New algorithm for computing the coefficients in the heat kernel expansion

Volume 225, number 3 PHYSICS LETTERS B 20 July 1989

N E W A L G O R I T H M F O R C O M P U T I N G T H E C O E F F I C I E N T S IN THE HEAT KERNEL EXPANSION

V.P. G U S Y N I N Institute for Theoretical Physics, 252 130 Kiev-130, USSR

Received 2 May 1989

A new covariant method for computing the coefficients in an asymptotic expansion of the heat kernel is suggested. The first two nontrivial coefficients for the second and fourth order minimal differential operators on a riemannian manifold are calculated in an arbitrary space dimension. The algorithmic character of the method suggested allows one to calculate the coefficients by computer using an analytical calculation system.

The coefficients in the asymptotic expansion of the heat kernel play a very important role in investigating the effects of quantum fields in curved spacetime. They determine one-loop divergences of the effective action, the axial and trace anomalies [ 1-3 ], the indices of elliptic operators [ 4], and in some cases, they allow the exact calculation o f functional determinants [ 5 ] for a certain type of differential operators.

The expansion we are interested in reads [4,6 ]

( x l e - ' A I x ) = ~ Em(x la ) t ~m-n)/zr, t-,O+, (1) m

where A is a positive elliptic differential operator of the order of 2r acting on sections o f a vector bundle over a n-dimensional compact r iemannian manifold without boundary. Summation in ( 1 ) is carried out over all inte- ger nonnegative m. The coefficients E m ( x l A ) (called Schwinger-De Witt or Seeley-Gilkey coefficients) are proven to be local polynomial invariants o f the operator A.

There exist two methods to compute the coefficients Em. One of them, that of De Witt [ 1 ], is based on a certain ansatz for the heat kernel matrix elements. The advantage of the method consists in explicit covariance with respect to gauge and general-coordinate transformations. However, the use of a definite ansatz for the heat kernel complicates generalization to higher than the second-order differential operators and also to nonminimal operators ~ ~ (see, for example, refs. [ 7,8 ]. The second method used, chiefly, by mathematicians [ 4,6,9 ], is based on applying the technique of pseudodifferential operators [ 10 ]. It is free of drawbacks inherent in the De Witt method. However, its generalization to the case of curved manifolds results in certain technical difficulties due to the lack of explicit covariance with respect to general-coordinate transformations. This requires the use of appropriate gauges for affine and bundle connections and a normal coordinate system at intermediate stages of calculation [ 11 ].

In this paper we suggest a new algorithm for computing the coefficients based on a covariant generalization ofpseudodifferential operator techniques to the case of curved manifolds (the intrinsic symbolic calculus) [ 12]. This method possesses the explicit covariance with respect to general-coordinate and gauge transformations, allows one to calculate coefficients for both minimal and nonminimal operators of any order, and can be gen- eralized to manifolds with torsion and to supermanifolds. An important advantage of the method proposed implies also its algorithmic character which makes it possible to calculate the coefficients Em by computer.

~ For nonminimal differential operators the highest power of the derivatives is not the power of operator [~.

0370-2693/89 /$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

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Let us express the heat operator e-'A through the operator A resolvent by the formula of functional calculus

e-~A= f ~-~n e - t ~ ( A - 2 ) - 1, (2) C

the contour C goes counterclockwise around the spectrum of an elliptic operator A. The main point is to repre- sent the matrix elements of the resolvent ( A - 2 ) - 1 in the form

A ~ f d"k ei,(x, x,, k)a(X, X', k; 2) (3) G ( x , x ' , 2 ) - ( x l I x ' ) = ( 2 z ~ ) " ~

where l(x, x', k) is a phase function, a(x, x', k; 2) is an amplitude [ 10], g=de t gu~, everywhere the euclidean metric is used. In the case of flat space the phase l(x, x', k) = k , ( x - x ' )u is for each x' a linear function o fx and k:

O l(x ,x ' ,k)=k~, O axe, ~ l(x, x', k) = (x -x ' )u , l(x, x', k)I~=~' = 0 .

In the standard calculus ofpseudodifferential operators one uses the phase k, ( x - x ' ) ~ even in the case of curved manifolds. But non-invariance of the phase k z ( x - x ' )~ with respect to general coordinate transformations implies the lack of manifest covariance of the whole technique of pseudodifferential operators; the results produced by this technique are expressed in terms of of nontensorial functions and their partial derivatives.

For the covariant generalization to the case of curved manifolds we require the real function l(x, x', k) to be a biscalar with respect to general-coordinate transformations. The requirement for the ruth symmetrized covariant derivative to vanish at the point x' with m >/2, i.e.

{V~,Vz~...V,.,.,}/lx=x.--- [{V,.V.=...Vu~}I] =k,.,, m= 1,

=0, m # l , (4)

proves to be the generalization of a linearity condition at x'. In eq. (4) the brackets {...} denote symmetrization in all indices and the square brackets-taking the coincidence limit [f(x, x') ] - f ( x , x')I~=x.. The covariant derivative Va includes, in general, different connections (affine connection, spinor connection, gauge fields)~2. Also we require the phase l(x, x', k) to be a linear homogeneous function of k. One can show that the function l(x, x', k) determined by the conditions (4) really exists [12] in some coordinate neighbourhood with local coordinates x. Since we are interested in the diagonal matrix values of operators the local properties of the function (4) are sufficient to perform the calculations.

Further we need the coincidence limits for nonsymmetrized covariant derivatives [Va, Va2...Vum/] . AS we shall now show, these are all polynomials in the torsion and curvature tensors T¢',~ and R~',ap of the given connection and their covariant derivatives. The quantities [Vu~Vu2...Vum/] are obtained directly from eq. (4) reducing all terms to a unified indices ordering with use made of the Ricci identity for the commutator of covariant derivatives on tensors

[Vu, V. ]fm...~,. = - L R~.,.~L,...u,-,~., ..... u.+Tau~V:oCu,..... + W~..fu,...t,., i = 1

(5)

where

• ;. ~ +Fx Fpv_F~ Fpa, T;u~=Faa.-F ~

~2 On mixed objects which have both fiber (left understood) and base indices the covariant derivative is given by the rule (/) _ _ I 0 ) ah Vj.,..,,=(O,,+~o.)f,,...,,-Y;'_~F~.,.f~,.,., ,~.,+,.....,,whereF~..isanaffineconnection, .-~ . X~b+Aulsthebundleconnection,

Z.j, are the generators of the representation of the rotation group SO (4) to which f,,...u,, belongs, A. is the gauge potential.

234


and Wu~ = 0~a~-0~co~ + [ mu, o9~ ] is the bundle curvature. Taking into account the scalar character of the function l for the lowest coincidence limits we find from eqs. (4) and ( 5 )

[VuV~l]=½k,~T"u~, (6)

and if the torsion T"u~ = 0 (the result being straightforward but more complicated otherwise )

[ V , , V ~ v d ] = 2 - ~ k ~ R (au~), (7)

[VuV.VaVol ] = - ~k~ { 5V.R"(a.~) + 5V.R ~(a.o) + V¢aR ~u.~) + V(aR'~.u~)}, (8)

[ v ~ v . v ~ v ~ v d ] , . . . . =-g-6k~{25V~V~R (,,~)+25VuVaR (,,~p)-5VuV~R'~(,,p~)-5V~,VpR (~)+4V(~Va)R (~u,)

+ 4V~V(~R'~a~o) + 4V~V(~R'~p),a + 4VaV(,,R ~)+,~ + 4V~ V(,,R'~o) + 4V(oVp) R '~(~) }

+~sk.{1 '* ~ " a " 7R (a~)R (~o)+ 17R~(p~a)Ra(~p) - 15R~(pu~)R~a~ - 15R (B.p)R o ~ - 7 R ( ~ ) R (~pa)

- 7R'~(p~)RP(~o~) + 3RC'(~ao)Ra~pu+ 3R"(ap~)R ~a~ + 3R~(~ap)Ra~o~,+ 3R"(a~p)R~o~,

-4R`~(~p)R ~( ~+a ) -4R`~( ~ )R ~(°~) + 2R`~(~)R ~ ( ~ ) + 2 R ~ ( ~ ) R ~( ~ ) + 2 R ~ ( ~ ) R ~( ~°)

+ 2R'~(~o)R ~( ~uo) } . (9)

The brackets (#... u) mean symmetrization in extreme indices with the coefficient 1 / 2. Further we shall consider only the case of a riemannian manifold without torsion, but as is seen from eq. (6) the method admits the generalization to manifolds with torsion, although the calculations of coincidence limits become very cumber- some with growing of a number of derivatives. The resolvent of the operator A satisfies the equation

1 (A(x, V u ) - 2 ) G ( x , x ' , 2 ) = - ~ = ~ ( x - x ' ) , (10)

therefore, to fulfil (10) it is sufficient to require that the amplitude a(x, x ' , k; 2) satisfies the equation

(A(x, Vu+iVfl) - 2 ) a ( x , x ' , k; 2) =I(x , x' ) . ( 11 )

The function I(x, x' ) with bundle space indices will be defined by the conditions analogous to eq. (4)

[I.a] =~.a, [{V.iVu2...V~m}a~'I~fl]=O, m>~l, (12)

Similar to the case of the function l(x, x', k) eq. (12) allows one to find the coincidence limits for nonsymmetrized covariant derivatives (torsion T ' % = 0):

[ v , , v d l = ½ m ~ , , [ v ~ v ~ v d ] = - -~v(~ m ~ > ,

[VuV.Va V fl] = ~ (V.V~ W;,o + V.V~ W~,, + V~V~ Wuo ) - ~ ( W~oRP(~, n + W.eRP(az~ ) + W~,RP(.~,o) + W.pR'(..a))

+ l (Wu~ Wa~,+ WuaW,,,+ WuoW~,~ + W,,aW~,,,+ W,,,WN + W,~oW~,~) . (13)

The functions l(x, x' , k) and I(x, x ' ) introduced by means of eqs. (4) and (12) play a very important role in the so-called intrinsic symbolic calculus which was pioneered by Widom [ 12 ]. Their use allows one to make the covariant generalization of pseudodifferential operator techniques to the bundle space. In fact, the influence of the base manifold curvature and torsion and the bundle curvature is made manifest through these functions. The role of the functions l(x, x', k) and I(x, x' ) is, to some extent, analogous to that played by the geodesic interval and the function of parallel transport in the De Witt method [ 1 ]. We note, however, that the functions introduced are defined also for manifolds without metric g~,,, whereas the geodesic interval is defined only for manifolds equipped with metric.

To generate the expansion ( 1 ) we introduce in eq. ( 11 ) an auxiliary parameter e according to the rule l~l/~, 2--.2/e ~" and expand

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2r 6e(x,x',k;)~)= g2r+mam(X,x',k;.,~), A(x, Vv+iVul/g)= ~ gm-2rAm(x, V u, Vul). ( 1 4 )

m=O m=O

Substituting eq. (14) into eq. ( 11 ) and equating the terms with identical powers ofe we obtain recursion equations to determine the coefficients am:

(Ao - 2 ) a o = I ,

(Ao--2)al + A j a 0 = 0 ,

2r

(Ao--J.)am"F ~ Aiam_i=O, m>~2r. (15) i= 1

For example, in the case of operator A = - [] + X (X is a matrix valued function, r = 1 ) the recursion equations ( 15 ) take the form

(v, qv~l-,~)ao = i ,

(WIV,,I-2 )a, - i ( [ ] l + 2WlV,,)ao = O,

(VulVul-2)am-i(Dl+2VulVu)am_~ "~'(--['~"~'X)am_2=0 , m>~2. (16)

Using the formulae ( 2 ) - (4) we have for the diagonal matrix elements of the heat kernel

(xle-'AIX)=m=O ~ ~ d"k ¢id2 -,4 (2~-x/-g J ~ - e [am](x,k, 2), (17)

where a,~ are obtained consecutively from the recursion relations ( 15 ) (we have put the parameter e to be equal to one). In taking the coincidence limit for a m we need the coincidence limits for derivatives of functions l and I (eqs. ( 6 ) - ( 9 ) and ( 13 ) ). As it fo l lows f rom the recurs ion equat ions ( 15 ), [ am ] are h o m o g e n e o u s funct ions of the degree - m - 2r in variables (k,/~ l /2r) :

[am] (x, lk, 12rj.) =l -(m+2r) [am] (X, k, 2) . (18)

Hence, changing in (17) the variables k ~ k / t ~/>, 2~2 / t we arrive at the expansion ( 1 ) where the coefficients Em are expressed through the integrals from [ am ] of the form

f d ' k ¢ id2 -4 Em(xlA) (2---7-x/~ j c ~--~-n e [ a m ] ( X , k , ~ ) ~ J ( t a m ] ) . (19)

Because [ a2m ÷ 1 ] contain odd powers of k the odd coefficients E2m + ~ equal to zero. Integration over 2 and k in ( 19 ) is easily performed using the standard integral

l (n , l , s ,p , r )= f d~k f 1 F(n/2r+(s+p) /r ) ( 2 n ) " , ~ (k2)pk~'ku~'"k"'~s id2 e -'~" 2zc [ (k2)r-~,] l - (47c) n/2 2S.rF(n/2+s)F(l) g{u,u2...u2s},

c

(20)

where &~,.2....~} is the symmetrized combination of metric tensors, for instance, g{.,u~u~u4} =guau:gu~.. + g~,.3&,2i,, + g~,u.gu2m.

As an illustration we consider the second order minimal operator A = - [] + X for which the recursion relations are given by (16). The calculation of the coincidence limits for the derivatives of am is reduced to computing the coincidence limits for the derivatives of ao and, finally, for the necessary values [ a m ] w e have

236


1 [fro] = [a] = k2_2 , [a2] = -2[a]3k'~ka(l~'ro,#+lo,~ra)- [a] zX,

[O'4] =4[alSkC~kak~kaF,~a~6-2[a]4k'~ka{G~,a-3(l%,~a+l. %a)X- 2V.VaX} + [a] 3(Ia a# ~ - [-IX+X 2) , (21)

where

F~p~ = 2 ( 1%,~p~ + l~ °~p~ + l ~ %r~ + l~y %~) + 3 (l °o~ + l~ ~p) ( l ~ w~ + lv ~ ~)

+ 12l ~°~fl~oT~ + 4l,~ ~"~ (l~o.~ + 2lo~) + 4 I ~ , (22)

G,~ = l,~ ".%p + l"v~%~ + I" Yo,~B + 1%% ( l/~p + l~,,p ) + l"~y,~l."~ + 21"°r,~l~or~ + 2 (1°%. + 21"~% ) I po

+4(lJ,,p+ 2lVo p)I.~+ 2(l%~' + 21~ ~)Iop+ 2(I.a Y+l. yra+I'~y,~a) , (23)

and we have embedded the following notations

[V~V..,.V~I] =k'~l~....~,~, [V,,V....V~I]-Iu....~.

Thus, we find

1 { 2 (I,~f~+UB )_ X } 1 E2(x) =J ( [ a2 ] ) = (4zt)./2 2F(3) F ~ - (42~) "/2 ( ~ R - X ) , (24)

1 5 I D ~ , s o o o l l 1 2 1 t ~ } E4(x)=J([a4])= (47r)n/2 [3gOax ..o, oro-~-~6R'~aR.a+vA-sR2+sgrqR-gRX+sX - g O X + ~ W Wo,#_ •

(25)

The relevance of the De Witt-Seeley-Gilkey coefficients is well known, both in geometry [4,6,9] and in theoretical physics [ 1-3 ].

We now apply the method presented to compute the coefficients for the fourth-order operators of the form

ZJ: [~21t- VltUVuVu + NuV u + X . (26)

Such operators arise, for instance, while studying quantum gravity with quadratic in curvature terms in a lagran- gian [ 13 ]. This modified gravity is regarded now as promising candidate for a quantum gravity because it is renormalizable [ 14 ] and exhibits asymptotic freedom [ 13,15 ]. Writing down the corresponding recursion relations and solving them we obtain the following expressions for the coincidence limits [am] (one can find a more detailed consideration in ref. [ 16 ] ):

1 [fro] = [a] = (k2)2_2,

[a2] = 2 [ a ] 2 ( 1 - 4 [ a ] (k 2 ) 2)kuk"(l'~,~u.+lu'~,~.) + [al2k~k"Vu.,

[a4]=4[a]3k"kak~ka{ (1-12[a](k2)Z +16[a]Z(kZ)4)F,~a + l V,~V~}

+2 [a]3kZk'~ka{4( 1 - 2 [a] (k z) ~)G.a + 4( Va~+ V~a)I.r+ [] V.a + 2V~Na + ( V.~+ Vr.) (l°~a+Ir%)

+ (Va;+ V ;~) (/.aho +/~zoa) }_ [a]~{ ( l _ 4 [ a ] (k~) 2)I.'~aa+ V'~aI,~a+X}. (27)

To obtain the coefficients Em we integrate over 2 and k by means of formulae ( 19 ) and (20). We find

1 F ( ( n - 2 ) / 4 ) ( 1 V) (28) Ez(x lA)=J([az] ) - (4n) . /22F((n_2)/2) ~R+~n ,

237


E4(xI,J)=J( [a4] )

1 F((n+4) /4) {(n_Z)[9~R,PY~R,~p~_~R,~PR,~+~6R2+~VqR+~W~W,~p ] = (4~r) n / ~ 2 F ( ( n + 2 ) / 2 )

n + 4 2 n + l 1 1 + 6 ( n + 2~ [] V - ~ n+~ V~VP V~,~) + 4 ( n + 2~ VZ+ 2 ( n + 2 ~ V~"~)V~,~) + ~ VR- ½ V~"~)R,p

- V t-a~ W ~ + WN~ - 2 X } , (29)

V(~P)=½(V"~+VP~), Vt"~=½(V~P-VP"), V=V%.

We note that as compared to the coefficients for the second order operators those for the four th-order ones (and higher) imply fairly nontr ivial dependence on the space d imens ion n.

The coefficients (28) , (29) for the opera tor (26) were also calculated in refs. [ 7,15,17,18 ]. In ref. [ 17 ] they were calculated for arbi t rary space dimension, and in refs. [ 7,15,18 ] for the d imension n = 4. However, in refs. [ 17,18 ] there are no total der ivat ive terms and in ref. [ 15 ] these terms are given with wrong numerical values. The expressions (28) , (29) , obta ined in the present paper , for the coefficients E2 and E4 for the four th-order opera tor (26) are most complete among those cited. In four d imensions they are consistent with analogous expressions in ref. [ 7 ] for the case o f V u" being symmetric .

By the method proposed it is possible to compute the next coefficients in the heat kernel expansion, but the terms get compl ica ted as one goes further and computa t ions by hand become impractical .

The great advantage o f the suggested method is its algori thmic character which allows one to perform all calculations by computer . Using an analytical calculation system ( " H E C A S " ) [ 19 ] a program for comput ing the coefficients Em was developed for any order min imal operators with pr incipal symbols being the scalar values. In part icular, the coefficients Ez, E4, E6 were computed for the second-order operator. The results o f computa t ions are prepared for publicat ion.

The method can be also general ized to the case of manifolds with torsion, nonmin imal differential operators and supersymmetr ic operators.

The author is grateful to I.L. Buchbinder , P.I. Fomin, V.A. Miransky, V.V. Roman 'kov and Yu.A. Sitenko for fruitful discussions and valuable remarks.

References

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L.S. Brown, Phys. Rev. D 15 (1977) 1469; J.S. Dowker and R. Critchley, Phys. Rev. D 16 (1977) 3390; S.M. Christensen and M.J. Duff, Nucl. Phys. B 154 (1979) 301.

[4] M.A. Atiyah, R. Bott and V.K. Patodi, Invent. Math. 19 (1973) 279; P.B. Gilkey, The index theorem and the heat equation (Publish or Perish, Boston, 1974); V.N. Romanov and A.S. Schwarz, Teor. Mat. Fiz. 41 (1979) 190.

[ 5 ] R.I. Nepomechie, Ann. Phys. 158 (1985) 67; I.L. Buchbinder, V.P. Gusynin and P.I. Fomin, Yad. Fiz. 44 (1986) 828; V.P. Gusynin and V.V. Romankov, Yad. Fiz. 46 (1987 ) 1832; S. Blau, M. Visser and A. Wipf, Phys. Lett. B 209 (1988) 209.

[6] P.B. Gilkey, J. Diff. Geom. 10 (1975) 601.

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[7] H.W. Lee and P.Y. Pak, Phys. Rev. D 33 (1986) 1012; H.W. Lee, P.Y. Pak and H.K. Shin, Phys. Rev. D 35 (1987) 2240.

[8] R. Endo, Prog. Theor. Phys. 71 (1984) 1366. [9] R.T. Seeley, Proc. Symp. Pure Math., Amer. Math. Soc. 10 (1967) 288.

[ 10 ] M.A. Schubin, Pseudodifferential operators and spectral theory (Nauka, Moscow, 1978 ); E. Treves, Introduction to pseudodifferential and Fourier integral operators, Vols. I, II (Plenum, New York, 1982).

[ 11 ] Yu.N. Obukhov, Nucl. Phys. B 212 (1983) 237; G. Cognola and S. Zerbini, Phys. Lett. B 195 (1987) 435.

[ 12] H.A. Widom, in: Topics in functional analysis, eds. I. Gohberg and M. Kac (Academic Press, New York, 1978); H.A. Widom, Bull. Sci. Math. 104 (1980) 19.

[ 13 ] J. Julve and M. Tonin, Nuovo Cimento B 46 (1978) 137. [14] K. Stelle, Phys. Rev. D 16 (1977) 953. [ 15 ] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. B 104 ( 1981 ) 377; Nucl. Phys. B 201 ( 1982 ) 469. [ 16 ] V.P. Gusynin, Seeley-Gilkey coefficients for the fourth-order operators on a riemannian manifold, preprint ITP-88-156E (Kiev,

1989). [ 17] P.B. Gilkey, Duke Math. J. 47 (1980) 511. [ 18 ] A.O. Barvinsky and G.A. Vilkovisky, Phys. Rep. 119 ( 1985 ) 1. [ 19 ] S.N. Grudtsin and V.N. Larin, Analytical calculation system HECAS, Input language description, IHEP preprint 87-26 (Serpukhov,

1987).

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New algorithm for computing the coefficients in the heat kernel expansion