Heat Kernel on Homogeneous Bundles over Symmetric Spaces

0.35 MB·English

by  Ivan G. Avramidi

7 N M T (February 2, 2008)
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Heat Kernel
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on Homogeneous Bundles
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over Symmetric Spaces
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1 Ivan G. Avramidi
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0 DepartmentofMathematics
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h New Mexico Instituteof MiningandTechnology
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a Socorro,NM 87801,USA
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E-mail: [email protected]
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WeconsiderLaplaciansactingonsectionsofhomogeneousvectorbun-
r
a dlesoversymmetricspaces. Byusinganintegralrepresentationoftheheat
semi-groupwefindaformalsolutionfortheheatkerneldiagonalthatgives
a generating function for the whole sequence of heat invariants. We show
explicitly that the obtained result correctly reproduces the first non-trivial
heat kernel coefficient as well as the exact heat kernel diagonals on two-
dimensional sphere S2 and the hyperbolic plane H2. We argue that the
obtainedformal solutioncorrectly reproduces theexact heat kernel diago-
nal afterasuitableregularizationand analyticalcontinuation.
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 1
1 Introduction
The heat kernel is one of the most powerful tools in mathematical physics and
geometric analysis (see, for example the books [24, 17, 26, 13, 27] and reviews
[2, 18, 12, 14, 31]). The short-time asymptotic expansion of the trace of the heat
kernel determines the spectral asymptotics of the differential operator. The co-
efficients of this asymptotic expansion, called the heat invariants, are extensively
usedingeometricanalysis,inparticular,inspectralgeometryandindextheorems
proofs[24, 17].
There has been a tremendous progress in the explicit calculation of spectral
asymptotics in the last thirty years [23, 2, 3, 4, 5, 30, 33]. It seems that further
progress in the study of spectral asymptotics can be only achieved by restricting
oneself to operators and manifolds with high level of symmetry, in particular,
homogeneous spaces, which enables one to employ powerful algebraic methods.
In some very special particular cases, such as group manifolds, spheres, rank-
onesymmetricspacesandsplit-ranksymmetricspaces,itispossibletodetermine
the spectrum of the Laplacian exactly and to obtain closed formulas for the heat
kernel in terms of the root vectors and their multiplicities [1, 18, 19, 20, 26, 22].
The complexity of the method crucially depends on the global structure of the
symmetric space, most importantly its rank. Most of the results for symmetric
spaces areobtainedforrank-onesymmetricspaces only[18].
It is well known that heat invariants are determined essentially by local ge-
ometry. They are polynomial invariants in the curvature with universal constants
that do not depend on the global properties of themanifold [24]. It is this univer-
sal structure that we are interested in this paper. Our goal is to compute the heat
kernel asymptotics of the Laplacian acting on homogeneous vector bundles over
symmetric spaces. Related problems in a more general context are discussed in
[7, 9, 11].
2 Geometry of Symmetric Spaces
2.1 Twisted Spin-Tensor Bundles
In this section we introduce the basic concepts and fix notation. Let (M,g) be an
n-dimensionalRiemannianmanifoldwithoutboundary. Weassumethatitiscom-
plete simply connected orientable and spin. We denote the local coordinates on
M by xµ,withGreekindicesrunningover1,...,n. Lete µ bealocalorthonormal
a
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 2
framedefining abasisforthetangentspaceT M so that
x
gµν = δabe µe ν, (2.1)
a b
We denote the frame indices by low case Latin indices from the beginning of the
alphabet, which also run over 1,...,n. The frame indices are raised and lowered
by the metric δ . Let ea be the matrix inverse to e µ, defining the dual basis in
ab µ a
thecotangent spaceT M, so that,
x∗
g = δ ea eb . (2.2)
µν ab µ ν
TheRiemannianvolumeelementisdefined as usualby
dvol = dx g1/2, (2.3)
| |
where
g = detg = (dete µ)2. (2.4)
µν a
| |
Thespinconnectionωab isdefined in termsoftheorthonormalframeby
µ
ωab = eaµeb = ea ebµ
µ µ;ν µ;ν
−
= eaν∂ eb ebν∂ ea +e eaνebλ∂ ec , (2.5)
[µ ν] [µ ν] cµ [λ ν]
−
where the semicolon denotes the usual Riemannian covariant derivative with the
Levi-Civitaconnection. Thecurvatureofthespinconnectionis
Ra = ∂ ωa ∂ ωa +ωa ωc ωa ωc . (2.6)
bµν µ bν ν bµ cµ bν cν bµ
− −
TheRicci tensorand thescalarcurvatureare defined by
R = e µeb Ra , R = gµνR = e µe νRab . (2.7)
αν a α bµν µν a b µν
Let be a spin-tensor bundle realizing a representation Σ of the spin group
T
Spin(n), the double covering of the group SO(n), with the fiber Λ. Let Σ be
ab
thegenerators of theorthogonalalgebra (n), theLie algebra of theorthogonal
SO
groupSO(n), satisfyingthefollowingcommutationrelations
[Σ ,Σ ] = δ Σ +δ Σ +δ Σ δ Σ . (2.8)
ab cd ac bd bc ad ad bc bd ac
− −
The spin connection induces a connection on the bundle defining the co-
T
variantderivativeofsmoothsectionsϕ ofthebundle by
T
1
ϕ = ∂ + ωab Σ ϕ. (2.9)
µ µ µ ab
∇ 2 !
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 3
The commutator of covariant derivatives defines the curvature of this connection
via
1
[ , ]ϕ = Rab Σ ϕ. (2.10)
µ ν µν ab
∇ ∇ 2
Asusual,theorthonormalframe,ea ande µ,willbeusedtotransformtheco-
µ a
ordinate(Greek)indicestotheorthonormal(Latin)indices. Thecovariantderiva-
tive along the frame vectors is defined by = e µ . For example, with our
a a µ
∇ ∇
notation, T = e µe νe αe β T .
a b cd a b c d µ ν αβ
∇ ∇ ∇ ∇
The metric δ induces a positive definite fiber metric on tensor bundles. For
ab
Dirac spinors, the fiber metric is defined as follows. First, one defines the Dirac
matrices, γ , as generators of the Clifford algebra, (represented by 2[n/2] 2[n/2]
a
×
complexmatrices),
γ γ +γ γ = 2δ I , (2.11)
a b b a ab S
where I is the identity matrix in the spinor representation. Then one defines the
S
anti-symmetrizedproductsofDiracmatrices
γ = γ γ . (2.12)
a1...ak [a1··· ak]
Thenthematrices
1
Σ = γ (2.13)
ab ab
2
are the generators of the orthogonal algebra (n) in the spinor representation.
SO
TheHermitianconjugationofDiracmatrices defines aHermitianmatrixβ 1 by
γ = βγ β 1, (2.14)
a† a −
whichdefinesaHermitianinnerproductψ¯ϕ = ψ βϕinthevectorspaceofspinors.
†
Wealsofind thefollowingimportantrelation
Rab γ γcd = 2Rab I = 2R I , (2.15)
cd ab ab S S
− −
whereR isthescalarcurvature.
Inthepresentpaperwewillfurtherassumethat Misalocallysymmetricspace
withaRiemannianmetricwiththeparallelcurvature
R = 0, (2.16)
µ αβγδ
∇
1TheDiracmatricesγ andthespinormetricβshouldnotbeconfusedwiththematricesγ
ab AB
andβ definedbelow.
ij
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 4
whichmeans,inparticular,thatthecurvaturesatisfiestheintegrabilityconstraints
Rfg Re Rfg Re +Rfg Re Rfg Re = 0. (2.17)
ea bcd eb acd ec dab ed cab
− −
Let G be a compact Lie group (called a gauge group). It naturally defines
YM
the principal fiber bundleoverthe manifold M with the structure groupG . We
YM
consider a representation of the structure group G and the associated vector
YM
bundlethroughthisrepresentationwiththesamestructuregroupG whosetyp-
YM
ical fiber is a k-dimensional vector space W. Then for any spin-tensor bundle
T
wedefinethetwistedspin-tensorbundle viathetwistedproductofthebundles
V
and . The fiber of the bundle is V = Λ W so that the sections of the
W T V ⊗
bundle are represented locallybyk-tuplesofspin-tensors.
V
Let beaconnectiononeformonthebundle (calledYang-Millsorgauge
A W
connection) taking values in the Lie algebra of the gauge group G . Then
YM YM
G
thetotalconnection onthebundle isdefined by
V
1
ϕ = ∂ + ωab Σ I +I ϕ, (2.18)
µ µ µ ab W Λ µ
∇ 2 ⊗ ⊗A !
and thetotalcurvatureΩofthebundle isdefined by
V
[ , ]ϕ = Ω ϕ, (2.19)
µ ν µν
∇ ∇
where
1
Ω = Rab Σ + , (2.20)
µν µν ab µν
2 F
and
= ∂ ∂ +[ , ] (2.21)
µν µ ν ν µ µ µ
F A − A A A
isthecurvatureoftheYang-Millsconnection.
Wealsoconsiderthebundleofendomorphismsofthebundle . Thecovariant
V
derivativeofsectionsofthisbundleisdefined by
1
X = ∂ + ωab Σ X +[ ,X], (2.22)
µ µ µ ab µ
∇ 2 ! A
and thecommutatorofcovariantderivativesis equalto
1
[ , ]X = Rab Σ X +[ ,X]. (2.23)
µ ν µν ab µν
∇ ∇ 2 F
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 5
In the following we will consider homogeneous vector bundles with parallel
bundlecurvature
= 0, (2.24)
µ αβ
∇ F
whichmeans thatthecurvaturesatisfies theintegrabilityconstraints
[ , ] Rf Rf = 0. (2.25)
cd ab acd fb bcd af
F F − F − F
2.2 Normal Coordinates
Let x be a fixed point in M and be a sufficiently small coordinate patch con-
′
U
taining the point x . Then every point x in can be connected with the point x
′ ′
U
by a unique geodesic. We extend the local orthonormal frame e µ(x ) at the point
a ′
x to alocal orthonormalframee µ(x)at thepoint xbyparallel transport
′ a
eaµ(x) = gµν (x,x′)eaν′(x′), (2.26)
′
eaµ(x) = gµν′(x,x′)eaν (x′), (2.27)
′
wheregµ (x,x )istheoperatorofparallel transportofvectorsalong thegeodesic
ν ′
′
from the point x to the point x. Of course, the frame e µ depends on the fixed
′ a
point x as a parameter. Here and everywherebelow the coordinateindices of the
′
tangentspaceat thepoint x are denotedbyprimedGreek letters. Theyare raised
′
and lowered by the metric tensor g (x ) at the point x . The derivatives with
µν ′ ′
′ ′
respect to x willbedenotedby primedGreek indicesas well.
′
Theparametersofthegeodesicconnectingthepoints xand x ,namelytheunit
′
tangent vectorat the point x and the length of the geodesic, (or, equivalently,the
′
tangent vector at the point x with the norm equal to the length of the geodesic),
′
provide normal coordinate system for . Let d(x,x ) be the geodesic distance
′
U
between thepoints xand x andσ(x,x )beatwo-pointfunctiondefined by
′ ′
1
σ(x,x ) = [d(x,x )]2. (2.28)
′ ′
2
Thenthederivativesσ (x,x )andσ (x,x )arethetangentvectorstothegeodesic
;µ ′ ;ν ′
′
connecting the points x and x at the points x and x respectively pointing in op-
′ ′
positedirections;oneisobtainedfrom anotherbyparallel transport
σ = g ν′σ . (2.29)
;µ µ ;ν
− ′
Hereand everywherebelowthesemicolondenotesthecovariantderivative.
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 6
Theoperatorofparalleltransportsatisfies theequation
σ;µ gα = 0, (2.30)
µ β
∇ ′
withtheinitialconditions
gα = δα. (2.31)
β′(cid:12)x=x′ β
(cid:12)
It can beexpressedinterms oftheloc(cid:12)al parallelframe
(cid:12)
gµ (x,x ) = e µ(x)ea (x ), (2.32)
ν ′ a ν ′
′ ′
gµν′(x,x′) = eaµ(x)eaν′(x′). (2.33)
Now,letus definethequantities
ya = ea σ;µ = ea σ;µ′, (2.34)
µ µ
− ′
sothat
σ;µ = e µya and σ;µ′ = e µ′ya. (2.35)
a a
−
Noticethat ya = 0 at x = x . Further, wehave
′
∂ya
= eaµ′σ , (2.36)
∂xν − ;νµ′
sothat theJacobian ofthechangeofvariablesis
∂ya
det = g 1/2(x )det[ σ (x,x )]. (2.37)
∂xν! | |− ′ − ;νµ′ ′
Thegeometricparametersya arenothingbutthenormalcoordinates. Byusing
theVan Vleck-Morettedeterminantdefined by2
∆(x,x ) = g 1/2(x )g 1/2(x)det[ σ (x,x )], (2.38)
′ − ′ − ;νµ ′
| | | | − ′
wecan writetheRiemannian volumeelementintheform
dvol = dy ∆ 1(x,x ). (2.39)
− ′
Let (x,x ) be the operator of parallel transport of sections of the bundle
′
P V
fromthepoint x to thepoint x. It satisfies theequation
′
σ;µ = 0, (2.40)
µ
∇ P
2DonotconfuseitwiththeLaplacian∆definedbelow.
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 7
withtheinitialcondition
= I . (2.41)
V
P(cid:12)x=x′
(cid:12)
Anyspin-tensorϕcan benowex(cid:12)panded inthecovariantTaylorseries
(cid:12)
∞ 1
ϕ(x) = (x,x ) ϕ (x )yc1 yck . (2.42)
P ′ k! ∇(c1 ···∇ck) ′ ···
Xk=0 (cid:2) (cid:3)
Therefrom it is clear, in particular, that the frame components of a parallel spin-
tensorare simplyconstant.
InsymmetricspacesonecancomputetheVanVleck-Morettedeterminantex-
plicitlyinterms ofthecurvature. Let K bean n matrixwiththeentries
×
Ka = Ra ycyd. (2.43)
b cbd
Then[5, 2, 13]
a
∂ya √K
= eb , (2.44)
∂xν sin √K b ν
and,therefore,  
√K
∆(x,x ) = det . (2.45)
′ TM
sin √K
Thus, the Riemannian volume element in symmetric spaces takes the following
form
sin √K
dvol = dy det . (2.46)
TM
 √K 
The matrix (sin √K)/√K determines the orthonormal frame in normal co-
ordinates, and the square of this matrix determines the metric tensor in normal
coordinates,
sin2 √K
ds2 = dyadyb. (2.47)
 K 
Let usdefine an endo-morphismvalued1ab-form ˜ by theequation
a
A
= ˜ ea σ;µ′ . (2.48)
ν a µ ν
∇ P PA ′
Then for bundles with parallel curvature over symmetric spaces one can find it
explicitly[5,2, 13]
I cos √K b
˜ = yc − . (2.49)
a bc a
A −F  K 
 
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 8
Thisobjectdeterminesthegaugeconnectionin normalcoordinates,
I cos √K b
= yc − dya. (2.50)
bc a
A −F  K 
 
This means that all connections on a homogeneous bundle are essentially the
same. In particular, the spin connection one-form in normal coordinates has the
form
I cos √K c
ωa = Ra yd − dye. (2.51)
b bcd e
−  K 
Remarks. Two remarks are in order here. First, strictly speaking, normal
coordinates can be only defined locally, in geodesic balls of radius less than the
injectivity radius of the manifold. However, for symmetric spaces normal coor-
dinates cover the whole manifold except for a set of measure zero where they
becomesingular[18]. Thisset is preciselytheset ofpointsconjugateto thefixed
point x (where ∆ 1(x,x ) = 0) and of points that can be connected to the point
′ − ′
x by multiplegeodesics. In any case, this set is a set of measure zero and, as we
′
will showbelow,it can bedealt with by someregularization technique. Thus, we
will use the normal coordinates defined above for the whole manifold. Second,
for compact manifolds (or for manifolds with compact submanifolds) the range
of some normal coordinates is also compact, so that if one allows them to range
over the whole real line R, then the corresponding compact submanifolds will be
coveredinfinitelymanytimes.
2.3 Curvature Group of a Symmetric Space
Weassumedthatthemanifold M islocallysymmetric. Sincewealsoassumethat
it is simply connected and complete, it is a globally symmetric space (or simply
symmetric space) [32]. A symmetric space is said to be compact, non-compact
or Euclidean if all sectional curvatures are positive, negative or zero. A generic
symmetricspacehas thestructure
M = M M , (2.52)
0 s
×
where M = Rn0 and M is a semi-simple symmetric space; it is a product of a
0 s
compactsymmetricspace M and anon-sompactsymmetricspace M ,
+
−
M = M M . (2.53)
s +
× −
Ivan G.Avramidi: Heat Kernel onHomogeneousBundles 9
Ofcourse,thedimensionsmustsatisfytherelationn +n = n,wheren = dimM .
0 s s s
Let Λ be the vector space of 2-forms on M at a fixed point x . It has the
2 ′
dimensiondimΛ = n(n 1)/2,and theinnerproductinΛ is defined by
2 2
−
1
X,Y = X Yab. (2.54)
ab
h i 2
TheRiemanncurvaturetensornaturallydefines thecurvatureoperator
Riem : Λ Λ (2.55)
2 2
→
by
1
(RiemX) = R cdX . (2.56)
ab ab cd
2
Thisoperatorissymmetricandhasrealeigenvalueswhichdeterminetheprincipal
sectional curvatures. Now, let Ker(Riem) and Im(Riem) be the kernel and the
rangeofthisoperatorand
n(n 1)
p = dimIm(Riem) = − dimKer(Riem) . (2.57)
2 −
Further, let λ, (i = 1,...,p), be the non-zero eigenvalues, and Ei be the cor-
i ab
responding orthonormal eigen-two-forms. Then the components of the curvature
tensorcan bepresented intheform [10]
R = β Ei Ek , (2.58)
abcd ik ab cd
whereβ isasymmetric,infact, diagonal,nondegenerate p p matrix
ik
×
(β ) = diag(λ ,...,λ ). (2.59)
ik 1 p
Of course, the zero eigenvalues of the curvature operator correspond to the flat
subspace M , the positive ones correspond to the compact submanifold M and
0 +
the negative ones to the non-compact submanifold M . Therefore, Im(Riem) =
−
T M .
x s
InthefollowingtheLatinindicesfromthemiddleofthealphabetwillbeused
todenotetensorsinIm(Riem);theyshouldnotbeconfusedwiththeLatinindices
fromthebeginningofthealphabetwhichdenotetensorsin M. Theywillberaised
and loweredwiththematrixβ and itsinverse
ik
(βik) = diag(λ 1,...,λ 1). (2.60)
−1 −p