wd }. Consider the vector space Va,π with the basis {|b, wi i : b ∈ a, 1 ≤ i ≤ d}. For any fixed b ∈ a, fix kb ∈G such that −1 b = kb akb−1 and let ka = e. It is easy to see that kgbg −1 gkb is always in Z(a). For any w
∈ W, b ∈ a and gh∗ ∈ D(G), define E −1 −1 gh |b, wi = δh,b gbg , π(kgbg−1 gkb )w ∗
(4.22)
This action gives a representation of D(G) [16]. All irreps of D(G) are of the above form and are indexed by conjugacy classes of G and irreps of the centralizer of a fixed element in the corresponding conjugacy class. It can also be showed that if π is an irrep of Z(a), so is (a, π). The trivial representation of D(G) is denoted as (e,1). The charge conjugation of (a, π), denoted as Conj(a, π)) is isomorphic to (a−1 , π ∗ ). The a label of the irrep is called the magnetic charge, while π is called the electric charge of the particle. 0
0
Fusion Rules: Let (a, π) and (a0 , π ) be two irreps of D(G). (a, π) ⊗ (a0 , π ) is also a
– 39 –
representation of D(G) and is isomorphic to the sum of irreducibles 0
(a, π) ⊗ (a0 , π ) =
M
N
(h,ρ)
(a,π)(a0 ,π 0 )
(h, ρ)
(4.23)
(h,ρ)
where N
(h,ρ) (a,π)(a0 ,π 0 )
are non-negative integers.
The S matrix is defined by
S(a,π)(a0 ,π0 ) =
1 |Z(a)||Z(a0 )|
X
0 −1
trπ (ha
h−1 )trπ0 (h−1 a−1 h)
(4.24)
h:ha0 h−1 ∈Z(a)
Z The non-negative integers NXY in the fusion rule can be computed using the Verlinde
formula. Let us go through the essential steps involved in finding the fusion coefficients in the anyonic model defined by the quantum double of a given group G which might have got buried in the mathematical details given above. 1. Identify the conjugacy classes in the group G. These labels the possible magnetic charges of the anyons. 2. Find the centralizer for a representative of each conjugacy class. 3. Find the irreducible representations of the centralizers. These labels the possible electric charges of the anyons. 4. Find the modular S-matrix using the above formula. 5. Find the fusion coefficients using the Verlinde formula. Let us look at some examples which should make the above procedure transparent. 4.2.3
Quantum Double of Z2
Z2 is an abelian group with 2 elements.
Z2 = {e, a}
– 40 –
(4.25)
where a2 = e. Since Z2 is an abelian group, the conjugacy class b of a general element b ∈ Z2 is just the set {b} itself. Hence, the possible magnetic charges are {e} and {a}. Again, since the group is abelian, the centralizer Z(b) for any b ∈ Z2 is Z2 . There are two irreducible representations of Z2 . One is the trivial representation of 1 dimension which we will denote by 1 and the 1 dimension non trivial representation defined by e → 1 and a → −1 which we will denote by 1∗ . Thus, the possible anyonic types in this theory are (e, 1), (e, 1∗ ), (a, 1) and (a, 1). Let us calculate the elements of the modular S-matrix using the given formula. First of 0
all we note that |Z(b)| = 2 for any b ∈ Z2 . Also, the condition ha h−1 ∈ Z(a) is trivially 0 −1
satisfied for all h ∈ Z2 . Also, ha
0 −1
h−1 = a
and h−1 a−1 h = a−1 Hence, the general
formula for the elements of the modular S-matrix reduces to 1 X 0 −1 trπ (a )trπ0 (a−1 ) 4
S(a,π)(a0 ,π0 ) =
(4.26)
h∈Z2
Let us choose a = e and π = 1, which is the particle with trivial magnetic and electric charge. We discusses that the physics interpretation of the modular S-matrix involves braiding two anyons. But, the result of braiding an anyons and another anyon with trivial charges should be trivial. Indeed, we see that 1 X 0 −1 tr1 (a )trπ0 (e) 4 h∈Z2 1 X = 1·1 4
S(e,1)(a0 ,π0 ) =
(4.27) (4.28)
h∈Z2
=
1 2
which is trivial except for the factor 21 . 0
Now, let us choose a = e, π = 1∗ and a = a. Then,
S(e,1∗ )(a,π0 ) =
1 X tr1∗ (a)trπ0 (e)(4.30) 4 h∈Z2
– 41 –
(4.29)
=
1 X −1 · 1 4
(4.31)
h∈Z2
=−
1 2
(4.32)
The nontrivial phases are produced when anyons with magnetic and electric charges are braided around each other. In the above case, the first anyon has trivial magnetic charge and non-trivial electric charge, and the second anyon has non-trivial magnetic and electric charges. The extra minus sign is produced by the Aharanov-Bohm effect between the non-trivial electric and magnetic charges. Similarly, we can show that S(a,1)(a,1∗ ) = − 21 Also, we can show that S(e,1∗ )(e,1∗ ) = S(a,1)(a,1) = S(a,1∗ )(a,1∗ ) =
1 2.
In this case we get
a positive sign because we get a -1 from the braiding of the magnetic charge of the first anyon with electric charge of the second and also another -1 from the braiding of the electric charge of the first with the magnetic charge of the second. Note that the modular S-matrix is symmetric. Hence, the full matrix can be written as
1 1 S= 1 1
1 1 −1 −1
1 −1 1 −1
1 −1 −1 1
(4.33)
Finally, we can obtain the fusion rules using the Verlinde formula. Since this is an abelian group, the fusion of anyons is essentially like matrix multiplication. From the fusion coefficients we indeed see that a non trivial fusion is given by (a, 1) ⊗ (a, 1∗ ) = (e, 1∗ )
(4.34)
We observe that the resulting magnetic charge is obtained just by the group multiplication of a representative element of the conjugacy class and the resulting electric charge is essentially the composition of the representation maps. The number of anyonic species possible in this model is 4 which is equal to |Z2 |2 . This is a general result for abelian groups. For an abelian group G, there are |G| distinct magnetic charges possible corresponding to the conjugacy class of each element in the group which is
– 42 –
the element itself. Also, for every element, the centralizer is the group itself. The possible electric charges are the different irreducible representation of the group. Also, we know that for an abelian group G, the number of irreps is given by |G|. Hence, the total number of anyonic species in an anyonic model defined by the quantum double of an abelian group G is |G|2 4.2.4
Quantum Double of S3
The permutation group on three elements is a non-abelian group and hence, the anyonic theory based on D(S3 ) has some non-trivial properties. The group S3 is defined by
S3 = {e, (12), (23), (31), (123), (132)}
(4.35)
The conjugacy classes are the identity {e}, the set of two cycles {(12), (23), (31)} and the set of three cycles {(123), (132)}. Hence, the possible magnetic charges are {(e, (12), (123)}. The centralizers for the magnetic charge labels are
Z(e) ' S3 , Z(12) ' Z2 = {e, (12)}, Z(123) ' Z3 = {e, (123), (132)}
(4.36)
S3 has three irreducible representations. The trivial representations denoted 1, the sign representation denoted sign in which every permutation is mapped to its sign, {e → 1, (12) → −1, (23) → −1, (31) → −1, (123) → 1, (132) → 1} and also the 2-dimensional representation π given by
e→
(31) → where ω = e
2πi 3
! 1 0 , (12) → 0 1 ! 0 ω , (123) → ω 0
! 0 1 (23) → 1 0
ω 0 0 ω
0 ω ω 0
! (132) →
!
ω 0 0 ω
(4.37)
! (4.38)
.
For Z2 , we have two irreducible representations, the trivial rep 1, the rep -1 given by {e → 1, (12) → −1}.
– 43 –
For Z3 , we have three irreps, the trivial rep 1, the rep ω given by {e → 1, (123) → ω, (132) → ω 2 } where ω = e
2πi 3
and the rep ω ∗ given by {e → 1, (123) → ω 2 , (132) → ω}.
Thus, in the anyonic model obtained from the quantum double of S3 , we have the following 8 anyonic types {(e, 1), (e, sign), (e, π), ((12), 1), ((12), −1), ((123), 1), ((123), ω), ((123), ω ∗ )}
(4.39)
With the above labels, we can compute the S matrix to be
1 1 2 3 S= 3 2 2 2
1 1 2 −3 −3 2 2 2
2 2 4 0 0 −2 −2 −2
3 −3 0 3 −3 0 0 0
3 −3 0 −3 3 0 0 0
2 2 −2 0 0 4 −2 −2
2 2 −2 0 0 −2 4 −2
2 2 −2 0 0 −2 −2
(4.40)
4
Note that Quantum double of finite groups does not exhaust all possible anyonic models. A general anyonic model is described by a modular tensor category, and the quantum double is just one example of it. In fact, the simplest non-abelian anyonic model, called the Fibonacci anyonic model, cannot be obtained from the quantum double of any finite group. This model has only two types of charges labelled by 1 and τ and the non-trivial fusion rules are τ ⊗τ =1⊕τ
(4.41)
The quantum double of a group with one element is trivial, and even for order two, the number of types of possible charges is 4. Thus, the Fibonacci anyonic model cannot be obtained from the quantum double of a finite group. The following relation holds in an MTC, D−1
X
θi d2i = e
i
– 44 –
cπi 4
(4.42)
where D =
qP
2 i di
and θi are the topological spins [17]. ctop = c mod 8 is called the
topological central charge. For a quantum double MTC, the LHS of equation 4.42 is 1 [16], and hence c is a multiple of 8. Thus, a quantum double MTC has a topological central charge 0. But, ctop of the Fibonacci anyonic model is known to be non-zero [18]. This again shows that it cannot be obtained from a quantum double MTC. The Fibonacci model belongs to another series of anyonic models based on algebraic objects called quantum groups about which we will study in the next section. 4.3
Quantum Groups and their Representations
In the previous subsection, we learned that one could construct modular tensor categories form the representation category of quasitriangular Hopf algebras. Given a finite group, we looked at the construction of a specific Hopf algebra called the quantum double, whose representation category defines a series of anyonic models. Now, we will consider semisimple lie groups and again construct a modular tensor category from the representations of an algebra called quantum group which defines another series of anyonic models. We follow the exposition in [19]. 4.3.1
Universal Enveloping algebra U(g) of Lie algebra g
For a given lie algebra g, one can construct a unital, associative algebra called the universal enveloping algebra U(g). Definition: A complex lie algebra g is a vector space over C with a bilinear operation [ , ] : g × g → g satisfying the properties
(i) Antisymmerty : [x, y] = −[y, z] ∀x, y ∈ g
(4.43)
(ii) Jacobi Identity : [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 ∀x, y, z ∈ g
(4.44)
In the following definition we will assume g to be finite dimensional with basis {X1 , ...Xn }. For elements Xi , Xj ∈ g, [Xi , Xj ] can be written as
[Xi , Xj ] =
X k
– 45 –
cijk Xk
(4.45)
where cikj are the structure constants of g. Definition: Given an n-dimensional Lie algebra g, with basis {X1 , ...Xn } and structure constants cijk , the universal enveloping algebra U(g) is the associative, non-commutative algebra generated by an identity element 1 and {x1 , ...xn } subject to the constraints
xi xj − xj xi =
X
cijk xk ,
1 ≤ i, j ≤ n
(4.46)
k
One can make this a Hopf algebra by adding the following maps,
∆x = x ⊗ 1 + 1 ⊗ x, x = 0, Sx = −x
∀x ∈ U (g)
(4.47)
Before going further, let us specialize to the case of a complex simple Lie algebra g. Let t be the cartan subalgebra of g and let t∗ be its dual. Let αi ∈ t∗ be a set of positive simple roots. We have the Cartan matrix defined by
aij =
2(αi , αj ) (αi , αi )
(4.48)
where (, ) is the symmetric bilinear form on t∗ . Consider the Cartan generators Hi and the generators X±i corresponding to the simple roots such that the lie algebra g is generated by [Hi , Hj ] = 0, [Hi , Xj ] = ±aij X, [X+i , X−j ] = δij Hi
(4.49)
Recall that the remaining roots can be obtained by the action of the Weyl group on the fundamental roots. This action terminates because of the Serre relations
[X±i , ]1−aij (X±j ) = 0 i 6= j
(4.50)
This allows us to consider the universal enveloping algebra U(g) as being generated by the above generators and an identity element 1 with the constraints 4.49 and 4.52.
– 46 –
Quantum Group Uq (g)
4.3.2
Quantum Group Uq (g) is obtained by deforming the universal enveloping algebra U(g) by a formal variable q. The deformation of U(g) with a formal variable q is an algebra generated hi
by {qi 2 , xi , x−i , } with the following relations
[hi , hj ] = 0, 1−aij
X k=0
where
n m qi
qihi − qi−hi qi − qi−1
(4.51)
1 − aij 1−a −k x±i ij x±j xk±i = 0 ∀i 6= j (−1) k qi
(4.52)
[hi , x±j ] =ij x±j ,
[x+i , x−j ] = δij
k
is defined in terms of [n]qi =
qin −qi−n qi −qi−1
and [n]qi ! = [n]qi [n−1]qi ...[1]qi . Expression
4.52 is the q-deformed Serre relation. The Hopf algebra structure is given by hi
∆hi = hi ⊗ 1 + 1 ⊗ hi , (hi ) = (x±i ) = 0,
−hi 2
∆x±i = x±i ⊗ qi 2 + qi Shi = −hi ,
⊗ x±
(4.53)
Sx±i = −qi±1 x±i
(4.54)
It can be shown that Uq (g) is isomorphic to U(g) except when q is a root of unity [20]. Hence, the most interesting case is when q is a root of unity. In fact, in this case, it can be shown that the category of representations of Uq (g), denoted Z(U (g)), is semi-simple. Moreover, it can be proven to be a modular tensor category. Hence, Z(U (g)) defines a series of anyonic models for each g. Strictly speaking, one also imposes the unitarity condition on the MTC. Given a Lie algebra g and q, where q is the lth root of unity. The MTC Uq (g) is unitary if m|l, where m is the maximal number of edges between any two nodes of the Dynkin diagram [21]. To be precise, one also has to truncate the representations of Uq (g) to a set of ‘good’ (type II) representations to obtain an MTC [22]. In the next section, we will look at the Quantum group Uq (sl(2)) which would make the above definitions more transparent.
– 47 –
4.3.3
Quantum Group Uq (sl(2))
Recall that the lie algebra of sl(2) is given by
[H, X± ] = ±2X± , [X+ , X− ] = H
(4.55)
The universal enveloping algebra U(sl(2)) is generated by {1, h, x+ , x− } with the constraints
[h, x± ] = ±2x± , [x+ , x− ] = h
(4.56)
Let q be a nonzero parameter. The quantum group Uq (sl(2)) is a noncommutative algebra h
h
generated by 1, x+ ,x− ,q 2 ,q − 2 satisfying h
h
q± 2 q∓ 2 = 1 h
h
q 2 x± q − 2 = q ±1 x± ,
(4.57) qh
q −h
− q − q −1
[x+ , x− ] =
(4.58)
Uq (sl(2)) is a Hopf algebra with the following comultiplication, counit and antipode, h
h
h
∆q ± 2 = q ± 2 ⊗ q ± 2 , h
q ± 2 = 1 , Sx± = −q ±1 x± ,
h
h
∆x± = x± ⊗ q 2 + q − 2 ⊗ x±
(4.59)
x± = 0
(4.60)
h
h
Sq ± 2 = q ∓ 2
(4.61)
When we consider the field to be the field of all rational functions of an indeterminate q over C, C[[t]], then the generators are {h, x± } and the Hopf algebra is quasitriangular with
R=q
h⊗h 2
X (1 − q −1 )n n=0
where [n] =
q n −q −n q−q −1
[n]!
h
h
(q 2 x+ ⊗ q − 2 x− )n q
n(n−1) 2
(4.62)
and [n]! = [n][n − 1]...[1]. The TQFT corresponding to the quantum 2πi
group Uq (sl(2)) for q = e k+2 is the same as Chern-Simons-Witten (CSW) theory based on the su(2)k algebra [24]. su(2)k is also the symmetry algebra of a CFT called the WessZumino-Witten (WZW) model based on the Lie group SU (2). In Chapter 5, we will see
– 48 –
that one can define braiding and fusion matrices in a CFT. It turns out that the braiding and fusion matrices of Uq (sl(2)) is exactly same as that of su(2)k WZW theory! The algebra su(2)k in WZW is called current algebra, and we study the fields in the theory by finding the highest weight representations of the current algebra. Again, we will find that we do not have to consider all the highest weight representations. The representations of interest are called integrable representations. Definition: Given a current algebra gk and a highest weight vector λ, then the representation corresponding to λ is integrable if
k − (λ, θ) ≥ 0
(4.63)
where θ is the is the highest root normalized to be of length 2. P Note that the highest root θ is the unique root for which, in the expansion θ = i mi αi P where αi are the simple roots, the sum i mi is maximized. For the special case of gk = su(2)k , a spin-l state is integrable if 2l ≤ k. From the above criterion, it is clear that we have only a finite number of integrable representations in the theory. The non-integrable representations (for 2l > k) correspond to trivial fields, and hence we are only concerned with integrable representations. This is, in fact, a general result. Non-integrable fields vanish identically in any theory which is invariant under current algebra [23]. We will discuss this fact again when we consider the representations of su(n)k algebras in section 6.3.1. CSW and WZW theories are closely related to each other, and we will explore these theories and their relationship in detail in Chapter 6. In the next chapter, we will derive the highest weight representations of Virasoro algebra and study how braiding and fusion matrices arise in CFTs.
– 49 –
5
Fusion and Braiding in 2D Conformal Field Theories
It turns out that fusion and braiding arise naturally when we consider four-point correlation functions in a CFT indicating a relationship between conformal and topological field theories. Section 5.1 briefly describes the Witt algebra, its central extension into Virasoro algebra and its representations. Then we will identify the null states in the representation and look at how those constrain the correlation functions in a CFT. We will see that the product of primary fields has an expansion similar to the fusion rules of labels in an anyonic theory. Then we will look at the Verlinde formula and discuss its relationship to modular invariance. Finally, in section 5.6 we will describe how fusion and braiding matrices arise in a CFT. 5.1
The Virasoro Algebra and its Representations
Let us consider conformal field theory on the complex plane. It can be proved that infinitesimal conformal tranformations in 2D are given by holomorphic functions [3]. Under a conformal tranformation, the metric tensor tranforms as
ds2 = dzdz →
∂f ∂f dzdz ∂z ∂z
(5.1)
Thus, the conformal algebra in 2d has an infinite number of generators. We remind the reader that under a coordinate transformation z → f (z), a primary field φ(z, z) transforms by definition as
0
φ(z, z) → φ (z, z) =
∂f ∂z
h
∂f ∂z
h φ(z, z)
(5.2)
Another important aspect of a CFT is the operator-state correspondence. It says that all states in the theory can be created by operators acting locally on a small neighbourhood of the origin. It is a one to one mapping between the operators φ(z, z) and states
|φi = lim φ(z, z) |0i z,z→0
– 50 –
(5.3)
where |0i is the CFT vacuum. It is crucial to note that such a one to one correspondence is possible in CFT since we have radial quantization, and the theory evolves radially away from the origin, which corresponds to the limit z, z → 0. Since this limit corresponds to a particular point, the whole Hilbert space of the theory can be obtained by operators acting at this point. In a general QFT, one can obviously think of any state as the action of an operator on the vacuum with the limit t → −∞. However, since t → −∞ is not a single point on the Minkowski space, one cannot associate a given state with a unique local operator acting at a point. Hence, the relation between states and operators is not one to one in a general QFT. The operator product expansion provides an algebraic structure in quantum field theories. For any two local operators in a QFT at nearby points x1 and x2 , their product can be written in terms of a series of local operators at x1 or x2 with some functions depending on x1 − x2 as coefficients. It reflects the locality properties of QFTs. For a 2D CFT, let Oi denote all the local operators in the theory. The operator product expansion is given by
Oi (z, z)Oj (w, w) =
X
k Cij (z − w, z − w)Ok (w, w)
(5.4)
k
This series, in general, has singular behaviour. OPEs are essential because they contain the transformation behaviour of operators under a conformal transformation. For example, for the stress-energy tensor, the OPE is given by [3],
T (z)T (w) =
c/2 2T (w) ∂w T (w) + + + ... (z − w)4 (z − w)2 z−w
where ... denotes non-singular terms. 5.1.1
The Witt algebra and its central extension
The 2D conformal algebra is called the Witt algebra and is given by
[lm , ln ] = (m − n) lm+n
– 51 –
(5.5)
lm , ln = (m − n) lm+n lm , ln = 0
(5.6)
where n, m ∈ Z. In a quantum theory, we have a projective unitary representation of the symmetry group acting on the Hilbert space of the theory. Going from non-projective representations to projective representations results in the so-called central extension of the algebra. For finite algebras, the central extension is trivial, and hence mostly in quantum field theories with some symmetry corresponding to a Lie group, we do not have to worry about central extensions. However, the Witt algebra is infinite dimensional, and hence it has a non-trivial central extension. Algebra obtained from the central extension of the Witt algebra is called the Virasoro algebra which is given by [3]
[Lm , Ln ] = (m − n)Lm+n +
c (m3 − m)δm+n,0 12
(5.7)
There is a similar commutation relation with Ln → Ln and [Lm , Ln ] = 0. c is called the central charge or the conformal anomaly. 5.1.2
Highest Weight Representation of Virasoro Algebra
Define the heighest weight state |hi with properties
L0 |hi = h |hi ; Ln |hi = 0 ∀n > 0
(5.8)
From the Virasoro algebra we can show that,
L0 (L−m |hi) = (h + m) |hi
(5.9)
Hence, L−m raises the eigenvalue of L0 by m. Thus, it acts like a raising operator. A basis for the representation space is given by
L−k1 ...L−kn |hi
(1 ≤ k1 ≤ ... ≤ kn )
– 52 –
(5.10)
The L0 eigenvalue of the above state is h + k1 + ... + kn . N =
Pn
i=1 ki
is called the
level of the descendant. The space of highest weight states and its descendants is called a Verma module V (c, h), where c stands for the central charge of the theory and h is the L0 eigenvalue of the highest weight state. A particular subclass of CFTs have only a finite number of primary fields and hence, only a finite number of representations of the Virasoro algebra. Such theories are called minimal models. (Note that even though one might define minimal models for CFTs with more general algebras, in our case, minimal models refers to the theories with just the Virasoro algebra.) 5.2
CFTs based on Kac-Moody algebras
We saw that the Virasoro algebra is an infinite dimensional algebra. Since Lie groups play an important role in physics, a natural question is whether one can define a conformal field theory based on the Lie algebra of some given Lie group. In fact, given a Lie algebra g, one ∧
can extend it to the so called Kac-Moody algebra g given by
a [jm , jnb ] = i
X
c f abc jm+n + kmδ ab δm+n,0
(5.11)
c a are the Laurent modes of the current j(z) and from k is called the level of the theory. jm
5.11, we can see that the zero modes from a subalgebra which is none other than the Lie algebra [j0a , j0b ] = i
X
f abc j0c
(5.12)
c
One can construct an energy-momentum tensor for this theory using the Sugawara construction [25]. dimg X 1 T (z) = N (j a j a )(z) 2(k + Cg )
(5.13)
a=1
where N () denotes normal ordering and Cg is the Coexter number of the Lie algebra. The central charge of the theory is given by
c=
k dimg k + Cg
– 53 –
(5.14)
These theories will play an important role when we discuss the relationship between conformal and topological field theories. One can consider a subset of theories in which there are only a finite number of highest weight representations. In fact, there are other general algebras using which we can define a CFT. A rational conformal field theory (RCFT) is a CFT in which there are only a finite number of primary fields with respect to this general algebra. As we mentioned above, if the symmetry algebra is Virasoro, then the RCFT is called a minimal model. 5.3
Null states and constraints on correlation functions
It is not guaranteed that the representations that we constructed in 5.1 are irreducible. Suppose we consider a state |ψi ∈ V (c, h) such that
Ln |ψi = 0 ∀n > 0,
|ψi = 6 |hi
(5.15)
Consider a general state in the Verma module L−k1 ...L−kn |hi and consider the inner product hψ| L−k1 ...L−kn |hi = hh| Lk1 ...Lkn |ψi∗ = 0
(5.16)
Hence, |ψi is a null vector. The descendants of the null vector are also null vectors. It can be shown that since |hi itself is not a null vector, the space of states L−k1 ...L−kn |ψi is a proper subspace which is invariant under the action of the Virasoro generators [25]. Hence, the existence of null vectors in the Verma module renders it reducible. Let the space of null states in V (c, h) be N (c, h). To make the representation irreducible, we have to quotient out N (c, h) out of V (c, h), i.e, we identity |xi, |yi ∈ V (c, h) if |xi−|yi ∈ N (c, h). One can study the conditions for the existence of null states in the module which gives constraints on the form of h. Using the operator-state correspondence, we know that for every descendent state L−n |hi, there is a corresponding descendent field Φ−n (z) of a primary field Φ(z). Quotienting out the singular states from the Verma module corresponds to assigning the value zero to the corresponding field. Considering correlations functions of primary fields along with a field
– 54 –
corresponding to a null vector imposes constraints on the possible conformal weights of the primary fields in the operator product expansion. 5.4
Fusion Algebra
Let Φi be the field corresponding to the highest weight state |hi i and let [Φi ] denote the field and its descendants. [Φi ] is called a conformal family. Focusing on the conformal weights and the constraints on it that we discussed in the previous section, the operator product expansion of any two fields from a corresponding pair of conformal families and the possible fields in that expansion can be written as an algebra.
[Φi ] × [Φj ] =
X
Nijk [Φk ]
(5.17)
k
Nijk are called fusion coefficients, and they indicate the number of ways in which a field in [Φk ] can be obtained by fusing two fields in [Φi ] and [Φj ], respectively. k . The identity This is called the fusion algebra, and it is commutative. Hence, Nijk = Nji k = δ . The associativity of element of the algebra is the vacuum field Φ0 and hence Ni0 ik
OPE of primary fields implies that the above algebra is associative. This implies that X
l Njk Nilm =
X
l
m Nijl Nlk
(5.18)
l
For RCFTs, we can define matrices Ni of finite size with elements (Ni )jk = Nijk . Then, the associativity constraint and be written as
Ni Nk = Nk Ni
5.5
(5.19)
Verlinde Formula and Modular Invariance
Suppose we can diagonalise the matrices Ni using a matrix S. If the eigenvalues of Ni are αil . Then, Nijk = (SDS − 1) =
X lm
– 55 –
Sjl αil δlm (S −1 )mk
(5.20)
k =δ , Since Ni0 ik
Sin =
X
=
X
k Ni0 Skn
k
S0l αil (S −1 )lk Skn
lk
= S0n αin ⇒
αil =
Sil S0l
(5.21)
Thus, we have Nijk =
X Sjl Sil (S −1 )lk
(5.22)
S0l
l
This is the Verlinde formula which gives the fusion coefficients in terms of the S-matrix. For a modular invariant CFT, the Verlinde formula tells us something deep and profound. The fusion rules give local information about the CFT, while for a modular invariant theory, the S matrix is one of the generators of the modular group, which is a global property of the theory, i.e, the CFT being consistently defined on a torus. The Verlinde formula is the remarkable result that these two characteristics of the theory are closely related. 5.6
Fusion and Braiding Matrices
Consider the four point function
G(z, z) = hφi (z1 , z 1 )φj (z2 , z 2 )φk (z3 , z 3 )φl (z4 , z 4 )i
(5.23)
It can only depend on the crossing ratios
x=
z12 z34 , z13 z24
x=
z 12 z 34 z 13 z 24
(5.24)
We can use the OPE for φi φj and φk φl to reduce 5.23 to
G(z, (z)) =
X
kl
p p kl Cij Ckl Fij (p|x)F ij (p|x)
p
where Fijkl are called conformal blocks.
– 56 –
(5.25)
Alternatively, one can use the OPE for φj φk and φi φl . This amounts to exchanging the fields φj and φl which is equivalent to x → 1 − x. Thus, we get
G(z, (z)) =
X
jk
p Cilp Cjk Filjk (p|1 − x)F il (p|1 − x)
(5.26)
p
Also, one can use the OPE for φj φl and φi φk . This amounts to exchanging the fields φi and φl which is equivalent to x → x1 . Thus, we get G(z, (z)) =
X p
1 jl 1 p p jl Cik Cjl Fik (p| )F ik (p| ) x x
(5.27)
Since the order of fields is irrelevant in the correlation function, we should equate 5.25, p 5.26 and 5.27 which gives some constraints on the constants Cij . This is called the crossing
symmetry conditions and it can be depicted pictorially as [25] j j X
k
k p
p p Cij Ckl
X
p
p
q Cilq Cjk
q
i
l i
l
Figure 20. Crossing Symmetry
In minimal models, the number of conformal families is finite and the conformal blocks form a finite dimensional vector space. In this case, the different conformal blocks in 5.25, 5.27 and 5.27 differ only by a choice of the basis of this vector space and hence, they should be related by unitary transformations. We have,
Fijkl (p|x) =
X
(jk)
B(il)
1 jl pq Fik (q| )
q
x
(5.28)
and Fijkl (p|x) =
X
(jk)
F(il)
q
– 57 –
jl pq Fik (q|1
− x)
(5.29)
The unitary matrices B and F are called Braiding and Fusion matrices, respectively. Using five-point functions, one can prove that F and B matrices should satisfy some consistency conditions which are precisely the Pentagon and Hexagon equations that we saw in anyonic models. The constraints that the F and B matrices have to satisfy are also known as duality identities. Let us summarise what we have just found. We saw that the existence of null vectors in Verma modules constraints the conformal weights of primary fields in OPEs. We then defined the Fusion algebra and the Verlinde formula. We also found that crossing symmetry in 4-point functions leads to the concept of Fusion and Braiding matrices. Since fusion rules and the existence of matrices satisfying the Pentagon and Hexagon equations are the fundamental properties of an MTC, for every given RCFT, one can construct an MTC from it. This strongly indicates that conformal and topological field theories are deeply connected to each other. In the next section, we will give an axiomatic definition of topological field theories and eventually understand how topological invariance is related to the duality relations in a CFT.
– 58 –
6
3D Topological Field Theory and 2D Conformal Field Theory
In this chapter, we will explore the relationships between 3D topological quantum field theory (TQFT) and 2D conformal field theory. We will first go through an abstract and geometrical definition of a TQFT. After trying to get some intuition for this definition, we will give a more abstract definition of a 2D CFT. We will describe a 2D RCFT called the Wess-Zumino-Witten model and a 3D TQFT called Chern-Simons-Witten theory, and study their relationships. 6.1
Topological Quantum Field Theory
A topological quantum field theory can be naively defined as a quantum field theory in which the physics does not depend on the metric of the manifold on which the theory is defined. Thus, in the TQFTs that we are concerned with, the action does not depend on the metric. This immediately means that the stress-energy tensor of the theory is zero. As we mentioned in the introduction 1, TQFTs compute the topological invariants of a theory. We looked at topological quantum field theories which can be constructed from the Quantum Double of a finite group and Quantum group at a root of unity. Below, we will look into a more abstract definition of a TQFT which will eventually help us to find the relationships between topological and conformal field theories. 6.1.1
Atiyah Axioms
In this section, by a manifold, we mean a compact, topologically oriented manifold with boundary and vector spaces are defined over the field k. A (d+1)-dimensional topological field theory is given by the following set of data satisfying a set of axioms: 1. To any d-dimensional manifold Σ, we associate a finite-dimensional vector space HΣ . 2. To any (d+1)-dimensional manifold, we associate a vector vM ∈ H∂M where ∂M denote the boundary of M.
– 59 –
3. For any homeomorphism of d manifolds f : Σ1 → Σ2 , we associate an isomorphism of the corresponding vector spaces f∗ : HΣ1 → HΣ2 . 4. The following isomorphisms hold: ∗ HΣ ' HΣ
(6.1)
H∅ ' k
(6.2)
HΣ1 F Σ2 ' HΣ1 ⊗ HΣ2
(6.3)
where Σ denotes the manifold with the opposite orientation. 5. Gluing: If M1 and M2 are glued along a d-manifold Σ to form M, then
vM = hvM1 , vM2 i
(6.4)
6. Completeness: The states vM for all M such that δM = Σ spans HΣ . Let us consider the most straightforward example where d=0, which corresponds to a 1D TQFT. From a physical point of view, if we view a d+1 TQFT as defined on d space + 1 time, then 1D TQFT is like quantum mechanics. We associate a Hilbert space Hp with a point p (0D manifold). For a map p → p denoted by Hp
Hp
we can take the corresponding isomorphism of vector spaces to be the Euclidean time evolution operator e−tH : Hp → Hp . After that extremely short digression on 1D TQFT, let us consider the more complicated case of d=1. There is only one 1D connected oriented manifold, which is the circle. Let us associate the vector space H0 with the circle. Then, we can associate a vector in H0 , which we denote as 1, with the disc D1 . Since H∅ = k, we associate the map 1 : k → H0 with the diagram
– 60 –
H0
k
Similarly, we can associate the trace map tr : H0 → k with
H0
k
Also, we can associate a multiplication map × : H0 ⊗ H0 → H0 with H0
H0 H0
The commutativity of the multiplication map follows from the homeomorphism of the following diagrams,
a
b
a×b
a×b a
b
while the associativity follows from the homeomorphism of the diagrams a
b
c
a
(a × b) × c
b
a × (b × c)
c
where a, b, c ∈ H0 . One can also show that tr(a × b), a, b ∈ H0 is non-degenerate using the homeomorphism of the following diagrams,
– 61 –
a
a b
b
This is, in fact, the properties of a Frobenius algebra. A Frobenius algebra is a commutative and associative algebra with a unit and a non-degenerate linear map. It can be proved that 2D TQFTs and Frobenius algebras are in one-to-one correspondence [16]. For d=2, we have a 3D TQFT, which by definition gives invariants of three-manifolds. It can be proven that every closed connected orientable 3 manifold can be obtained by a procedure called surgery on a set of knots in S 3 [29]. Also, it has been shown in [32] that an MTC uniquely defines a 3D TQFT satisfying the above axioms. Hence, anyonic theories are indeed TQFTs. These are the TQFTs that we are most interested in from the point of view of finding relationships between TQFTs and CFTs. Rather than giving a general description, in the following sections, we will look at a particular type of 3D TQFT and its relationship to 2D CFTs. We will then comment on results that relate 3D TQFTs and 2D CFTs in general. 6.2 6.2.1
Modular Functor of Conformal Field Theory Vertex Operators in CFT and the geometric picture
Recall that chiral fields depend only on the z coordinate. The operator product expansion of two chiral fields results in another chiral field. Thus, the set of chiral fields in a CFT is closed under the OPE. Same is true for the anti-chiral fields. This algebra is defined mathematically in terms of vertex operator algebras. Vertex algebras model the algebraic structure of operators on a Riemann surface. We follow [30] and define the oper i ator jk : Hj ⊗ Hk → Hi . Geometrically, we will denote this on a three-holed sphere with representation spaces on the three holes as given in figure 21. We assume that these operators commute with contour deformations on the surface i.e, it
– 62 –
j . z
∞. i k
. 0
Figure 21. Geometrical representation of a Chiral Vertex Operator
has to satisfy h X i i i n n+1−k z Ln (∞) (β ⊗ γ) = z Lk−1 β ⊗ γ + β ⊗ Ln (0)γ jk z jk z k
(6.5)
The z-dependence of these operators is completely specified using i d i (β ⊗ γ) = (L−1 β ⊗ γ) dz jk z jk z i jk
(6.6)
is called a chiral vertex operator. It is an operator associated with a three-holed sphere
which can be depicted diagramatically as in figure 21. In RCFTs, we have finite dimensional i . vector spaces of such operators denoted by Vjk
l .w
k
.
z
∞. i
j .0
Figure 22. Operator defined on a four holed sphere i of operators We can consider generalisations of this. We can define a vector space Vjkl
– 63 –
Hj ⊗ Hk ⊗ Hl → Hi . Operators in this space are naturally associated with a four-holed sphere (figure 22). i can be written in terms of V i ’s. This essentially boils down to the fact The spaces Vjkl jk
that the four holed sphere can be got by sewing two three holed spheres. Consider the i and the corresponding sewing of three holed spheres as given in decompositions of Vjkl
figure 23. j
k
p i
l
j
k
p i
l
j
k
p l
i
p i ∼ l Figure 23. First figure corresponds to the decomposition Vjkl ⊗Vki . Second one corresponds = Vjp p p i ∼ l i ∼ to the decompoisition Vjkl = Vkp ⊗ Vji , and the third one corresponds to Vjkl = Vpil ⊗ Vkj
The different ways of sewing the 3-holed sphere to get the 4-holed sphere correspond to i . Hence, the above decompositions should be related by unitary different bases for Vjkl
– 64 –
matrices. These are none other than the fusion and braiding matrices that we discussed i are the same as the conformal blocks of the theory. before. Thus, the spaces Vjkl
6.2.2
Segal’s Definition
In the previous section, we saw that chiral vertex operators are related to holed spheres and that RCFTs have finite dimensional vector spaces of such operators. Alternatively, we can think of associating vector spaces to Riemann surfaces. Segal gave an abstract mathematical definition of CFTs [31]. He defined a CFT as a modular functor from the category of labelled, oriented Riemann surfaces to the category of finite-dimensional vector spaces over C. According to Segal, a CFT is defined by the following data and conditions, Data: 1. A finite set I of labels with a distinguished element 0∈I and an involution i → i where i is the conjugate label. Also, 0 = 0. 2. A map from every oriented surface Σ with punctures Pr ,1 ≤ r ≤ n, with each puncture Pr equipped with a direction vr and label ir to vector spaces.
Σ(P1 ,...,Pn ) → HΣ
(6.7)
where each Pr actually stands for (Pr , vr , ir ). 3. A linear transformation Hf : HΣ1 → HΣ2 associated to an isomorphism f : Σ1 → Σ2 . The set I is the index set for the finite number of representations in the theory. The map defined in the second data associates a vector space with a given Riemann surface with punctures. Those vector spaces are the conformal blocks. Finally, the linear transformations defined in the third data are the duality transformations. The definition also includes some conditions. Conditions: 1. Functoriality: Hf only depends on the isotopy class of f.
– 65 –
2. Involution: If Σ the Riemann surface Σ with reversed orientation and the involution ∗. of the representations, then, HΣ ' HΣ
3. Multiplicativity: HΣ1 tΣ2 ' HΣ1 ⊗ HΣ2 4. Gluing: If Σ is obtained from
1
and Σ2 by gluing along a puncture with conjugate
labels j and j, then, the following isomorphism holds,
HΣ =
M
∈ HΣ1 ⊗ HΣ2
(6.8)
j
where Σ1 has a label j in one of its punctures while Σ2 has j in one of it’s. 5. Normalization: HS 2 ' C In the next section, we will study several interesting relationships between topological and conformal field theories. We will compare the axiomatic definitions of a TQFT and CFT and look at some conjuctures regarding the same. 6.3
Relationships between Topological and Conformal Field Theories
Before looking into the general relationships between a 3D topological field theory and 2D conformal field theory, let us study a specific example of topological field theory called the Chern-Simons-Witten (CSW) theory. This theory is related to a rational conformal field theory called the Wess-Zumino-Witten (WZW) model. 6.3.1
Wess-Zumino-Witten Model
The WZW model can be defined explicitely using a Lagrangian. Given a compact simplyconnected Lie group G and its lie algebra g, the action for WZW is given by k S= 16π
Z
d2 xT r(∂ µ h−1 ∂µ h) + kΓW ZW
– 66 –
(6.9)
where h is a function living on G. It should be values in a unitary representation of G so that the action is real. The trace is taken in that representation, and
ΓW ZW
−i = 24π
Z
−1 α
d3 y αβγ T r(h
∂ h h
−1 β
−1 γ
∂ h h
∂ h)
(6.10)
B
where B is a 3-manifold whose boundary is our original 2D space. The overline on h indicates the extension of the function to the 3-manifold. The solution for the classical field equation is given by h(z, z) = f (z)f (z)
(6.11)
where f (z) and f (z) are arbitrary functions. Also, the currents JZ = ∂z hh−1 ,
Jz = g −1 ∂z g
(6.12)
are conserved. It can be proved that the Laurent modes of Jz above follows the commutation relation a [jm , jnb ] = i
X
c f abc jm+n + kmδ ab δm+n,0
(6.13)
c
which is the Kac-Moody algebra that we described in section 5.2. In fact, the WZW model is an RCFT based on affine Lie algebra, which is a special case of Kac-Moody algebras. A Kac-Moody algebra is an affine Lie algebra when the generalized Cartan matrix is positive definite and has corank 1. Laurent modes of Jz also satisfies a similar relation leading to another independent affine Lie algebra. As mentioned in section 5.2, the energy-momentum tensor can be obtained by the Sugawara construction. [25] has an extensive account of WZW models. While the Laurent modes of the stress-energy tensor obtained using the Sugawara construction generates the Virasoro symmetry algebra, the WZW model also has a bigger symmetry which stems from the invariance of the theory under the transformation h(z, z) → Ω(z)h(z, z)Ω(z)−1
– 67 –
(6.14)
where Ω and Ω are arbitrary group valued functions of x + y and x − y respectively. Like in the case of CFTs with Virasoro symmetry, we will define the primary fields in WZW to be those which transform as φ(z, z) → Ω(z)φ(z, z)Ω(z)−1
(6.15)
and this is assumed to be a symmetry of the theory. We note that the transformation of h is a special case of the above equation for Ω and Ω in the fundamental representation of the algebra. In general, they may belong to general representations R and R, respectively. In section 5.1.2, we looked at the highest weight representation of the Virasoro algebra. Let us go through a similar analysis for theories invariant under a current algebra. The fields in the theory transform into each other under the application of Ln and Jna . From the definition of a primary field φ, we have
Ln φ = Jna φ = 0
for n > 0
(6.16)
Also, let the L0 and J0a eigenvalues of the field be ∆ and τ a , respectively. Analogous to the discussion in 5.1.2, we obtain the descendant states by acting on φ(zz) with operators a
a ,L L−n , J−m −n , J −m . The highest weight representation of the combined Virasoro ad Kac-
Moody algebras is given by a
a
1 a1 J 2 J−n L J a2 . . . J −n . . . L−m1 L−m2 . . . φ(z, z) 1 −n2 −m1 −m2 1 −n2
(6.17)
in which the highest weight vector is the primary field φ(z, z) with weights (∆, ∆) corresponding to (L0 , L0 ) and representations (R, R) corresponding to the Kac-Moody algebra. As in section 5.3, we should be careful about the null states in the representation and mod them out of it to make the representation irreducible. In this case, there are three types of possible null states 1. Purely Virasoro 2. Combined Virasoro and current algebra
– 68 –
3. Purely current algebra The most interesting case is the null states from pure current algebra. It turns out that the fields corresponding to non-integrable representations (recall the definition of integrable representations given in section 4.3.3) are trivial, and we have to only consider significantly less number of representations than those given by the construction 6.17 [23]. If the primary field φint is the highest weight on an integrable representation, then the tower of descendants obtained from it is finite-dimensional. To summarise, the important take-home message is that every representation of the affine algebra of interest in a field theory is an irreducible integrable representation. For details on the first and second case, one could refer [26] and [27]. 6.3.2
Chern-Simons-Witten Theory
Given a gauge group G, there is a Lie algebra valued field called the connection A. A is not an observable as it is not gauge invariant. However, the curvature F defined by
F = dA + A ∧ A
(6.18)
transforms as F → U −1 F U where U ∈ G and hence T r(F n ) := T r(F ∧ ... ∧ F ) is gauge invariant. We define the 2d forms called Chern forms
P2n (F ) = T r(F n )
(6.19)
It can be proved that Chern forms are closed [28]. Hence, using Poincare lemma, it can be locally written as the exterior derivative of a (2n-1) form,
P2n (F ) = dQ2n−1 (A, F )
(6.20)
Q2n−1 (A, F ) is called the Chern-Simons form. It can also be proved that under a gauge transformation, the Chern-Simons form transforms as a total derivative [8]. Witten wrote down a theory in 3 dimensions whose action is the integral of Q3 (A, F ). This is called the
– 69 –
Chern-Simons-Witten theory whose action is given by k S= 4π
Z
2 tr(A ∧ dA + A ∧ A ∧ A) 3
(6.21)
where k∈ Z for the consistency of the theory. Varying the action with respect to the gauge field, we get the classical equation of motion
F = dA + A ∧ A = 0
(6.22)
Note that since F vanishes, there are no local gauge invariant observables as they are products of F’s and their derivatives. However, we can have non-local gauge invariant observables. One important example is the Wilson loop operator. Consider a closed and oriented curve C in M. Given an irreducible representation R of G, one can define the Wilson loop operator as Z WR (C) = TrR P exp Ai dxi
(6.23)
C
where P denotes path ordering. Note that we didn’t use a metric in this definition and hence the Wilson loop operator respects general covariance. The expectation value of WR (C) is defined in the usual way Z hWR (C)i =
DA exp (iL)WR (C)
(6.24)
C is a general closed oriented curve in a 3D manifold. It can be as simple as the unknot or a very complicated knot. Given a knot, one would like to know whether it can be unknotted. Jones came up with a polynomial (called the Jones polynomial) which can be associated with a given knot which shows whether it can be unknotted or not. Such quantities are called knot invariants. They are characteristic of a given knot and helps us to distinguish it from other knots. Witten proved that the expectation value hWR (C)i defined above gives
– 70 –
the Jones polynomial for C when M = S 3 . Also, Z DA exp (iL)
(6.25)
gives the invariants of the 3D manifold M. We will consider the canonical quantization of the theory living on a 3 dimensional manifold M = Σ × R where Σ is a Riemann surface. Canonical quantization of the theory with R as time will produce a Hilbert space HΣ . If we denote the gauge field components in the time and spatial directions by A0 and Ai , respectively, then, the action can be written as k S= 4π
Z
Z dt
tr ij Ai
Σ
∂ Aj + A0 Fij ∂t
(6.26)
where Fij = ∂i Ai − ∂j Ai + [Ai , Aj ]. We can see that A0 acts as a Lagrange multiplier to impose the constraint that Fij = 0. We can impose the canonical commutations relations
[Aai (x), Abj (y)] =
2πi ij δ ab δ 2 (x − y) k
(6.27)
where a and b are group indices. Alternatively, we can use the complex structure on Σ to write the commutation relation as
[Aaz (z, z), Azb (z, z)] =
2π ab δ δ(z − w)δ(z − w). k
(6.28)
We can consider the Hilbert space to be the space of holomorphic functionals of the gauge field and impose conditions on them obtained from the constriant Fij = 0. Due to this constraint, the Hilbert space HΣ will be finite dimensional. 6.3.3
2D Duality Identities and 3D General Covariance
HΣ obtained in the previous section is actually the vector space of conformal blocks in the WZW model. In the modular functor definition of a CFT, Segal associates a vector space
– 71 –
to a Riemann surface Σ, and Witten showed that for CFTs with a current algebra of a compact group G at level k, the vector spaces in the definition of Segal can be obtained from the CSW theory. In general, we have several Wilson loops WRi (Ci ) in M which are cut by Σ. Hence, in this case, we have a Riemann surface Σ with finitely many points pi marked on it, and since each Wilson loop WRi (Ci ) comes with a representation Ri , each marked point pi also has the representation Ri associated with it. This is indeed the general picture of Riemann surface with marked points that we saw in the Segal’s definition where now the marked points correspond to the insertion of primary field or its descendants corresponding to the representation Ri . Again, the vector space associated to it is obtained by the canonical quantization of CSW on Σ. We saw that the Wess-Zumino-Witten model with Lie group G at level k defined on the Riemann surface Σ corresponds to the Chern-Simons-Witten theory with gauge group G, with the integer multiplying the action chosen to be k, defined on the 3D manifold M = Σ × R. More precisely, the vector space of conformal blocks in the WZW model on Σ is the Hilbert space HΣ of the CSW theory defined on Σ × R. The crucial insight that went into this result is that 2D duality identities are the same as 3D general covariance. A simpler instance of this correspondence is between U(1) CSW theory at level k ∈ 2Z and free boson RCFT with radius R2 = k. The relationships between CSW and WZW theories are, in fact, an example of the close relationship between RCFTs and TQFTs in general. First of all, we note that for d=2, the definition of a TQFT looks very similar to the definition of 2D CFT by Segal. To obtain a 2D modular functor from the definition of a 3D TQFT, one has to find the finite set of labels and the fusion algebra. On the other hand, to obtain a 3D TQFT from a 2D modular functor, one has to use the data in the modular functor and find the vector vM ∈ H∂M . In [33–35], it was proved that this can be done. Thus, one can define a modular functor for a given RCFT and then obtain a 3D topological field theory from it. From this point of view, we can see that the emergence of fusion and braiding in an RCFT and the consistency conditions like the Pentagon and Hexagon equations are a natural consequence of general covariance in 3D.
– 72 –
7
Entanglement Entropy in Topological Quantum Field Theories
Topological and conformal field theories are important arenas for doing calculations which help us understand the subtle properties of quantum field theories. The conformal symmetry in CFTs helps us in deriving analytic results whose QFT counterparts are too hard to derive. TQFTs, being independent of the metric, and because of the lack of dynamics are ‘simpler’ than CFTs. A broad and long-term goal of studying these theories is to understand the nature of quantum field theories better. Entanglement is one of the crucial properties of quantum systems. Hence, understanding the details of a quantum theory includes studying the entanglement structure of the theory. However, the infinite number of degrees of freedom in a QFT often makes this process hard. The problems become a bit easier to handle in a CFT. Explicit formulas for the entanglement entropy of a region in a CFT has been obtained [36]. In general, these formulas depend on the UV cutoff. We would expect that the problem becomes simpler in a TQFT. The idea of topological entanglement in a TQFT was first introduced in [37]. We will be interested in a recent analysis of entanglement in TQFTs or more precisely in CSW theories [38]. The basic idea is the following. As we discussed in section 6.3.2, the Hilbert space of CSW theory is finite. Given a link, we will construct a state associated with it, which belongs to a finite-dimensional Hilbert space. We will then define a reduced density matrix in a suitable manner for which we will calculate the Von-Newman entanglement entropy. Then, we will discuss the information about the link that the entanglement entropy contains. 7.1
Manifold with torus boundaries and link states
Consider a 3D manifold with boundary Σn with n disjoint boundary components. The path integral of the theory as a functional of data on the boundary defines a wavefunction on Σn . Since Σn is made up of disjoint components, we can associate a Hilbert space with each of the components, and the wavefuction obtained lives in the tensor product of these
– 73 –
Hilbert spaces. We will consider Σn in which each disjoint component is a torus. The 3D manifold corresponding to Σn is obtained by surgery on S 3 .
S3
Len
Figure 24. Surgery on S 3 : The black lines represent the link and the white part around it represents its thickening. We remove this tubular region from S 3 to obtain Mn .3
We will follow the notation in [38] and consider a link Ln in S 3 which is made up of n knots Li where 1 ≤ i ≤ n. Surgery on S 3 involves thicknening this link to form tubes, and then removing it from S 3 . Let Len be the thickened link. Then, the 3D manifold Mn with n disjoint torus boundaries is given by
Mn = S 3 − Len
(7.1)
The path integral of the CSW theory on Mn gives the state |Ln i called the link state. N This state belongs to ni=1 Hi where Hi is the Hilbert space of CSW theory defined on the ith torus in the boundary Σn . Each Hi is a finite dimensional Hilbert space. We choose two cycles m and l of the torus which generates its fundamental group. A basis for Hi is found by performing the CSW path integral on the solid torus with a Wilson line in the representation Rj placed in the bulk running parallel to the longitude cycle l. Index j denotes an integral representation of the gauge group G at level k. This gives a state |ji on the torus. If G=U(1), the integral representations are given by integer valued charges 3
Picture taken from [38]
– 74 –
0 ≤ j ≤ k and dim(H(T 2 , U (1))) = k. For G=SU(2), the integral representations are labelled by their spin j = 0, 12 , . . . , k2 and dim(H(T 2 , SU (2))) = k + 1. Carrying out the same procedure with the conjugate representation gives us hj|. This construction of the Hilbert space of CSW theory on a torus is given in [8]. The link state can be written as
|Ln i =
X
CLn (j1 , ..., jn ) |j1 , ..., jn i
(7.2)
j1 ,j2 ,...,jn
where CLn (j1 , ..., jn ) are the coloured link invariants of CSW theory with the representation Rj∗i placed along the ith component of the link. CLn (j1 , ..., jn ) = hWRj∗
1
(L1 ),...,Rj∗ (L2 ) i
(7.3)
2
Now we can consider a partition of the link in the following form
Ln = (L1 t L2 t ... t Lm ) t (Lm+1 t Lm+2 t ... t Ln )
(7.4)
and define the reduced density matrix
ρ=
1 TrL1 ,...,Lm (|Ln i hLn |) hLn |Ln i
(7.5)
The Von-Neumann entanglement entropy is given by
S(L1 ,...,Lm |Lm +1,...,Ln ) = −TrLm+1 ,...,Ln (ρ lnρ)
(7.6)
Note that if the link consists of n unlinked knots, then the coloured knot invariants factorise, and we get a separable state |Ln i = |L1 i ⊗ ... ⊗ |Ln i
(7.7)
for which the Von-Neumann entanglement entropy is zero. Also, since the theory doesn’t have any local dynamics, contributions to S(L1 ,...,Lm |Lm +1,...,Ln ) comes purely from the topological properties of Mn .
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7.2
Entanglement entropy for the abelian case G=U(1)
Let us take G = U (1)k , and consider two component links. From [8], the link state is given in terms of the U (1) representation charges and the Gauss linking number between the knots in the link. 2 1 X 2πiq1 q2 l12 L = k e |q1 i ⊗ |q2 i kqq
(7.8)
1 2
S(L1 |L2 )
= ln
k gcd(k, l12 )
(7.9)
For the Hopf link, l12 = 1 and S(L1 |L2 ) = ln(k) which shows that the Hopf link is maximally entangled.
Figure 25. Hopf Link
For 3-component links we get
S(L1 |L2 ,L3 )
= ln
k gcd(k, l12,l13 )
(7.10)
For an n-component link, we have the general formula
S(m|n−m)
m k = ln |KerG|
where G is the linking matrix for (m|n − m) partition, l1,m+1 · · · lm,m+1 . .. .. G= .. . . l1,n
···
– 76 –
lm,n
(7.11)
and lij is the Gauss linking number between knots Li and Lj modulo k. |KerG| is the number of solutions to the equation
G · ~x = 0 (mod k)
7.3
(7.12)
Entanglement entropy for non-abelian case G=SU(2)
In this case, the coloured link invariants are given by the coloured Jones polynomials. For the special case of Hopf link, the link state is given by
|Hopfi = √
X 1 Sj1 ,j2 |j1 i ⊗ |j2 i k+1 j j
(7.13)
1 2
where S is the unitary modular S-matrix. r Sj1 j2 =
2 (2j1 + 1)(2j2 + 1)π sin k+2 k+2
(7.14)
On tracing over the second loop, and calculating the Von-Neumann entropy, we get
S Hopf = ln(k + 1)
(7.15)
which once again shows that the Hopf link is maximally entangled. Let us now consider the following three component link.
Figure 26. A three component link which is the connected sum of two Hopf Links
Reduction on any one of the knots gives the same entanglement entropy given by the
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quantum dimensions. d−2 j S = −pi ln(pi ), where pj = P −2 j 0 dj 0 The quantum dimensions are given by dj =
S0j S00 .
(7.16)
Note that, in this case, trace over any
link gives a separable state, and hence the link state corresponding to the above link is like a GHZ state. However, the three knots in the above link are clearly different, as erasing the second one leads to an unlink, whereas erasing the other two leads to a Hopf link. Thus, entanglement entropy does not distinguish between the different components of the link. While the link state itself contains much information about the link, the entanglement entropy contains only a coarse-grained information about it. However, it can be shown that the relative entropy can be used to distinguish between the knots in the link [38]. From quantum information theory, we know that, in bipartite systems, there is a concept of a maximally entangled state. It has also been proved that one can create any given state from a maximally entangled state using LOCC operations [39]. In the case of CSW theories, the maximally entangled state is the link state corresponding to the Hopf link. For tripartite systems, the entangled states are classified into GHZ and W states. We saw that the link in figure 26 is a GHZ type state. In [40], it was shown that all links which can be drawn on a torus are W type states.
– 78 –
8
Conclusion
We looked at the properties of anyonic models and how they can be defined in terms of MTCs. We looked at two special cases of realisations of MTCs which were essential for our discussion. A more general account of the classification of MTCs is given in [18]. We also explored how fusion and braiding matrices arise in CFTs. The definition of a CFT in term of Vertex Operator Algebra and the derivation of duality relations for a general RCFT is given in [41]. [42] has a poof of the Verlinde formula for an RCFT. A general version of the formula for Vertex Operator Algebras was proved in [43]. We went through the rigorous definitions of TQFTs and CFTs and argued that 2D CFTs and 3D TQFTs are intimately related to each other by studying the explicit example of the correspondence between CSW theories and WZW models, the key ingredient being the realisation that 2D duality relations correspond to general covariance in 3D. The following conjectures were stated in [30] regarding the relationships between RCFTs and TQFTs. Conjecture 1: Any RCFT is equivalent to a CSW theory defined by the pair (G, λ) with G a compact group and λ ∈ H 4 (BG, Z). Conjecture 2: Any three-dimensional TQFTs is a CSW theory for some appropriate (super)-group. Since an MTC gives a unique TQFT, the above conjectures mean that we should be able to find a CSW theory for every MTC. It also states that we should be able to find an RCFT corresponding to every MTC. We saw that at least when G is a connected semisimple Lie group (CSW in the sense of the original Witten’s paper [8]), CSW theories are related to WZW models. Thus, if a given MTC is related to a CSW theory in the sense of Witten, there is a natural candidate for an RCFT associated to it. The resolution of these conjectures is still an open question. Though there is substantial evidence for the correspondence between MTCs and CSW theories, it has not yet been proven that every MTC gives rise to a CSW theory since we do not yet have a complete classification of CSW theories. In [44], the authors considered two MTCs which may not correspond to any CSW
– 79 –
TQFTs. The MTCs which cannot be constructed from CSW theories are called exotic. We also discussed how we could define topological entanglement entropy in CSW theories. Further work in this direction has been done in [45, 46]. Some of the interesting aspects of TQFTs that we did not discuss include its application in Condensed Matter Physics and Quantum Computation [5]. TQFTs are a rich class of theories which has ignited significant research within the mathematics and mathematical physics community. The fact that TQFTs can be defined mathematically rigorously has helped in the communication of mathematicians and physicists and helped them to understand each other’s fields better. The rigorous study of TQFTs and RCFTs carried out by mathematicians has allowed us to understand the relationships between the two in a more general setting than that of the CSW-WZW correspondence. On the other hand, CSW theories have impacted mathematics in several ways. Most importantly, it has provided an infinite number of topological invariants and an inherently 3D way of computing knot and link invariants.
– 80 –
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