Local heat kernel asymptotics for nonminimal differential operators

Physics Letters B 270 ( 1991 ) 29-36 North-Holland PHYSICS LETTERS B

Local heat kernel asymptotics for nonminimal differential operators

V.P. Gusynin and E.V. Gorbar Institute for Theoretical Physics, SU-252 130 Kiev-130, USSR

Received 24 May 1991

The method for computing the local coefficients in the asymptotical heat kernel expansion is extended to the case of nonminimal differential operators. The lowest expansion coefficients are calculated for the second-order nonminimal operators on one- forms over a riemannian space in arbitrary dimensions. Unlike the coefficients for the minimal second-order operators those for nonminimal ones turn out to be essentially dependent on the space dimension.

It is well known [ 1-3] that for a positive elliptic differential operator A of the order 2r there exists at t ~ 0 + an asymptotic expansion of diagonal matrix elements o f the heat kernel

( x l e x p ( - t A ) l x ) ~ ~ Em(X[A)t (m-n)/2r. (1) t~O+ m=O

An operator A is defined, in general, as acting on the bundle o f k-tensors with the values in a vector space V, the base M being the compact n-dimensional r iemannian manifold without a boundary. The expansion coefficients Em (x IA) are endomorphisms of the fibre at x and prove to be the local covariant quantities o f given dimen- sionality constructed from the coefficient functions of the operator A, curvatures and their covariant derivatives. They play an important role in theoretical physics, geometry and topology. The lowest coefficients in the expansion ( 1 ) determine divergences o f the one-loop effective action, the axial and trace anomalies [4,5 ], the indices of elliptic operators [ 3 ], and so on.

Usually, the expansion coefficients for the second-order minimal differential operators are computed by sub- stituting the DeWitt ansatz

(xlexp(-tA)lx')=exp((7(X'X')) ~ l / 2 ( x ' x t ) 2 t ,,=0 ~ E'n(X'X'lA)t(m-n)/2 (2)

into the heat equation that leads to recursion relations for Era. In eq. (2) a(x, x' ) is the geodesic interval and ~ ( x , x ' ) is the Van Vleck-Morette determinant. The most complete results were obtained for the operator A = - [] + X a c t i n g on functions (Xis a matrix in internal space). We refer to DeWitt [ 1 ] for calculating E2, E4 and Sakai and Gilkey [6,7 ] for similar formulas for E6 (these authors used differential geometry methods) . Recently, Avramidi [ 8 ] computed E8 which has formidable complexity, Xu [ 9 ] and Amsterdamski et al. [ 10 ] derived E6 and E8, respectively, for the particular case o f a scalar field with nonminimal conformal coupling. For a similar calculation o f E2, E4 in the case o f the Riemann-Car tan manifold see ref. [ 11 ]. Unfortunately, the DeWitt technique does not apply to higher-order operators and nonminimal operators whose leading term is not a power of the Laplace operator (at tempts to generalize the DeWitt method for treating minimal fourth- order operators have been made in ref. [ 12 ] ).

In recent papers [ 13 ] a new algorithm for computing the heat kernel coefficients was developed basing on the Widom [ 14 ] generalization o f pseudodifferential operator techniques to curved manifolds. The method is ex- plicitly gauge and geometrically covariant and permits calculation of the coefficients for the most general oper-

0370-2693/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved. 29

Volume 270, number 1 PHYSICS LETTERS B 7 November 1991

ators of arbitrary order defined on curved manifolds including manifolds with torsion (and even manifolds with metric noncompatible connection). The results of computing the E2, E4 coefficients for the fourth-order minimal operators in arbitrary space dimensions are presented in ref. [ 13 ] (for a generalization to the manifolds with torsion see ref. [ 15 ] ). In this paper the technique of ref. [ 13 ] is adopted for nonminimal operators on riemannian spaces. Such operators arise in quantum gravity [ 16-18 ], the study of gauge fields in curved space [ 19 ] and gauge field quantization in background fields [ 20 ]. For example, the quantization of the electromag- netic field A u in a background gravitational metric gu~ leads to the differential operator

HU~=-gUq~+ ( 1 - 1 ) W W + R u~ (3)

(the euclidean metric is used everywhere). The standard way for computing the coefficients for the operator (3) is to reduce it to the minimal differential operator choosing the Feynman gauge ot = 1 in which (3) coincides with the Hodge-de Rham operator A = ~d + d~ on one-forms. In this gauge we can employ the DeWitt ansatz (2) and calculate the expansion coefficients (see ref. [ 21 ] ). Some very artificial methods were developed for calculating in arbitrary gauge [ 17,19 ] using the fact of the ot parameter dependence of the operator H u~. We will show that the method of ref. [ 13 ] permits the direct computation of the coefficients for nonminimal operators.

In fact, we consider a slightly more general expression for a second-order nonminimal differential operator

HUV= _gU~['-] + aVuW + XUV , (4)

where the covariant derivative involves the connection of an internal space as well as the Levi-Civita connection, and X u~ is also a matrix over the internal space indices. Following refs. [ 13,14 ] we define the operator exp ( - tH) through the operator H resolvent

e x p ( - t H ) = -~n e x p ( - t2) ( H - 2 ) - ' , (5) c

where the contour C goes counterclockwise around the spectrum of an operator H and then, for the matrix elements of the resolvent ( H - ; t ) -~, we use the representation in the form

Guv(x ,x ' ,2 ) - (x , lz, ~ Ix', v ) = f dnk~---- exp (2n)nx/g( x' ) [il(x,x' ,k)]tru,,(x,x' ,k;2). (6)

In eq. (6) l(x, x', k) is a phase function and au,,(x, x', k; 2) is an amplitude. The phase l(x, x', k) must be a covariant generalization to curved manifolds of the fiat-space phase ku(x -x ' )u, so we require the real function l(x, x', k) to be biscalar with respect to general coordinate transformations and a linear homogeneous function in k. The generalization of the linearity condition in x is the requirement for the ruth symmetrized covariant derivative to vanish at the point, i.e.

{Vu, Vu~...Vu,.} llx=,, - [{V,,,Vu=...Vu=} l] =ku,, m= 1,

= 0 , r n ~ l . (7)

In eq. (7) the curly brackets denote symmetrization in all indices and the square brackets mean that the coincidence limit is taken. The local properties of the function (8) are sufficient for obtaining the diagonal heat kernel expansion.

The resolvent of the operator H satisfies the equation

( H - 2 ) G = I (8)

and therefore, in order to fulfill ( 8 ) it suffices to require that the amplitude tr,~ (x, x ' , k; 2) satisfies the equation

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[gu~(VPlVfl-i[]l-2iVPlVp - [] - 2 ) + a (iVUVal- VulVal+iVulVa+iVAIW+ VuV a) + Xua] aa~(x, x ' , k; 2)

=IU~(x,x') . (9)

Although the amplitude has two Lorentz indices, we note that only the first one is acted on by the covariant derivatives. The function IU~(x, x' ) with the Lorentz and the bundle space indices will be defined by the conditions

[ IU,]=6u.~ , [{Vu, Vuv..Vu,,}IU~]=O, m>~l, (10)

the unity in eq. (10) is the unit matrix in the internal space. The functions l(x, x', k) and IU~ (x, x' ) introduced by means of eqs. ( 7 ) and (10) play an important role in the intrinsic symbolic calculus developed in re£ [ 14 ]. They permit us to generalize covariantly the pseudodifferential operator technique to curved manifolds.

To generate the expansion ( 1 ) ( r= 1 ) we introduce the auxiliary parameter E into eq. ( 1 ) according to the rule l~l/E, 2--,2/~ 2 and expand the amplitude in formal series over e:

aUp : ~ E2+mtTrrtue . m~O

Equating now the terms with identical powers of ~ we obtain the recursion equations to determine the coefficients emuv:

DUaeoA~ = I U~ ,

DU~a~ +i [ _gUa( D l+ 2WIVp ) + a (VUVAI+ VulVA+ VA/V u) ]a0A~ = 0 ,

DUaa,,a~ + i [ _gU~( [2l+ 2VP/Vp ) + a (VuV~/+ VulVA+ VA/V v) ]am-1 ~ + ( -gUaD +aV uVA + xua) e,~_2 A~ = 0 ,

m>~2, (11)

where

DUa=gUa( VP/Vp l - 2) -aVglVXl ( 12 )

and further we set the parameter ~ equal to one. The main difference from the case of minimal operators is that for obtaining a,~u~ we must invert now the matrix D uv and differentiate it. O f course, this increases the algebraic labour but does not cause essential difficulties in the case of the method of ref. [ 13 ].

For the diagonal matrix elements (x , / t I exp ( - tH) Ix, v) we have

( x , l~ l exp ( - tH) . x , v )= ~ ~ d"k ! id2 , expt - t 2 ) [amu~] (x, k, 2) (13) m=o (27t)"v/g

so, to get the expansion ( 1 ) we need to solve the recursion relations ( 11 ) and take the coincidence limit. We find

1 ( akak~ [a°u~]=Mu~"~' Mu~m [Ou}]= ~ gu~+ ( 1 - - ~ - , ~ / '

[alum] =i(2Mu AkP-aMu,~U'g~-aMu Pk A) [VpaoA~] ,

[amu~] =i(2Mu AkP-aM~k'~g~'°-aMu PkA) [Vaam- ~ A.I

+ (M u pg,*a_aM u ,~gpa)[V,~Vpa,,,_z p~]- (M'Xa,,,_z)u~. (14)

Calculating (14) needs the coincidence limits for nonsymmetrized covariant derivatives of l and I of the form [ Vu,... Vu, , l], [ Vu,...Vu, .I]. Such quantities are obtained directly from eqs. ( 7 ), ( 1 O) reducing all terms to unified

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indices ordering, and using the Ricci identity for the commutator of covariant derivatives on objects with fiber (left understood) and base indices:

k [V u, V~ ]gu,...uk = - ~ , RZu,u~gu,...u,-,zu,+,...uk + Wu~%,,...u,,

i=1

where

R~puv=OuF~pv-OvF~pu + F~auFapv-FZ~vFapu, W~v=Ouwv-OvoJu + [09 u, coy]

are the riemannian and the bundle curvatures, respectively (FXu~ and o~u are the affine and bundle connec- tions). For example, taking into account the scalar character of the function l, for the lowest coincidence limits we obtain

[ V f l ] = k , , [VuVfll=O, [V~,V~Vfll=-2k=R'~(au~), (15)

where the brackets (2... v) mean symmetrization in extreme indices with the coefficient ½. In the same way we get the coincidence limits for the nonsymmetrized covariant derivatives of the function I ~ (x, x' ):

[ V u I ~ ] = 0 , [VuV~IzV]= ' P (16) - ~R ~ , + ½ Wu~.6~, [VuV~Vzlp ° 1 2 o 2 o = - ~V(uR p~)a + ~V(u W~)z6v

(for the coincidence limits of four and five covariant derivatives of the functions l, I see refs. [ 13,15 ] ). The calculation of the coincidence limits for the derivatives of a.,u~ is reduced to computing the coincidence limits for the derivatives of Gout:

[Vuao.z ] = 0 , [7uV.ao~o] = - M x r( Z g ~ k " - a k ~ a ~ - a k . d T )MvokPlu.ap + Mz~[ VuV.I~v] , (17)

etc., and finally we obtain the necessary coincidence limits

[a~ .~ ] = 0,

[a2u~] --{-26'~(M3)~g~Z+a[26'~(M2)~ ~M~ ~+ (M 2) "~M~pg°Z+M~ '~ (M2) p~g ~a]

+a2(Mu~M=~+ M~ '~Mp °)M~ ~} kPkrl,~,,z~

2a[~o(M )u~g +a~,~(M )~ +,~(M~M +M~,~M~)]

+ a2[ga~( M~,~M=~ + Mu ~'Mo,,) + d~( Mu~M'*~ + Mu =Mt~ ~) 1} kOk °

- (g'~rMu a-aMu ~gZ~)} [V~V~ao~] - (MXM)u~, (18)

where

[vuv,...v~/] = k%,..~,~.

Expressions for [amu, ] are polynomials in k and in the matrix Mu~, the [ a~,,u~ ] containing an even power of k and the [a2m+,u~] containing the odd ones. The odd coefficients [azm+~u,] make no contribution after the integration over k. Besides, as in the case of the minimal operators, [ amu~ ] are homogeneous functions of the variables k and 2

[amu~] (x, lk, 12),) =l-(m+2)[am] (X, k, ),) ,

which after changing the variables k--.k/t ~/2, ),~),/t in eq. (13) leads to the expression ( 1 ) where the coefficients Emu~ (x[H) are given by the integrals of [ a,.u~ ] of the form

f &k Cid2 Emu,,(x[H)= J (2z0.x/~ J ~ e x p ( - i t ) [amu~] (x, k, )`) -~¢( [amu~] ) • (19)

The most general integral which we come across under the integration over ), and k is of the type

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J F (k2)pku, kuv..ku2 , ] d 'k exp( - '1 ) L(ka- '1) ' [ (1-a)k2- '1]"_] ~ (k2)"ku, ku~'"k~,~y id~

--- (2n) 'x /g c -~n (k2- '1) ' [ (1-a)k=- '1] '' " (20)

The contour integral can be reduced to the form

~ i d 2 exp( - ' 1 ) i d 2 e x p ( - i 2 ) ( k2 - '1 ) t [ (1 -a )k2- '1] m = 2n ( k 2 - i 2 ) t [ ( 1 - a ) k 2 - i 2 ] m'

C - - o o

and using the formula from ref. [ 22 ]

- o o

( f l - i x ) - u ( y - i x ) - ~ e x p ( i p x ) d x = O , p > 0 ,

Ref l>O, R e T > 0 , R e ( / l + v ) > l ,

2n exp (tip) ( - 1 )u+ ~-. v(~+ v)

wefind

,F , (v ,p+v; ( y - f l ) p ) , p < 0 ,

~ i d2 exp( - ' 1 ) exp( - k 2) 2n ( k 2 - 2 ) t [ ( 1 - a ) k 2 - 2 ] m - F ( l + m ) lF ' (m ' l+m;ak2)" R e a < l , (21)

C

where 1F l (a, c; z) is the confluent hypergeometric function. In order to carry out the integration over k we note that for any function f ( k 2) we can perform an angular integration by means of

(2n)nx/~kulkm'"km, f ( k )=go,,m...u2s} (4n),/22,F(½n+s) dke(k2) (n-E)/2+f(k2) 0

(kE=gU"kuk~) ,

where g{u,~2...u2,} is the symmetrized sum of metric tensor products, for instance,

g{m*,2mu*} = gmu2gmu* + gmmgmu4 + gulu4gu~m .

Then, using the formula [ 22 ]

(22)

o o

~ e x p ( - s t ) t b-' iF1 (a, c; kt )d t=F(b)s-bF(a, b, c; k s - ' ) , Isl > Ikl, 0

we at last obtain

L( - ' 1 ) [ ( - a ) k -21 ] F(½n+s+p) J F-k2 ~ - ~ ' ~ m =go,,u2...u~a(4n),,/E2~F (½n+s)F(l+m) F(m, p+s+ ½n, l+m; a ) ,

a < l . (23)

Here F(a, b, c; z) is the Gauss hypergeometric function. We note that the limitation a < 1 is not a defect of our method. The matrix Mu~('1 = 0 ) = k2g,~ - akuk~ being a principal symbol for the differential operator (4) has two different eigenvalues '1, = k 2 and ,t 2 = ( 1 - a ) k 2 which correspond to transverse and longitudinal eigenvec- tors, respectively. So, the condition a < 1 is needed for a principal symbol to be positive-definite and the differential operator (4) to be elliptic.

From eqs. (18), (19) and (23) now follows

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1 Eou~(Xl H) = J( [crou~ ] ) - (47t),/2 gu~[ 1 + ½aF( 1, i n + 1, 2; a) ] ,

E2uv(xlH) = J ( [ 0"2/,iv ] )

- (4~)./2 ~,~R. ~[+½af(1 , ½n+ 2, 4; a)+~a3F(2, ½n+2,4;a)+~a2(½n+2)F(2, in+3 , 5;a)

+ha2(a-3)F(3, ½n+2, 4; a ) + ~ a 3 ( a - 1 ) (½n+2)F(3, ½n+3, 5; a)

+T~.~al 3 (~n+l 2)(½n+3)F(3,½n+4,6;a)]

+Ru .a( l + 5+a(~n+2) F(1, ½n+ l,3;a)+[½_, ga(~n+' 1 ) ]F( 1, ½n+2, 4; a) 4 . 9 \

(24)

. • n+ 2 +a(n+ 3 ) + F(2,½n+l,3;a)+a 12 F(2, ½n+2, 4; a)

+ia(½n+2)F(2, ½n+3, 5 ; a ) + - - a ( a - 3 )

12 F(3, i n+2 , 4; a)+4~a2(a-l)(½n+2)F(3, ½n+3, 5;a)

+~-~-6a2(½n+2)(½n+3)F(3, ½n+4, 6; a ) )

-Wu,,'2(l+[l+½a(ln+l)]F(1,½n+l,3;a)+-~F(2,½n+l,3;a)+~a2(½n+l)F(2,½n+2,4;a) )

-X(u,,)[ 1 + ½aF( 1, ½n+ 1, 3; a) + ~2a2F(2, ½n+2, 4; a) ] -Xtu,q [ 1 + ½aF( 1, ½n+ 1, 3; a) ]

-gu~X~.la2F(2, ½n+2, 4; a ) ] ,

Eqs. (24), (25) can be expressed through elementary functions if we use the formula [23 ]

F(1,b,m;z)=(m_l)! ( - z ) l - " ( m-2(b--m+l)k ) (1--b), ,_l ( l - - z ) ' - b - l - - k=O 2~ k! zk '

(25)

(a)k=a(a+ 1 )...(a+k- 1 ) is the Pochhammer symbol, m= 1, 2 .... and m-b~ 1, 2, 3 ..... and the formulas for differentiating the hypergeometric function

d" _ _ d n dz F(a,b,c;z)- (a)~(b)n F(a+n,b+n,c+n;z) dz-----~[za+~-IF(a,b,c;z)]=(a)~za-tF(a+n,b,c;z).

(c).

After some manipulations we get the final form for the lowest coefficients:

Eou~(xlH)= (4~),,/2gu~ l + - n [ ( 1 - a ) - " / 2 - 1 1 ' (26)

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E z u ~ ( x l H ) = l { [ 1 ( 4 ~ ) n / 2 guvR -~ +

- 2 ( l - a ) l - " / 2 - l a nZ-4n+12)]12

, ( +Ru~2(½n_l)3 (½n-1)(1-a)-"/z-½"½n(½n+l)(1-a) ~-"/2+n

l ( +--4--) +W~,~(½n_l)z ~n(l-a)~-"/E-2(1-a)'-"/2-1a 3n 8

I 1 ( (1--a)X-"/2-- l -X~u~) 1 + 2 ( ½ n _ 1 ) 3 (½n-1)(1-a)-"/2+n a

1 ( ( 1 - a ) a ~ / 2 - 1 n 2 2 ) ] -Xtu~J[ 1+ ( ½ n - - 1 ) 2 "

l ( -gu~Xaa 4(ln-l)3 (½n-l)(1-a)-"/2-2 ( l - a ) l - " / : - l a

l ( 4(½n-1)3 (½n-1)(1-a)-"/2+½"½n(½n+l)(l-a)l-"/2

( 1 - a ) l - " / 2 - 1 a 5 n 2 - 8 n - 12) 12

n2"-~n-2)l

We note that as compared to the coefficients for the minimal second-order operators those for the nonminimal ones have a fairly nontrivial dependence on the space dimension n. This fact was known earlier only for the fourth-order operators [ 24,13 ].

It is not difficult to verify that for the operator (3) (a = 1 - 1/or, Xu~ = Ru~, Wu~ = 0) the coefficients (26) and (28) coincide at n = 4 with those obtained in ref. [ 19 ]. However, we must mention that the method in ref. [ 19 ] is strongly tied to the concrete form of the operator (3) and it does not work for more general operators like (4). Another interesting operator at which eqs. (26), (27) can be applied is a conformaUy covariant operator on vectors, it plays an important role in the study of conformal geometry and follows from (4) if we let [25 ]

n(n--4) gu~R+ 2__2__R ~, Wu~= 0 a=4/n, X,,~=4(n_l)(n_2) n - 2 u

Recently, Gilkey, Branson and Fulling [ 26 ] have considered a natural second-order differential operator on k- forms over M

D = ~ 2 d ~ + ~ ' 2 8 d - E , (28)

where ci and ~are constants and E is a zeroth-order operator (endomorphism) which arises from an action of the curvature on the bundle of exterior k-forms. At k= 1 the operator (28 ) can be identified with the operator (4) if we let a2= 1 - a , b~2= 1, Eu,,=Ru,,-Xu,,, Wu,,=O. Using our results (26), (27) we can calculate the global invariants

Era(D)= f t r E m ( x l D ) , M

for example,

Eo(D) = [n+ ( 1 - a ) - n / 2 _ 1 ]vo l (M) , (29)

El(D)= ![{(ln-1)+~[ (1-a)l-"/2-1]} R+ (l + l [ (1-a)-"/2-1l)tr E] (30)

Eqs. (29), (30) are consistent with the results obtained in ref. [ 26 ] by the functorial method.

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The authors are grateful to V.V. Romankov for cooperation at the initial part of this work and to A.O. Barvinsky, I.L. Buchbinder and Yu.A. Sitenko for discussions and valuable remarks.

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[26 ] P.B. Gilkey, T.P. Branson and S.A. Fulling, Heat equation asymptotics of"nonminimal" operators on differential forms, J. Math. Phys. ( 1991 ), to appear.

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Local heat kernel asymptotics for nonminimal differential operators