initiated with Patera 2003; general, systematic study launched with Klimyk-Patera 2006-08. C-functions, or Weyl-orbit sums, are. Cλ(a) = â wâW e2Ïiãwλ,aã ,.
Weyl Orbit Functions and
Conformal Field Theory Mark Walton
Lie Theory and Mathematical Physics CRM, Montr´eal, 22 May 2014 Collaborator: Jiˇr´ı Hrivn´ ak (Prague); supported in part by NSERC.
IDEA
? Discretized Weyl-orbit functions X Sλ (a) = (det w )e 2πihw λ,ai w ∈W
for example
⇐ ⇒
Affine modular data, modular S-matrix of WZNW conformal field theory X 0 Sλ,µ ∝ (det w )e −2πihw λ,µi/M w ∈W
... ?
Incidentally, the relation between affine modular data and Weyl-orbit functions, of both S and C type, was exploited in Gannon-Jakovljevic-Walton 1995. Simple Lie algebra weight multiplicities were extracted using these objects and their symmetries. Quella has done work similar in spirit (2002).
OUTLINE
I
(Weyl-)Orbit Functions
I
Affine (Wess-Zumino-Novikov-Witten) Modular Data
I
Modified Multiplication, Galois Symmetry
I
Conclusion
Notation SPECIAL FUNCTIONS OF WEYL GROUPS G compact, simple Lie group, rank n, Lie algebra gn
1.1. Weyl groups of simple Lie algebras wish Π to = investigate to Weyl groups arising 2. SimpleWe roots: {αj | j ∈special {1, . .functions . , n} =: related I }, normalized |αlong |2 = from
simple Lie algebras. There are four series of simple Lie algebras An (n ≥ 1),
Weyl group Wsimple = h rjLie | jalgebras ∈ I i E6 , Primitive j generate Bn (n ≥reflections 3), Cn (n ≥ r2), Dnr(n ≥ 4) and five exceptional αj = E , E , F and G , each connected with a unique Weyl group [2, 19, 20, 29, 60]. They are completely classified by Dynkin diagrams (see Figure 1.1). A Dynkin
7 8 4 2 Coxeter-Dynkin diagram of gn encodes W .
An (n ≥ 1) Bn (n ≥ 3) Cn (n ≥ 2) Dn (n ≥ 4)
g 1
g 1 w 1 g 1
g g 2 3 g 2
...
w 2
...
g 2
...
...
g n
g w n−1 n w g n−1 n
gn − 1
g g g n−3n−2n
E6 g g 1 3 E7 g g 1
3
E8 g g 1
F4 g 1
3
g 2
g2 g 4 g2 g 4 g2 g 4
w 3
g 5
g 6
g 5
g g 6 7
g 5
w 4
g g 6 7
g 8
G2 g w 1
2
Notation
Fundamental weights Φ = {ωj | j ∈ I } weight space PR := R Φ ∼ = Rn Non-negative weights: P+,R := R≥0 Φ; positive weights P++,R := R>0 Φ Weight lattice P := Z Φ; dominant (regular) integral weights: P+ := N0 Φ (P++ := N Φ). Roots ∆, long (short) roots ∆` (∆s ), positive roots: ∆+ , etc.
ORBIT FUNCTIONS Weyl-orbit functions: program of study as “special functions” initiated with Patera 2003; general, systematic study launched with Klimyk-Patera 2006-08.
C -functions, or Weyl-orbit sums, are X Cλ (a) = e 2πihw λ,ai , w ∈W
where λ, a ∈ PR . So-named since gn = A1 → Cλ (a) = 2 cos (2πihλ, ai) W -invariance: Cw λ (a) = Cλ (wa) = Cλ (a) , ∀w ∈ W . Restrict to λ, a ∈ P+,R , the fundamental region of W .
S-functions are the antisymmetric Weyl-orbit sums Sλ (a) =
X
(det w ) e 2πihw λ,ai .
w ∈W
gn = A1 → Sλ (a) = 2i sin (2πihλ, ai) W -antisymmetry: Sw λ (a) = Sλ (wa) = (det w ) Sλ (a) , ∀w ∈ W . S-functions vanish on ∂P+,R , int(P+,R ) =: P˜+,R . Other “trig” possibility: Eλ (a) =
X
e 2πihw λ,ai
w ∈We
We ⊂ W subgroup of even elements. gn = A1 → Eλ (a) = e 2πihw λ,ai .
Non-simply-laced algebras, generalize to S ` - and S s -functions, via
ϕσλ (a) =
X
σ(w ) e 2πihw λ,ai .
w ∈W
σ(w ) is a sign homomorphism on W .
σ = 1, det ⇒ C -, S-functions, respectively. σ s (ri ) = −1 (+1), if αi is a short (long) simple root, respectively. Opposite definition for σ ` . σ = σ s , σ ` ⇒ ϕ = S s -, S ` -functions, respectively.
Ratios ⇒ characters, and “hybrid characters”: χλ (a) =
Sλ+ρ (a) , Sρ (a)
χ`λ (a) =
` Sλ+ρ ` (a)
Sρ`` (a)
,
χsλ (a) =
where ρ` =
1 X α = 2 ` α∈∆+
X
ωi ;
αi ∈Π∩∆`+
s
ρ similarly; and ρ is the usual Weyl vector. Focus here on the C - and S-functions, and characters χ.
s Sλ+ρ ` (a)
Sρs ` (a)
;
Properties of orbit functions
Restricting weight-labels λ ∈ P = ZΦ ⇒ affine Weyl symmetry: Cλ (wa) = Cλ (a) , Sλ (wa) = (det w ) Sλ (a) , or
ϕσλ (wa) = σ(w ) ϕσλ (a) ,
∀w ∈ W aff .
Affine Weyl group W aff = Q ∨ o W = hrj | j ∈ Iˆi ,
where Iˆ := {0, 1, . . . , n}.
0-th simple reflection: r0 a = rξ a +
2ξ hξ,ξi
, ξ highest root of gn .
Extended Coxeter-Dynkin diagram encodes the affine Weyl group W aff : J. Phys. A: Math. Theor. 42 (2009) 385208
J Hrivn´ak and J Patera
0
An , n ≥ 2
0
6
2
3
A1
E6 1
2
n
3
1
0
Bn, n ≥ 3
4
5
0
1
7
E7 1
n−1
2
n
0
1
2
3
4
5
6
8 Cn, n ≥ 2
0
1
n−1
2
Dn, n ≥ 4
n
n
0
E8 0
1
2
3
4
0
1
2
3
4
2
n−3 n−2 n−1
6
7
G2
F4 1
5
0
1
2
Figure 1. The extended Coxeter–Dynkin diagrams of simple Lie algebras and their numbering. The extension root carries number 0.
• The root lattice Q of G:
Fundamental domain of W aff , n o ω∨ ω∨ F = Conv 0, m11 , . . . , mnn . mj ∈ N known as marks, and ξ = m1 α1 + · · · + mn αn . Put m0 = 1. Φ∨ := {ω1∨ , . . . , ωn∨ } denote the dual fundamental weights, satisfying hωi∨ , αj i = δi,j . They are the fundamental weights of the dual Lie algebra gn∨ . Short ↔ long roots: Coxeter-Dynkin diagram of gn ⇒ diagram of gn∨ . Dual roots αj∨ satisfy hαi∨ , ωj i = δi,j (i, j ∈ I ).
Reflections rj∨ = rj (j ∈ I ) associated with the αi∨ generate the same Weyl group W , but a different affine Weyl group arises: caff = Q o W . W r0∨ a = rη a +
2η . hη, ηi
Highest dual root η =: −α0∨ = m1∨ α1∨ + . . . + mn∨ αn∨ defines the dual marks mj∨ , j ∈ I ; put m0∨ = 1. λ ∈ P ⇒ consider ϕσλ (a) a function on the fundamental domain F of W aff . Orbit function ϕσλ is an eigenfunction of the Laplace operator on F with Neumann (Dirichlet) boundary conditions on σ(r ) = +1 (σ(r ) = −1) hyperplanes.
CUBATURE FORMULAE FOR SIMPLE LIE GROUPS
gn = G2 example of a fundamental region F , shaded here:
ω ˇ2
α ˇ2
ω ˇ1
α ˇ1
Dual affine Weyl symmetry: Restricting weight-arguments a ∈ P ∨ = ZΦ∨ ⇒ affine dual Weyl symmetry: Cw λ (a) = Cλ (a) , Sw λ (a) = (det w ) Sλ (a) , ϕσw λ (a) = σ(w ) ϕσλ (a) ,
or
caff . ∀w ∈ W
caff , or dual fundamental domain, is The fundamental region of W F ∨ = Conv{0, mω∨1 , . . . , mω∨n }. 1
n
Generalizations of Chebyshev polynomials (Nesterenko-Patera-Tereszkiewicz 2011): Polynomials constructed 3 ways: • Most familiar: Xj := e 2πixj • When Cλ can be written as a sum of cosines, then it can be written in terms of basic ones using trig identities. Then these basic cosines can become the new variables. • Xj := Cωj (x) and S := Sρ (x). Decompose Xj Cλ → recursion relations defining the polynomials. (Also Xj := Sωi (x) and Xj := χωi (x), of course.)
Continuous orthogonality: hCλ , Cλ0 i := |F |−1
Z F
Cλ (x)Cλ0 (x) dx = |W λ| δλ,λ0
for λ, λ0 ∈ P+ . Discrete orthogonality: hf , g iM := |W | where FM :=
1 ∨ ∨ M P /Q
X
f (x) g (x)
x∈FM
∩ F is the discretized fundamental domain, or
“grid”; M controls the fineness, or resolution of the discretization.
gn = C2 example: grid F4 = 14 P ∨ /Q ∨ ∩ F . Dots in 14 P ∨ /Q ∨ , and |F4 | = 9 (fundamentalJ Hrivn´ region shaded). ak and F J Patera
J. Phys. A: Math. Theor. 42 (2009) 385208
η
α2 = α2∨
ω2 = ω2∨
ξ = ω1∨
F ω1 = 12 ω1∨
α1∨
α1
r0
r2 r1
Discrete orthogonality (Moody-Patera 2006): hCλ , Cλ0 iM = cG M n |W λ| δλ,λ0 , for λ, λ0 ∈ ΛM := MF ∨ ∩ P/MQ . cG = det(Cartan matrix)= order of the centre of G .
gn = C2 example: “dual grid” Λ4 = 4F ∨ ∩ Dots ∈ 4F ∨ ∪
P 4Q ,
P 4Q .
|Λ4 | = |F4 | = 9.
J. Phys. A: Math. Theor. 42 (2009) 385208
J Hrivn´ak and J Patera
4α2
∨ r0,4
η 4F ∨ α2 = α2∨
ω2 = ω2∨
ξ = ω1∨
1 2 ω2
r0∨
F∨
ω1 α1
r2
α1∨
4α1
r1
Figure 3. The cosets representants of P /4Q of C2 ; the cosets representants are shown as 32 black ∨
(Finite) orbit function transforms: Consider data with support F , or F˜. The Weyl-orbit functions provide useful expansion bases for the analysis of functions on F , or F˜. Digitized data, the values on the grid FM in F , or F˜M in F˜. M ∈ N will determine the resolution ∼ 1/M of the digital data of interest. Interpolate, by requiring f (x) =
X λ∈ΛM
Fλ Cλ (x) , ∀ x ∈ FM .
Discrete orthogonality ⇒ Discrete C -Transform Fλ =
X 1 f (x) Cλ (x) . n cG M |W | x∈FM
Cubature = quadrature in higher dimensions Cubature formula: a weighted sum of function evaluations used to approximate a multivariate integral. Z Ω
f (x) w (x) dx ≈
N X
wj Lj [f ] ,
j=1
Lj [f ] = f (xj ) , e.g . Exact (≈ → =) cubature formulas are possible if one restricts to a certain class of functions. Such exact cubature formulas are found using the orbit functions (H. Li & Y. Xu 2010, Moody-Patera 2011).
Define domain Ω = { (X1 (x), . . . , Xn (x)) : x ∈ F˜ } ⊂ Cn . Put m-degree of Xi as mi∨ , then m-deg(Sρ )= h − 1. For any function f on Ω, define f˜(x) := f ( χω1 (x), . . . , χωn (x) ) . Cubature formula: Polynomial f ∈ C[X1 , . . . , Xn ] with m-degree ≤ 2M + 1, have n X Z 1 2π 1/2 f K dX1 · · · dXn = f˜(x) K˜ (x) . cG M + h Ω x∈F˜M+h
Here K (x) := |Sρ (x)|2 and the Coxeter number h =
Pn
j=0
mj .
CUBATURE FORMULAE FOR SIMPLE LIE GROUPS Example of domain Ω for cubature formula (M = 8 for G2 ):
6
4
2
5
10
-2
Figure 3. The region Ω along with the 10 regular EFOs of the example.
2
Wess-Zumino-Novikov-Witten conformal field theories: AFFINE MODULAR DATA
Objects related to the modular data of Wess-Zumino-Novikov-Witten (WZNW) models bear a striking resemblance to the discretized Weyl-orbit functions. This modular data is associated with the affine Kac-Moody algebras of untwisted type, at a fixed level, and so is also called affine modular data.
As for any RCFT, the WZNW 1-loop partition function is invariant under the modular group SL(2; Z), with generators S and T . genus-1 conformal blocks ∼ = characters of the untwisted affine Kac-Moody algebra at fixed level, sometimes denoted gn,k . The affine characters form a finite-dimensional representation of the modular group (Kac-Peterson 1984).
Put
P+M = {
P
M and P++ ={
i∈Iˆ λi ωi
P
| λ i ∈ N0 ,
P
| λi ∈ N,
P
i∈Iˆ λi ωi
∨ i∈Iˆ λi mi
= M},
∨ i∈Iˆ λi mi
= M} .
P P Similarly, P+∨M = { i∈Iˆ λi ωi∨ | λi ∈ N0 , i∈Iˆ λi mi = M}, P P ∨M and P++ = { i∈Iˆ λi ωi∨ | λi ∈ N, i∈Iˆ λi mi = M} . ∨M Grid FM := P+∨ M /M. Grid interior F˜M := P++ /M.
M ˜ M = P++ Dual grid ΛM = P+M ; interior Λ .
Note that m0 + m1 + . . . + mn = m0∨ + m1∨ + . . . + mn∨ = h , the Coxeter number.
mj0 αj∨ , mj0 are known as co-marks. P Dual Coxeter number h∨ := 1 + j∈I mj0 .
Highest root of gn : ξ =
P
j∈I
Let ∆+ denote the set of positive roots of gn and put M 0 := k + h∨ . 0
M The affine characters can be labeled by weights in P++ , and the
Kac-Peterson modular S matrix has the form Sλ0 ,µ0 = RM 0
X w ∈W
0
M for λ0 , µ0 ∈ P++ .
0
0
(det w ) e −2πihw λ ,µ i/M
0
,
0
M But P++ = P+k + ρ; rewrite as
−2πihw (λ + ρ), µ + ρi k + h∨ w ∈W µ , = Rk+h∨ Sλ+ρ k + h∨
Sλ,µ = Rk+h∨
X
(det w ) exp
1
r
with λ, µ ∈ P+k , and Rk+h∨ = i k∆+ k |P/Q ∨ |− 2 (k + h∨ )− 2 . Modular S-matrix is unitary and symmetric, with symmetric affine Weyl symmetry: Sw .λ,µ = Sλ,w .µ = (det w ) Sλ,µ , ∀ w ∈ W aff ; shifted action w .λ := w (λ + ρ) − ρ.
Modified multiplication The affine Weyl symmetry → modified (truncated) multiplication of characters. Verlinde formula: X Sλ,σ Sµ,σ = Sρ,σ Sρ,σ k
(k)
ν Nλ,µ
ν∈P+
Sν,σ Sρ,σ
.
The ratios are (discretized) Weyl characters of integrable, highest-weight representations of gn :
Sλ,σ Sρ,σ
= χλ (σ) .
Products of characters decompose as tensor products of representations do: χλ (σ) χµ (σ) =
X
φ Tλ,µ χφ (σ) ,
φ∈P+ φ where Tλ,µ is the tensor product coefficient. Using the affine Weyl
symmetry to compare this with the Verlinde formula → (Kac-W) (k)
ν Nλ,µ =
X
w .ν (det w ) Tλ,µ .
w ∈W aff
The affine Weyl symmetry valid when the character is discretized results in a modified multiplication, so that the tensor product coefficients are modified, truncated to the fusion coefficients. The modified multiplications of discretized Weyl-orbit functions will work essentially the same way.
Galois symmetry
Galois symmetry is an important property of all RCFTs (Coste-Gannon 1994). The Galois symmetry of WZNW models is the motivation for a Galois symmetry of discretized orbit functions. Kac-Peterson Sλ,µ is a linear combination of roots of unity exp{−2πihw (λ + ρ), µ + ρi/(k + h∨ )} =: ϕ with rational coefficients. Let N ∈ N denote the minimum such that ϕN = 1 for all such ϕ.
Suppose gcd(`, N) = 1 for ` ∈ N. Define the Galois transformation t` by t` (ϕ) := ϕ` , and extend it linearly to sums of such terms. This transformation swaps one primitive root of unity for another. Equivalently, it maps λ + ρ to `(λ + ρ), possibly outside the dominant sector. Transforming back is possible, however, using the affine Weyl group: `(λ + ρ) = w` [λ] (t` [λ] + ρ) , w` [λ] ∈ W aff . The Galois symmetry of the modular S-matrix then follows: t` ( Sλ,µ ) = ` [λ] St` [λ],µ = ` [µ] Sλ,t` [µ]
ORBIT FUNCTIONS
The affine Weyl symmetry of the orbit functions can be summarized as follows. With a ∈ P, we have Cλ (wa) = Cλ (a) , w ∈ W aff ; caff ; Cwˆ λ (a) = Cλ (a) , w ˆ ∈W Sλ˜ (wa) = (det w ) Sλ (a) , w ∈ W aff ; caff ; Swˆ λ (a) = (det w ˆ ) Sλ (a) , w ˆ ∈W ˜ ∈ P ∨ /M. provided λ, λ
(1)
Modified multiplication of discretized Weyl-orbit functions
For any a ∈ PR , write Cλ (a) Cµ (a) =
X ν∈P+
Cλ (a) Sµ˜ (a) =
hC |CC iνλ,µ Cν (a) ,
X ν˜∈P++
Sλ˜ (a) Sµ˜ (a) =
X ν∈P+
˜ hS|CSiνλ, µ ˜ Sν˜ (a) ,
hC |SSiνλ, Cν (a) , ˜ µ ˜
˜ µ for all λ, µ, ν ∈ P+ , and all λ, ˜, ν˜ ∈ P++ .
M ˜ µ Similarly, if λ, µ, ν ∈ P+M , and all λ, ˜, ν˜ ∈ P++ , then
Cλ (a) Cµ (a) =
X M M ν∈P+
Cλ (a) Sµ˜ (a) =
X M M ν˜∈P++
Sλ˜ (a) Sµ˜ (a) =
X M M ν∈P+
for any a ∈ FM ⊃ F˜M .
hC |CC iνλ,µ Cν (a) , ˜ hS|CSiνλ, µ ˜ Sν˜ (a)
hC |SSiνλ, Cν (a) , ˜ µ ˜
We find M
M
hC |CC iνλ,µ =
˜ hS|CSiνλ, µ ˜ =
M
X caff w ˆ ∈W
X caff w ˆ ∈W
hC |SSiνλ, = ˜ µ ˜
ˆν hC |CC iw λ,µ ,
w ˆ ν˜ (det w ) hS|CSiλ, µ ˜ ,
X caff w ˆ ∈W
ˆν hC |SSiw . ˜ µ λ, ˜
Galois symmetry of Weyl-orbit functions
Let N denote the minimum positive integer such that
e 2πihλ,ai
N
= e 2πihNλ,ai = 1 ,
for all λ ∈ ΛM , a ∈ FM . Suppose gcd(`, N) = 1 for ` ∈ N. Define the Galois transformation t` by t` e 2πihλ,ai := e 2πi`hλ,ai , and extend it linearly to sums of such terms. This transformation swaps one primitive root of unity for another.
Applying the Galois transformation to the orbit functions, one gets t` Cλ (a) = C`λ (a) = Cλ (`a) , t` Sλ˜ (˜ a) = S`λ˜ (˜ a) = Sλ˜ (`˜ a) .
(2)
M ˜ ∈ P++ Now suppose that λ ∈ P+M , λ , a ∈ FM , and a˜ ∈ F˜M . Multiples of
these weights by a factor of ` will not also, in general, be part of the same sets.
They can all, however, be moved there by appropriate elements of the relevant affine Weyl group: w ˆ` [λ] ( ` λ ) =: t` [λ] ∈ P+M M ˜ `λ ˜ ˜ ∈ P++ w ˆ` [λ] =: t` [λ]
,
caff ; w ˆ` [λ] ∈ W
,
caff ; ˜ ∈ W w ˆ` [λ]
w` [a] ( ` a ) =: t` [a] ∈ FM
,
w` [a] ∈ W aff ;
w` [˜ a] ( ` a˜ ) =: t` [˜ a] ∈ F˜M
,
w` [˜ a] ∈ W aff .
(3)
Using the affine Weyl symmetries ⇒ the Galois symmetry of the orbit functions: t` Cλ (a) = Ct` [λ] (a) = Cλ (t` [a]) , t` Sλ˜ (˜ a) = ˆ` [λ] St` [λ] a) = ` [˜ a] Sλ˜ (t` [˜ a]) . ˜ (˜
(4)
Here we have defined the signs caff ; ˜ := det w ˜ , w ˜ ∈ W ˆ` [λ] ˆ` [λ] ˆ` [λ] ` [˜ a] := det w` [˜ a] , w` [˜ a] ∈ W aff .
(5)
Galois symmetry also produces relations involving the decomposition coefficients discussed above. For example, because the Galois transformation exchanges one root of unity for another, and because the coefficients t`
M
are rational, we have hC | SSiνλ, ˜ µ ˜
X Sλ˜ (a) t` Sµ˜ (a) = M ν∈P+
hC |SSiνλ, ˜ µ M ˜
t` Cν (a)
.
⇒ ˜ S ˜ (a) ˆ` [˜ ˆ` [λ] µ] St` [µ] ˜ (a) = t` [λ]
X M M ν∈P+
hC |SSiνλ, Ct` [ν] (a) . ˜ µ ˜
so that X
˜ ˆ` [˜ ˆ` [λ] µ]
hC |SSiνt [λ],t ˜
M
`
M ν∈P+
=
X M M ν∈P+
˜ ` [µ]
Cν (a)
hC |SSiνλ, Ct` [ν] (a) . ˜ µ ˜
Orthogonality relations for Cν (a) ⇒ ˜ ˆ` [˜ ˆ` [λ] µ] M hC |SSiνt [λ],t ˜ `
˜ ` [µ]
=
t [ν]
M
` hC |SSiλ, ˜ µ ˜
Similar relations for the other decomposition coefficients.
.
CONCLUSION (Future Work?)
(Generalized?) Orbit Functions
X ? ?
⇔
Conformal Field Theory
⇐ Modified multiplication, Galois symmetries (Hrivn´ak-W) ⇐ ⇒
Fusion generators and bases (Gannon-Walton 1999) Pasquier algebras, NIM-reps, boundary conditions, . . . ?
Chevalley groups
⇒
?
