Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA. ABSTRACT. We consider Laplace type partial differential ...
Heat Kernel on Homogeneous Bundles over Symmetric Spaces I VAN G. AVRAMIDI Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM 87801, USA
ABSTRACT We consider Laplace type partial differential operators acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. The obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytic continuation. This result is used to evaluate the nonperturbative low-energy effective action in quantum general relativity.
and the determinant
Here µ is a renormalization parameter and λ is a regularization parameter which supposed to be large and negative to make the operator L − λ positive.
Let C A BC be defined by C i ab = E i ab , C a ib = −C a bi = Da ib , C i kl = F i kl . We use the notation that capital Latin indices run from 1 to N = n + p.
∇µ Rαβγδ = 0 ,
2 Heat Kernel 2.1 Twisted Spin-Tensor Bundles Let (M, g) be a smooth compact (complete simply connected orientable and spin) Riemannian manifold of dimension n without boundary with a metric g. Let T be a spin-tensor bundle with fiber Λ realizing a representation Σ of the spin group Spin(n). Let GY M be a compact Lie (gauge) group. We consider a representation X : GY M → End (W ) of the Lie algebra GY M of the gauge group GY M in a vector space W and the associated vector bundle W through this representation with the structure group GY M whose typical fiber is W . Then for any spin-tensor bundle T we define the twisted spin-tensor bundle V via the twisted product of the bundles W and T with the fiber V = Λ ⊗ W .
2.2 Connection and Curvature Let ∇ be a connection on the vector bundle V defined with the help of the spin connection on T and a gauge connection on the bundle W. The curvature of the total connection on the bundle V is Rcd =
1 ab R cd Σab + X(Fcd ) , 2
A Laplace type operator L : C ∞ (V) → C ∞ (V) is a partial differential operator of the form L = −∆ + Q , µν
where ∆ = g ∇µ ∇ν is the Laplacian, and Q is a smooth endomorphism of the bundle V.
[ξA , ξB ] = C
ζ(s, λ) =
µ2s Γ(s)
Z∞ 0
dt ts−1 etλ Tr L2 exp(−tL),
C
AB ξC
Rabcd = βik E i ab E k cd , where E i ab is a collection of p anti-symmetric matrices and β = (βik ) is a symmetric nondegenerate p × p matrix. We denote the inverse of the matrix βik by β ik . We adopt the notation that Latin indices from the beginning of the alphabet run from 1 to n, and Latin indices from the middle of the alphabet run from 1 to p.
Next, we define the traceless n × n matrices Di = (Da ib ), by
Bab 0 0 0 Lie derivatives along Killing vectors ξA are defined by
Let B = (BAB ) be a matrix defined by (BAB ) =
. The twisted
where F j ik are the structure constants, and the p × p matrices Fi , by (Fi )j k = F j ik , which generate the adjoint representation of the holonomy algebra.
3.3 Homogeneous Vector Bundles Let ha b be the projection to the subspace Tx Ms of the tangent space Tx M and q a b = δ a b − ha b be the projection tensor to the flat subspace Rn0 . We decompose the gauge curvature according to Fab = Bab + Eab , where Bab = Fcd q c a q d b , Eab = Fcd hc a hd b . Then, one can show [4,5] that Bab takes values in an Abelian ideal of the gauge algebra GY M and Eab takes values in a representation of the holonomy algebra. The holonomy algebra can be embedded in the orthogonal algebra SO(n) via a representation Y : SO(n) → End (W ) of the orthogonal algebra SO(n) in W with generators Yab . This defines the product representation G = Σ ⊗ Y : SO(n) → End (V ) in the vector space V = Λ ⊗ W with the generators Gab = Σab ⊗ IY + IΣ ⊗ Yab . Then the matrices Ri = − 21 Da ib Gb a form a representation R : H → End (V ) of the holonomy algebra in V and the total curvature of the twisted spin-tensor bundle V is Rab
= −E i ab Ri + X(Bab ) =
1 cd R ab Gcd + X(Bab ) . 2
sinh (tX(B)) tX(B)
−1/2
The one-loop effective action in quantum general relativity (with the cosmological constant) has the form 1 ′ ΓGR (1) = − ζGR (0) , 2
[LA , LB ] =
C C AB LC + BAB .
where ζGR (s) is the gravitational zeta-function ζGR (s) =
4.3 Laplacian
µ2s Γ(s)
Z∞
dt ts−1 etλ ΘGR (t) ,
0
δab 0 and γ AB its 0 βik inverse. Finally, we show that the Laplacian can be expressed in terms of the Casimir operators as follows
Let γ = (γAB ) be a matrix defined by (γAB ) =
L2 − R2 ,
where L2 = γ AB LA LB , R2 = β ij Ri Rj = 41 Rabcd Gab Gcd .
and ΘGR (t) is the total heat trace (including the ghost contribution). By using the above results for the heat trace of Laplace type operators we obtain Z 1 1 ΘGR (t) = (4πt)−n/2 d vol exp R + RH t 8 6 M Z dω 1 × β 1/2 exp − hω, βωi ΨGR (t, ω) 4t (4πt)p/2 Rn reg
1 − 21 sinh [ F (ω)/2] 2 sinh [ D(ω)/2] × det H . det T M F (ω)/2 D(ω)/2
5 Heat Trace
[Di , Dj ] = F k ij Dk ,
det T M
6 Effective Action in Quantum Gravity
ext, we prove [4,5] that they form the algebra
Da ib = −βik E k cb δ ca , which form the holonomy algebra, H,
These integrals need to be regularized (for details, see [4,5]).
∆ =
3.2 Holonomy Group
tr W
× tr Λ exp −t R2 + Q exp [R(ω)] .
.
1 LA = ∇ξA + ξA a ;b Gb a . 2
A generic symmetric space has the structure M = M0 × Ms , where M0 = Rn0 , Ms = M+ × M− , and M+ and M− are compact and noncompact symmetric spaces respectively. The curvature tensor of a symmetric space can be presented in the form
1 − β ij F k il F l jk , 4
4.2 Algebra of Twisted Lie Derivatives
3.1 Symmetric Spaces
2.4 Heat Kernel Trace and Zeta Function One can show that for t > 0 the operators exp(−tL) form a semi-group of bounded trace-class operators on L2 (V) with a well defined L2 -trace Tr L2 exp(−tL), which enables one to define the zeta-function,
∇µ Q = 0 ,
which corresponds to homogeneous bundles over symmetric spaces.
where Fcd is the curvature of the gauge connection.
2.3 Laplace Type Operators
∇µ Rαβ = 0 ,
=
Ψ(t, ω) =
Then, one can show [4,5] that there is a set of Killing vectors ξA forming a Lie algebra G (which is a subalgebra of the total isometry algebra)
We restrict ourselves in this work to parallel curvatures
The heat kernel is one of the most powerful tools in mathematical physics and geometric analysis. We develop powerful algebraic methods to study the heat kernel for operators and manifolds with high level of symmetry, in particular, homogeneous spaces. We applied such algebraic methods first for the Laplacian with a parallel Abelian gauge connection in the Euclidean space in [1], and then generalized to scalar Laplacians on Riemannian manifolds with parallel curvature in [2,3]. Finally, in [4,5] we generalized the method to Laplacians acting on sections of arbitrary homogeneous vector bundles over symmetric spaces and applied the results to the calculation of the low-energy effective action in quantum gravity and gauge theories in [6].
RH
4.1 Killing Vectors
3 Homogeneous Vector Bundles 1 Introduction
where |β| = det βij , hω, βωi = βij ω i ω j , D(ω) = ω i Di , F (ω) = ω i Fi , R(ω) = ω i Ri , B = (B a b ),
4 Twisted Lie Derivatives
∂ ζ ′ (0, λ) ≡ ζ(s, λ) = − log Det (L − λ) . ∂s s=0
5.1 Heat Semigroup
Here
The key point in our method is the following integral representation of the heat semigroup exp(t∆) −1/2 sinh (tB) tB Z 1/2 sinh [C(k)/2] 1 2 dk |γ|1/2 det G × exp −tR + RG t 6 C(k)/2
exp(t∆) = (4πt)−N/2 det T M
RN reg
1 × exp − hk, γtB coth (tB)ki exp[L(k)] , 4t where |γ| = det γAB , hu, γvi = γAB uA v B , C(k) = CA k A , L(k) = LA k A , and 1 RG = − γ AB C C AD C D BC . 4
5.2 Heat Trace By using the above integral representation one can obtain finally the heat trace [4,5] Z 1 1 R + RH t Tr L2 exp(−tL) = d vol (4πt)−n/2 exp 8 6 M Z dω 1 × |β|1/2 exp − hω, βωi Ψ(t, ω) 4t (4πt)p/2 Rn reg
1 − 12 sinh [ D(ω)/2] sinh [ F (ω)/2] 2 det T M × det H F (ω)/2 D(ω)/2
ΨGR (t, ω) =
Ψ2 (t, ω) − 2Ψ1 (t, ω) ,
Ψ2 (t, ω) =
exp [−t(R − 2Λ)] tr T2 exp (tM2 ) exp [2D(ω)] ,
Ψ1 (t, ω) =
tr T M exp (tM1 ) exp [ D(ω)] ,
ab
=
D(a i(d δ b) c) ω i ,
(M1 )b a
=
2Ra b ,
(M2 )cd ab
=
4δ (a (c Rb) d) − Rcd g ab −
[D(ω)]cd
R 2 gcd Rab + gcd g ab , n−2 (n − 2)
and tr T2 denotes the trace in the vector space of symmetric 2-tensors.
References [1] I. G. Avramidi, A new algebraic approach for calculating the heat kernel in gauge theories, Phys. Lett. B 305 (1993) 27-34 [2] I. G. Avramidi, The heat kernel on symmetric spaces via integrating over the group of isometries, Phys. Lett. B 336 (1994) 171-177 [3] I. G. Avramidi, A new algebraic approach for calculating the heat kernel in quantum gravity, J. Math. Phys. 37 (1996) 374-394 [4] I. G. Avramidi, Heat kernel on homogeneous bundles, Int. J. Geom. Meth. Mod. Phys. 5 (2008) 1-23 [5] I. G. Avramidi, Heat kernel on homogeneous hundles over symmetric spaces, Comm. Math. Phys. 288 (2009) 963-1006 [6] I. G. Avramidi, Non-perturbative effective action in gauge theories and quantum gravity, arXiv:0903.1295 [hep-th]
