Non-perturbative Heat Kernel Asymptotics on Homogeneous Abelian Bundles

45 Pages·2008·0.26 MB·English

New Mexico Tech (October 27, 2008)
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Non-perturbative Heat Kernel Asymptotics
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2 on Homogeneous Abelian Bundles

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- Ivan G. Avramidi and Guglielmo Fucci
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m New Mexico Instituteof MiningandTechnology
[ Socorro,NM 87801,USA

2 E-mail: [email protected],[email protected]
v
9
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8 WestudytheheatkernelforaLaplacetypepartialdifferentialoperator
4
acting on smooth sections of a complex vector bundle with the structure
.
0
groupG×U(1)overaRiemannianmanifoldMwithoutboundary. Thetotal
1
8 connectionon thevectorbundlenaturallysplitsinto aG-connectionand a
0
U(1)-connection,whichisassumedtohaveaparallelcurvatureF. Wefind
:
v anewlocalshorttimeasymptoticexpansionoftheoff-diagonalheatkernel
i
X U(t|x,x′) close to the diagonal of M × M assuming the curvature F to be
r ofordert−1. Thecoefficientsofthisexpansionarepolynomialfunctionsin
a
theRiemann curvaturetensor(and thecurvatureoftheG-connection)and
its derivatives with universal coefficients depending in a non-polynomial
but analytic way on the curvature F, more precisely, on tF. These func-
tionsgenerateall termsquadraticand linearintheRiemann curvatureand
of arbitrary order in F in the usual heat kernel coefficients. In that sense,
we effectively sum up the usual short time heat kernel asymptotic expan-
siontoallordersofthecurvatureF. Wecomputethefirstthreecoefficients
(bothdiagonaland off-diagonal)ofthisnewasymptoticexpansion.
nphkahab01.tex;October28,2008;21:50
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 1
1 Introduction
The heat kernel is one of the most powerful tools in quantum field theory and
quantum gravity as well as mathematical physics and differential geometry (see
for example [18, 23, 11, 12, 10, 20, 22, 19] and further references therein). It is
of particular importance because the heat kernel methods give a framework for
manifestlycovariantcalculationofawiderangeofrelevantquantitiesinquantum
field theory like one-loop effective action, Green’s functions, effective potential
etc.
Unfortunately the exact computation of the heat kernel can be carried out
only for exceptional highly symmetric cases when the spectrum of the operator
isknownexactly,(see[17,19,20]andthereferencesin[8,15,14,13]). Although
these special cases are very important, in quantum field theory we need the ef-
fective action, and, therefore, the heat kernel for general background fields. For
thisreason various approximationschemes havebeen developed. One of theold-
estmethodsistheMinackshisundaram-Pleijelshort-timeasymptoticexpansionof
theheat kernel as t → 0 (seethereferences in[18, 2,23]).
Despite its enormous importance, this method is essentially perturbative. It
is an expansion in powers of the curvatures R and their derivatives and, hence, is
inadequate for large curvatures when tR ∼ 1. To be able to describe the situa-
tion when at least some of the curvatures are large one needs an essentially non-
perturbative approach, which effectively sums up in the short time asymptotic
expansion of the heat kernel an infinite series of terms of certain structure that
contain large curvatures (for a detailed analysis see [4, 9] and reviews [10, 12]).
For example, the partial summation of higher derivatives enables one to obtain
a non-local expansion of the heat kernel in powers of curvatures (high-energy
approximation in physical terminology). This is still an essentially perturbative
approach since the curvatures (but not their derivatives) are assumed to be small
and oneexpandsin powersofcurvatures.
On another hand to study the situation when curvatures (but not their deriva-
tives) are large (low energy approximation) one needs an essentially non-pertur-
bativeapproach. A promisingapproach to the calculation of the low-energy heat
kernelexpansionwasdevelopedinnon-Abeliangaugetheoriesandquantumgrav-
ityin[3,4,5,6,7,8,13,14,15]. Whilethepapers[3,4,6,7]dealtwiththeparal-
lelU(1)-curvature(thatis,constantelectromagneticfield)inflatspace,thepapers
[5, 8, 13] dealt with symmetric spaces (pure gravitational field in absence of an
electromagnetic field). The difficulty of combining the gauge fields and gravity
was finally overcome in the papers [14, 15], where homogeneous bundles with
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 2
parallelcurvatureon symmetricspaces was studied.
In this paper we compute the heat kernel for the covariant Laplacian with a
large parallel U(1) curvature F in a Riemannian manifold (that is, strong covari-
antlyconstantelectromagneticfield inanarbitrarygravitationalfield). Ouraimis
to evaluate the first three coefficients of the heat kernel asymptotic expansion in
powers of Riemann curvature R but in all orders of the U(1) curvature F. This is
equivalenttoapartialsummationintheheatkernelasymptoticexpansionast → 0
of all powers of F in terms which are linear and quadratic in Riemann curvature
R.
2 Setup of the Problem
Let M be a n-dimensional compact Riemannian manifold without boundary and
S be a complex vector bundle over M realizing a representation of the group
G⊗U(1). Letϕ beasectionofthebundleSand ∇bethetotalconnectiononthe
bundle S (including the G-connection as well as the U(1)-connection). Then the
commutatorofcovariantderivativesdefines thecurvatures
[∇ ,∇ ]ϕ = (R +iF )ϕ , (2.1)
µ ν µν µν
where R is the curvature of the G-connection and F is the curvature of the
µν µν
U(1)-connection (whichwillbealso called theelectromagneticfield).
In thepresent paperwe considera second-orderLaplacetypepartial differen-
tialoperator,
L = −∆, ∆ = gµν∇ ∇ . (2.2)
µ ν
Theheatkernel fortheoperatorL isdefined asthesolutionoftheheat equation
(∂ +L)U(t|x,x′) = 0 , (2.3)
t
withtheinitialcondition
U(0|x,x′) = P(x,x′)δ(x,x′) . (2.4)
where δ(x,x′) is the covariant scalar delta function and P(x,x′) is the operator of
paralleltransportofthesectionsofthebundleSalongthegeodesicfromthepoint
x′ to thepoint x.
ThespectralpropertiesoftheoperatorL aredescribedintermsofthespectral
functions, defined in terms of the L2 traces of some functions of the operator L,
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 3
suchas thezeta-functionζ(s) = Tr L−s, andtheheat trace
Tr exp(−tL) = dvol tr Udiag(t), (2.5)
Z
M
where dvol = g1/2dx is the Riemannian volume element with g = detg and tr
µν
denotesthefibertrace. Hereandeverywherebelowthediagonalvalueofanytwo
pointquantity f(x,x′)denotesthecoincidencelimitas x → x′,that is,
fdiag = f(x,x). (2.6)
It is well known [18] that the heat kernel has the asymptotic expansion as
t → 0(see also[2, 10,11, 23])
σ(x,x′) ∞
U(t|x,x′) ∼ (4πt)−n/2P(x,x′)∆1/2(x,x′)exp − tka (x,x′) , (2.7)
k
" 2t #
Xk=0
where σ(x,x′) is the geodesic interval (or the world function) defined as one half
the square of the geodesic distance between the points x and x′ and ∆(x,x′) is
the Van Vleck-Morette determinant. The coefficients a (x,x′) are called the off-
k
diagonalheat kernel coefficients.
The heat kernel diagonal and the heat trace have the asymptoticexpansion as
t → 0[2, 23]
∞
Udiag(t) ∼ (4πt)−n/2 tkadiag , (2.8)
k
Xk=0
∞
Tr exp(−tL) ∼ (4πt)−n/2 tkA , (2.9)
k
Xk=0
where
adiag = a (x,x) (2.10)
k k
and
A = dvol tr adiag . (2.11)
k Z k
M
Thecoefficients A are called the globalheat kernel coefficients; they are spectral
k
invariantsoftheoperatorL.
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 4
The diagonal heat kernel coefficients adiag are polynomials in the jets of the
k
metric, theG- connection and the U(1)-connection; in other words, in the curva-
turetensorsandtheirderivatives. Letussymbolicallydenotethejetsofthemetric
and theG-connectionby
R = ∇ ···∇ Ra b , ∇ ···∇ Ra , (2.12)
(n) (µ1 µn µn+1 µn+2) (µ1 µn µn+1)
n o
and thejetsoftheU(1) connectionby
F = ∇ ···∇ Fa . (2.13)
(n) (µ1 µn µn+1)
Here and everywhere below the parenthesis indicate complete symmetrization
overallindicesincluded.
By counting the dimension it is easy to describe the general structure of the
coefficients adiag. Let usintroducethemulti-indicesofnonnegativeintegers
k
i = (i ,...,i ), j = (j ,..., j). (2.14)
1 m 1 l
Letus alsodenote
|i| = i +···+i , |j| = j +···+ j . (2.15)
1 m 1 l
Thensymbolically
k N N−l
adiag = C F ···F R ···R , (2.16)
k (k,l,m),i,j (j1) (jl) (i1) (im)
XN=1Xl=0 Xm=0 Xi,j≥0
|i|+|j|+2N=2k
whereC aresomeuniversalconstants.
(k,l,m),i,j
Thelowerorderdiagonalheat kernel coefficients are wellknown[18,2, 11]
adiag = 1, (2.17)
0
1
adiag = R, (2.18)
1 6
1 1 1 1
adiag = ∆R+ R2 − R Rµν + R Rαβµν
2 30 72 180 µν 180 αβµν
1 1 1
+ R Rµν + R iFµν − F Fµν. (2.19)
µν µν µν
12 6 12
To avoid confusion we should stress that the normalization of the coefficients a
k
differsfrom thepapers [2, 10, 11].
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 5
InthepresentpaperwestudythecaseofaparallelU(1)curvature(covariantly
constantelectromagneticfield), i.e.
∇ F = 0 . (2.20)
µ αβ
That is, all jets F are set to zero except the one of order zero, which is F itself.
(n)
In thiscaseeq. (2.16)takes theform
k N N−l
adiag = C Fl R ···R , (2.21)
k (k,l,m),i (i1) (im)
XN=1Xl=0 Xm=0 Xi≥0
|i|+2N=2k
whereC are nowsome(other)numericalcoefficients.
(k,l,m),i
Thus,by summingup allpowersof F intheasymptoticexpansionoftheheat
kernel diagonalweobtainanew(non-perturbative)asymptoticexpansion
∞
Udiag(t) ∼ (4πt)−n/2 tka˜diag(t) , (2.22)
k
Xk=0
wherethecoefficients a˜diag(t)arepolynomialsinthejetsR
k (n)
k N
a˜diag(t) = f(k) (t) R ···R , (2.23)
k (m,i) (i1) (im)
XN=1Xm=0 Xi≥0
|i|+2N=2k
and f(k) (t)aresomeuniversaldimensionlesstensor-valuedanalyticfunctionsthat
(m,i)
dependon F onlyin thedimensionlesscombinationtF.
Fortheheattrace weobtainthena newasymptoticexpansionoftheform
∞
Tr exp(−tL) ∼ (4πt)−n/2 tkA˜ (t) , (2.24)
k
Xk=0
where
A˜ (t) = dvol tra˜diag(t) . (2.25)
k Z k
M
This expansion can be described more rigorously as follows. We rescale the
U(1)-curvature F by
F 7→ F(t) = t−1F˜, (2.26)
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 6
sothat tF(t) = F˜ is independentoft. Then theoperatorL(t)becomesdependent
on t (in a singular way!). However, the heat trace still has a nice asymptotic
expansionas t → 0
∞
Tr exp[−tL(t)] ∼ (4πt)−n/2 tkA˜ , (2.27)
k
Xk=0
where the coefficients A˜ are expressed in terms of F˜ = tF(t), and, therefore, are
k
independentoft. Thus,whatwearedoingistheasymptoticexpansionoftheheat
traceforaparticularcaseofasingular(ast → 0)time-dependentoperatorL(t).
Letusstressonceagainthattheeq. (2.23)shouldnotbetakenliterally;itonly
represents thegeneral structureof the coefficients a˜diag(t). To avoid confusionwe
k
listbelowthegeneral structureofthelow-ordercoefficients in moredetail
a˜diag(t) = f(0)(t), (2.28)
0
a˜diag(t) = f(1) αβµν(t)R + f(1) µν(t)R , (2.29)
1 (1,1) αβµν (1,2) µν
a˜diag(t) = f(2) αβµνσρ(t)∇ ∇ R + f(2) αβµν(t)∇ ∇ R
2 (1,1) (α β) µνσρ (1,2) (α β) µν
+f(2) αβγδµνσρ(t)R R + f(2) αβµν(t)R R
(2,1) αβγδ µνσρ (2,2) αβ µν
+f(2) αβµνσρ(t)R R (2.30)
(2,3) αβ µνσρ
with obvious enumeration of the functions. It is the universal tensor functions
f(i) (t) that are of prime interest in this paper. Our main goal is to compute the
(l,m)
functions f(i) (t)forthecoefficients a˜diag(t), a˜diag(t)and a˜diag(t).
(l,m) 0 1 2
Of course, for t = 0 (or F = 0) the coefficients a˜ (t) are equal to the usual
k
diagonalheat kernel coefficients
a˜diag(0) = adiag. (2.31)
k k
Therefore, by using the explicit form of the coefficients adiag given by (2.19) we
k
obtain the initial values for the functions f(i) . Moreover, by analyzing the corre-
(j,k)
sponding terms in the coefficients adiag and adiag (which are known, [18, 2, 22]),
3 4
one can obtain partial information about some lower order Taylor coefficients of
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 7
thefunctions f(i) (t):
(j,k)
1
f(0)(t) = 1− t2F Fµν +O(t3), (2.32)
µν
12
1
f(1) αβ (t) = δαδβ +O(t), (2.33)
(1,1) µν 6 [µ ν]
1
f(1) µν(t) = tiFµν +O(t2), (2.34)
(1,2) 6
1
f(2) αβµν (t) = gαβδµ δν +O(t), (2.35)
(1,1) σρ 30 [σ ρ]
1
f(2) αβ (t) = − tiF(α δβ) +O(t2), (2.36)
(1,2) µν 15 [ν µ]
1 1
f(2) γδ σρ(t) = g g gσ[γgδ]ρ − δ[γg gδ][ρδσ]
(2,1)αβ µν 180 µ[α β]ν 180 [α β][ν µ]
1
+ δγ δδ δσδρ +O(t),
72 [α β] [µ ν]
(2.37)
1
f(2) αβ (t) = δαδβ +O(t), (2.38)
(2,2) µν 12 [µ ν]
1 1 1
f(2) αβµν (0) = − tiFαβδµ δν − tiFµνδα δβ + δ[µtiFν][αδβ] +O(t2).
(2,3) σρ 36 [σ ρ] 30 [σ ρ] 9 [σ ρ]
(2.39)
Thisinformationcan beused tocheck ourfinal results.
NoticethattheglobalcoefficientsA˜ (t)haveexactlythesameformasthelocal
k
ones;theonlydifferenceisthatthetermswiththederivativesoftheRiemanncur-
vaturedonotcontributetotheintegratedcoefficientssincetheycanbeeliminated
byintegratingby parts andtakingintoaccount that F is covariantlyconstant.
Moreover,westudyevenmoregeneralnon-perturbativeasymptoticexpansion
for the off-diagonal heat kernel and compute the coefficients of zero, first and
second order in the Riemann curvature. We will show that there is a new non-
perturbative asymptotic expansion of the off-diagonal heat kernel as t → 0 (and
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 8
F = t−1F˜, sothat tF is fixed)oftheform
∞
U(t|x,x′) ∼ P(x,x′)∆1/2(x,x′)U (t|x,x′) tk/2b (t|x,x′) (2.40)
0 k
Xk=0
whereU isan analyticfunctionof F such thatfor F = 0
0
σ(x,x′)
U (t|x,x′) = (4πt)−n/2exp − . (2.41)
0 (cid:12) " 2t #
(cid:12)(cid:12)F=0
(cid:12)
(cid:12)
Hereb (t|x,x′)areanalyticfu(cid:12)nctionsoftthatdependon F onlyinthedimension-
k
less combination tF. Of course, for t = 0 they are equal to the usual heat kernel
coefficients, thatis,
b (0|x,x′) = a (x,x′), b (0|x,x′) = 0. (2.42)
2k k 2k+1
Moreover, we will show below that the odd-ordercoefficients vanishnot only for
t = 0 andany x , x′ butalsoforany t and x = x′,that is,on thediagonal,
bdiag (t) = 0. (2.43)
2k+1
Thus,theheatkernel diagonalhas theasymptoticexpansion(2.22)as t → 0with
a˜diag(t) = (4πt)n/2Udiag(t)bdiag(t). (2.44)
k 0 2k
3 Geometric Framework
Ourgoal isto studytheheat kernel U(t|x,x′)in theneighborhoodof thediagonal
as x → x′. Therefore, we will expand all relevant quantities in covariant Taylor
seriesnearthediagonalfollowingthemethodsdevelopedin[2,11,10,12]. Wefix
apoint,say x′, onthemanifold M andconsiderasufficientlysmallneighborhood
of x′, say a geodesic ball with a radius smaller than the injectivity radius of the
manifold. Then, there exists a unique geodesic that connects every point x to
the point x′. In order to avoid a cumbersome notation, we will denote by Latin
letters tensor indices associated to the point x and by Greek letters tensor indices
associatedtothepoint x′. Ofcourse,theindicesassociatedwiththepoint x(resp.
x′) are raised and lowered with themetricat x (resp. x′). Also, wewill denoteby
∇ (resp. ∇′)covariantderivativewithrespect to x(resp. x′).
a µ
I. G. Avramidiand G.Fucci : Non-perturbativeHeatKernel Asymptotics 9
We remind below the definition of some of the two-point functions that we
willneed inouranalysis. Firstofall,theworldfunctionσ(x,x′)isdefined as one
half of the square of the length of the geodesic between the points x and x′. It
satisfiestheequation
1
σ = uau = u uµ, (3.1)
a µ
2
where
u = ∇ σ, u = ∇′σ . (3.2)
a a µ µ
Thevariablesuµ are nothingbutthenormalcoordinatesat thepoint x′.
TheVan Vleck-Morettedeterminantisdefined by
∆(x,x′) = g−21(x)det[−∇a∇′νσ(x,x′)]g−12(x′) . (3.3)
This quantity should not be confused with the Laplacian ∆ = gµν∇ ∇ . Usually,
µ ν
themeaningof∆willbeclearfrom thecontext. Wefind itconvenienttoparame-
terizeitby
∆(x,x′) = exp[2ζ(x,x′)] . (3.4)
Next,wedefinethetensor
ηµ = ∇ ∇′µσ , (3.5)
b b
and thetensorγa inversetoηµ by
µ a
γa ηµ = δa, ηµ γb = δµ. (3.6)
µ b b b ν ν
Thisenables usto definenew derivativeoperatorsby
∇¯ = γa ∇ . (3.7)
µ µ a
These operators commute when acting on objects that have been parallel trans-
ported to the point x′ (in other words the objects that do not have Latin indices).
In fact, when acting on such objects these operators are just partial derivatives
withrespect tonormalcoordinateu
∂
∇¯ = . (3.8)
µ
∂uµ
Wealsodefine theoperators
1
D = ∇¯ − iF uα . (3.9)
µ µ µα
2