Dilation shift under duality and torsion of elliptic complex

NUCLEAR

P HY S I C S B

Nuclear Physics B399 (1993) 691—708 North-Holland

Dilation shift under duality and torsion of elliptic complex A.S. Schwarz

1

Department of Mathematics, University of California, Davis, CA 95616, USA

A.A. Tseytlin

2,*

TheoreticalPhysics Group, Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received 2 October 1992 (Revised 4 February 1993) Accepted for publication 4 February 1993

We observe that the ratio of determinants of 2d laplacians which appear in the duality transformation relating two sigma models with abelian isometrics can be represented as a torsion of an elliptic (DeRham) complex. As a result, this ratio can be computed exactly and is given by the exponential of local functional of 2d metric and target space metric. In this way the well-known dilation shift under duality is reproduced. We also present the exact computation of the determinant which appears in the duality transformation in the path integral.

1. Introduction Duality transformations in two-dimensional sigma models with abelian isometries have recently attracted much attention in connection with string theory (see e.g. refs. [1—41and references there) As is well known [1,6,3], the duality transformation which inverts the metric in the direction of an isometry should also include a shift of the dilation field (coupled to the curvature of the 2d metric). Let us consider a sigma model *•

1= ~fd2z~{(G~

+B~)(g~+ ~~)ôaX~

ôbf

+R~1,

(1)

which is invariant under a global isometry ~xI~ = eK~L.Then K~must be a Killing 1

2 *

*

E-mail: [email protected]. E-mail: [email protected] or [email protected]. On leave of absence from the Department of Theoretical Physics, P.N. Lebedev Physics Institute, Moscow 117924, Russian Federation. Present address: Theory Division, CERN, CH-1211 Geneve 23, Switzerland. For duality transformations in the non-abelian case see e.g. ref. [5].

0550-3213/93/$06.00 © 1993

—

Elsevier Science Publishers B.V. All rights reserved

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Dilaton shift

vector of G1~~ and the Lie derivatives of B~(and 4i) must vanish up to a gauge 1 y, transformation In general, can choose local coordinates (x~) is{xshifted x’} such that ~ term. B,~and fr are one independent of the coordinate y which =

by the isometry. Then the action (1) is at most quadratic in y(z). To obtain the dual model one replaces ~3aY by a “momentum” field Pa adding the constraint term 9~ab(apb aaPb), where 9 is a Lagrange multiplier. By formally doing the gaussian integral over Pa one finds an action (dual action I) for {j~L} (~1 9 x’} which has the form (1) with —

G

11 =A4~, G11

B11

=

]W~‘Ga,

=

)W’B1~, G~ G1~ =

B1~ B1~ ill =

—

—

(B1iGIJ

—

—

B!IBIJ),

GI~B1J), M

G11.

(2)

The integral Pa is,function however, ill-defined given that the coefficient M(x) of the 2 term is justover a local and not an elliptic differential operator. Therefore pone cannot assert that the partition functions corresponding to action functionals I and I are equal. However one may expect that the ratio of these partition functions should be given by the exponential of a local functional M= ~fd2z~(coA

+c

2A. 1 aaA

8~A+c2RA),

(3)

M~e

The coefficients c,~should depend on a particular way of defining the integral over Pa~The definition suggested in ref. [1] was based on changing the variables Pa “ôa~1 +Eabôb~2,

(4)

and computing the determinant of the corresponding second order differential operator Q defined on the scalar fields f”. The part of this determinant which depends on the conformal factor of the 2d metric gab is determined by the Weyl anomaly and can be easily extracted [1,3]. As a result, one finds that c 2 in (3) is equal to 1, implying that the dilation should be shifted under the duality by flog M. There remains, however, the question about other possible terms in the determinant of Q which are Weyl invariant but M-dependent. It is not clear why such terms in det Q should be local. This question was not resolved in refs. [1,3] We shall present the exact computation of log det Q in Appendix A and show that it is given by a local expression (3) with non-vanishing c1. The 3aA aaA term coming from log det Q is cancelled by the contribution of another determinant (ignored in ref. [1]) which appears in the process of duality transformation in the path integral. —

—

*

*

It was noted in ref. [3] that det Q may be exactly computable using the standard variational argument but a consistent derivation was not carried Out.

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693

In general, one would like to fix the structure of finite local counterterms (3) in such a way that the original and dual sigma models remain equivalent at the quantum level. The necessary condition for the equivalence is that (3) should compensate for the non-trivial ratio of the determinants which appear as a result of integrating over y and 9 in the original and dual theories,

f [dy] exp[_fd2zV~M(x)aaya~~yI f [d9]

(5)

exp{_fd2zv~M1(x)a~93a9]

Ignoring the zero mode factors (6) Z= [det 40]~”2[det _]1/2 However, since each of the determinants in (6) is a complicated non-local functional of the function M it is not a priori clear why the logarithm of their ratio (6) should have a local form. In sect. 3 below we shall prove that (6) is, in fact, given by a local expression which structure in general depends on choices of the scalar products in the corresponding spaces of y and 9. For the natural choice of scalar products we will find that z=exp(~’_Id2zv~~Rlog M)~

(7)

implying that the dilation is shifted under the duality transformation [1] çb=çb—~1ogM.

(2’)

The proof will be based on the interpretation of the partition function (5), (6) as the torsion of the elliptic complex of exterior differentials acting on forms (the DeRham complex) in d 2 in the case of non-trivial scalar products in the corresponding vector spaces. The result may be considered as a generalisation of the standard expression which gives the index of the DeRham complex in terms of the Euler number (see e.g. ref. [7]). =

2. Basic definitions and relations We shall start with summarising some relevant information (following ref. [8]) about a special combination of determinants of elliptic operators which is known as the torsion of an elliptic complex. Let us consider a complex (E,, 1) (i 0, 1, =

N; E 1

=

EN÷l 0), i.e. a sequence of vector spaces E1 and linear operators =

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T1 acting from the space E1 to the space E1~1and satisfying 7~~1 0. Let K, )~ denote a scalar product in E. and define the adjoint operators T,’~’:E~~1 E1 by (a, 1b)~±1 (Tj*a,b)j. If the self-adjoint operators 4~:E1 E1, =

—*

=

—

(8)

are elliptic differential operators the complex (E1, T1.) is called an elliptic complex. The torsion Z(E1, T1) of the elliptic complex (E1, 1) is given by the formula *

N

log Z(E,, T,)

=

~ (— 1)’(i + 1) log det li~,

(9)

i=0

where the regularised determinants of operators ii~ are defined by means of i-function. Z can be identified with the partition function of a degenerate functional (associated with (E1, T,)) with zero action [8]. Using that T~~1T.=0,det z.1~=det(T*T1)det(T~1T~1), det(T,T,*) =det(T1*T,), (10) one can represent Z in the form

z

=

~(det

T*T)~1)’~

(11)

Since T,’~’depend on a definition of the scalar products in E,, Z changes under a variation of the scalar products. If one redefines the scalar products by inserting the operators M1: E1 E~, —*

(,)=(M~,~1, T.*P=M_lT*M.+1.

(12)

As was shown in ref. [8], the variation of the partition function (9), (11) under the variation of the scalar products in E1 can be expressed in terms of the Seeley coefficients (i.e. local functionals appearing in the t — 0 asymptotics of exp( — and the zero modes of 4~.If ~A~=~M’

5M~

are the operators describing an infinitesimal change of the scalar products ~ log Z= ~(—1)’{~P0(8AJli1)

* **

—P(~A~I4~)].

~,

(13)

Note that we have changed the notation as compared to ref. [8]: our 7~is TN, of ref. [8]. The proof of this formula is based on representing the determinants in terms of proper-time integrals of the heat kernels and relating the variations of heat kernels corresponding to different operators.

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Here ~P,Jis the coefficient of t°in the asymptotic expansion Tr(~A~ eta)

~V’k(8AjIlij)t,

(14)

k

and P(~A~ I

~ (6~ f~(n),f~(n))~,

=

(15)

where f~ is a basis in the kernel of li,. Let us now specialise to the case when (E,, 7~)is the DeRham complex, i.e. when E, are the spaces of i-forms on a compact riemannian space 4~N with the metric gab and T1 d, are exterior differentials. The torsion (9), (11) can be considered as the partition function of the quantum theory of the antisymmetric tensor of rank N which has zero action [8—11].We are interested in the generalisation of the standard discussion to the case when the scalar products in E1 are non-trivial, namely contain additional scalar functions M1( z), =

(w,cr)~= ~

...

(16)

~

Then according to (12), (8) —

1 j

T*_5J

,J*.j,f i i+1’

li~=M[1 d7M 1±1d~+d~.1M1j1 d~1M~.

(18)

When M1 1 the partition function Z (11) is known as the Ray—Singer torsion tor4’ of the manifold 4~N [9,11] the variation of which under a change of the metric gab can be expressed in terms of the “anomalies” and zero modes of zi~ (tor%’ is equal to 1 if N is even). Note that det li~~ det liNt when M, + 1. The variation of Z (13) under a variation of M1 takes the form =

~ log Z

=

~

(— 1)1 (JdNz 5A~bN(z I lii)

—

~ (~A~ f~(n),f~(~))t),

2Ai. (19) M~~e Here bN(z I li~)is the local Seeley coefficient (integrand of 1Jt~)of the laplacian 4. and f~(Pz) are the zero modes of li~(for a discussion of the zero mode contribution see ref. [11]). Note that for M, 1 =

~(

_1)tf dNz bN(z I 4~)

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is the index of the DeRham complex which by index theorem is equal to the Euler number x of 4N Eq. (19) then implies that (up to the contribution of the zero modes) the power of the common constant scale of M, in Z is equal to ~A.’•If bN(z I 4,) and ~ are known explicitly, eq. (19) gives a system of functional equations ~ log Z ~ =F(Al,...,AN;

gab)

which, in principle, can be integrated to find the dependence of Z on M,..

3. Torsion of generalised DeRham complex in two dimensions The case of our interest is when the complex is defined on a two-dimensional Riemann space, i.e. the simplest non-trivial case of the above construction (N 2), =

0

—~

E0

E~ E~—*0.

--*

—*

d0 acts from the space of scalars E0 to the space of vectors (1-forms) E1 and d1 acts from E1 to the space of 2-forms E2. In two dimensions 2-forms can be identified with (“dual”) scalars (Wab EabW). In the present case =

=

T0* T0,

~

=

T1* T1

+

TOTO*,

~2

=

T1T1*.

The corresponding scalar products and laplacians are given by (16), (18) (w, o~ fd2z~/~Mowo~,(w, =

(to, 0>2

=

0)1

=

~

(20)

~

1d~’M 40=M~’ d~M1d0, 41=M~ 4~= —M45’ Va(M1Va), 4Iab

=

2d1+d0M1~’d~’M1, 42=d1M1’ d1~’M2, —V’~(Mj~ V~M2),

~2

—Mj~ VC[M

2(g~V —g~bV~)] Va(M~ VbMl). (21) 14S, where S is an operator of Since the determinants are invariant under 4 S also in the equivalent form multiplication by a function, ~2 can be represented —

—*

~2

_M 2Va(Mj~V~.,).

(21’)

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The partition function Z in (9), (11) is given by (for a moment we are ignoring possible zero mode contributions) Z=(det 40)”2(det 41)1(det 42)3~/2 (det ~2) —1/2 det 4~(det 4~)—3/2

(22)

=

Using the relation (10) (which is a consequence of T,~T, 0, i.e. of “d2 have =

=

0”) we

det 4~ det(T~T =

1)det(T0T~) det(T1T~)det(T~’T0) det =

~2

=

det 4~.

As a result, we can represent Z (22) in the form 2(det 42)1~~~2. (23) Z=(det 40)” Comparing (22), (23) with (5), (6) we conclude that Z in (23) gives a well-defined expression for the ratio of the partition functions of the two dual scalar theories in the general case of arbitrary functions in the scalar products (M M 1). Our aim will be to compute Z explicitly as a function of M,(z) and gab using the variational relation (19). Let us first ignore the contribution of the zero modes. Then (19) takes the form =

~ log Z

=

fd2z[~A0

b2(z I 4~) ~A1b2(z —

I~

+

6A2 b2(z

I ~2)1. (24)

2~Pgab can be The variationasofa particular Z under the Weyl rescaling of the metric represented case of (24) with 6A 0 3p, ~ =

~ log Z

=

fd2z

~p[b2(z

I 4~) b2(z I —

~gab =

=

0, 6A2

42)].

=

—op, i.e. (25)

This expression, being the difference of the conformal anomalies of the scalar theory and its dual, is also an obvious consequence of (23). We shall use the following standard result for the Seeley coefficients of the Laplace-type operators (21) in two dimensions [12]. Consider the elliptic differential operator 4

=

Igab~i91,— 2Aa3a +

Y,

where a, b 1, 2, g’~’(z)is positive definite, 2I isThen the n and Y are n x n matrix valued functions in R =

(26) Xn

identity matrix and A~

b

2(zIli)

=

~

Tr(~RI—~Aa_g~~AaAb_y),

(27)

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where gab is the inverse of grn’, R is the curvature of gab and VaAa (1/ ~ Aa). The operator (26) can be represented also as _gabD~D~+x,

4

=

(28)

where the covariant derivative Da contains both the Christoffel and A a connection terms. The equivalent form of (27) is *

b2(zI4)=~V~Tr(~IR-X).

(29)

The operators 4~and ~2 in (21) can be represented in the form (26) (with the metric rescaled by M1/M0 and M2/M1). Applying (27) and returning back to the original metric in (21) we find b2(z I 4~) ~—~/~[~R

b2(z

I ~2)

2(A +

~V

+

~V

1 A0) —

—

2A V 1

—

B~A1a’~A1], M~ ~

2(A =

~—ä[~R

2A 2 —A1) + V 1 ~~3aAiôaA]

(30)

(31)

Substituting (30), (31) into (25) and integrating the resulting equation we find the dependence of Z on the metric gab **

log Z=~fd2z%/~[RAl+~R(AO+A2_2Al)J +O(A0, A1’ A2).

(32)

For the case of equal M1 (A, A) this expression was found in ref. [3]. To determine the A,-dependent terms in (32) we need to integrate eq. (24). Since for general M, the vector operator 4~in (21) is not of the type (26) we shall do this in two steps. First, we substitute (30) and (31) into (24) and integrate over A0 and A2. Taking into account (32) we get =

log Z

=

~

~R(A0

A2

—

2A1)

1—A0—A2)

—

(A0—A2)V

+

+

2(2A +~(A0—A2)V +

0(A1).

2A 1

—

(A0+A2)3aAi 3aAj (33)

1~lS,where S is the operator of * **

Note that b2 isbyinvariant under the similarity transformation ~i —‘ S multiplication a function. We consider the case of the spherical topology so the dependence on the metric is determined by the dependence on its conformal factor.

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To determine the terms which depend only on A1 we consider the special case when all A, are equal to A. The ~1 in (21) takes the form 4lab

=

—e2’>’

=

—gabV +Rab

~Ce2A(g

bV

—

—

g~bV~) —

Vae_2A Vbe2A

2g~~dCA V~— 2VaV7bA.

(34)

Now (26)—(29) can be applied and one finds b

(35)

2(ZI4l)IA,A=~__%/~GR_R_2aaABaA). Substituting (30), (31) and (35) into (24) and assuming 6A, log Z

I

=

=

i3A we get (36)

~_Jd2z~RA.

The origin of the “anomalous” term (36) can be attributed to the presence of the Rab term in the vector operator (34) (which gives the —R contribution to (35)). Comparing (33) to (36) we determine that the 0(A1) term in (33) must be 3aAi 2A1 t 8a1~1 i.e. the final expression for Z (up to zero mode factors) is log Z

=

~

~R(A0 +A2

+

—

2A1)

—

(A0

2(2A +

~(A0

—

A2) V

2AI 1

—

A0

—

A2)

—

A2)V

— (A

0 + A2

—

2A1)aaAi 8~lA1)}.

(37)

It is easy to check that the derivative of (37) with respect to A1 is equal to (35) for A A. Note that in agreement with the index theorem the common constant scale of M~appears in log Z with the coefficient equal to one half of the Euler number. Let us now include the contribution of the zero modes. We shall assume the spherical topology so that ~1 does not have zero modes. The normalised zero modes of ~ and ~2 are =

f0=

(i

—1/2 d2zV~Mo)

—1/2

f2=M~1(fd2zv1~M~1)

.

(38)

Including the corresponding zero mode terms (see (19)) into (24) and integrating over A0 and A2 we get the following additional terms in Z (23): (log Z)zm

~

log(fd2zV~Mo)+ ~ log(fd2z~/~M~1).

(39)

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Dilaton shift

As a result, we can represent (23) in the form —1/2

1/2

z=z~(fd2Z~1~M0) (Jd2Z~M2_1)

(40)

,

where Z’ is the local expression given by (37). The case of equal functions M, is the one which is relevant for the duality transformation problem discussed at the beginning of the paper. In fact, given the sigma model (1) it is natural to define the scalar product in the tangent space at x~ as (i~x,6x’)

=

fd2z~~G~(x)6x~ t5x~v

=fd2z1/~M(x)~y

The scalar product corresponding to the dual (&~,~if)=fd2z~i

6y’+

sigma

...

(41)

.

model (2) is then

G~(x)&~~5j~’

=fd2z~M1(x)~9

~9’+

...

(42)

.

The scalar product on scalars 5y is thus determined by the function M0 M. As it is clear from the structure of the quadratic functions (actions) in (5), to be able to identify M with the scalar product function in the space of vectors we need to rescale 9 by a factor of M, i.e. we should identify M 19 with the 2-forms which appeared in the above discussion (so that 9 const. corresponds to the zero mode, etc). Then the scalar product on the 2-forms is also defined by M (i.e. all M~.are equal to M) and the operators 4 and D in (6) are equivalent to 4~and ~2• For M, M eq. (40) takes the form (see (36)) =

=

=

1~2

1/2

(fd2z~M) =

(det

~0Y

exp(~ Jd2z~R log M)(Jd2z~M1)

(det

42)_1/2.

(43)

Note that if M const. it drops out of (43) as it should since in the case 4~and ~2 in (21) are equal and independent of M (we have assumed that the 2-space has the topology of a sphere, i.e. the Euler number is two). =

AS. Schwarz, A.A. Tseytlin / Dilaton shift

701

As a result, we have proved that up to the ratio of the zero mode factors 1/2

fdy

(Id2zV~M)

I d9(Jd2zv1~M1)

(44)

1/2

(5) is indeed given by the local expression (7).

4. Concluding remarks In conclusion, let us make several comments. Eqs. (5), (7), (43) have a straightforward generalisation to the case when the sigma model (1) has a number of commuting isometries, i.e. when y and 9 have an additional index s. Then G5~ ~ is a matrix depending on the rest of the fields x’(z) and M in (7), (43) should be replaced by det M. The relation (43) proved above may be useful in trying to clarify the issue of modification of the leading order duality transformations (2), (2’) by higher loop effects [3]. Eqs. (33) and (43) may be of interest also from a mathematical point of view giving the explicit dependence of the torsion of the N 2 DeRham complex on the scalar products and providing the “M + const.” generalisation of the corresponding index theorem. It may be of interest to study higher dimensional analogs of (33), (43) (in particular, in connection with possible higher dimensional analogs of sigma model duality). The torsion (22), (23) can be interpreted as the partition function of the theory of a rank 2 antisymmetric tensor in two dimensions [8,11] (i.e. of a “topological” theory with zero action). What we have found is that if non-constant functions are included in the definition of the scalar products the resulting “effective action” (36), (37) is a local action of 2d gravity coupled to scalar(s). Thus the 2d scalar—tensor gravity appears as an “induced” theory. The methods of ref. [8] and of the present paper may find applications in other contexts (e.g. in gauged WZW theory) where a careful account of the dependence on the definition of the scalar products is important. The resulting ambiguity in a choice of local counterterms should be fixed by additional conditions depending on a particular theory. =

We are grateful to the Aspen Center for Physics for the hospitality while part of this work was done. A.T. would like to thank R. Metsaev for checking the perturbative calculation described after (A.18). A.S. was partially supported by NSF grant No. DMS-9201366. A.T. would like to acknowledge Trinity College, Cambridge and SERC for support.

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Appendix A. Integration over “non-dynamical” vector field and the determinant of the operator Q 1. Let us consider the following integral over the 2d vector field Z1 =f[dPa] exp(_~fd2zV~Ml(z)g~thpapb),

pa:

(A.1)

where M1 is a given function. We are assuming that the measure (scalar product) for Pa IS trivial (if M1 is present also in the scalar product it is natural to set Z1 1). This integral is not well-defined. Using different definitions (regularisations) of (A.1) one will get expressions which will differ by local (finite) counterterms. A choice of a particular definition is dictated by some additional conditions which the total theory (where (A.1) appears at some intermediate step) should satisfy. + 2Rab) is a Laplace operator on vecGiven that ~ d~’d1 + d0d~’ (—g~~V tors one may define (A.1), for example, as (cf. (34), (35)) =

=

=

=

exp[

—

~Tr(log M 1 e~1)]

=

ex~[_ ~

M1(_

+

~R

_R)1.

(A.2)

It is more natural, however, to use the following general idea: if M is a “bad” operator one can define its determinant by introducing an auxiliary operator P such that P *Mp and P ~ are “good” operators and setting det P*MP det M~ det p*p

(A.3)

The result will, in general, depend on a choice of P. However, the dependence on P can be calculated; see ref. [8]. Changing the variables from Pa to a pair of scalar fields f”, 1’ ab~ 1~anSe~~, (A.4) Pa

=

‘~a~’ + Ca

one can represent (A.1) as

2, J=[ZQ(Ml==1)]1, (A.5) ZQ~(det Q)” where J is the jacobian of the transformation and Q is the following laplacian defined on a pair of scalars: Z

1=JZQ,

Q~PtP, Pt~M~~P*M 1,

(A.6)

A.S. Schwarz, A.A. Tseytlin / Dilaton shift Qnm

=

703

_M1~lonmVa(Miaa) nmE”M~ aaMi~1’ —

2 =

—

2(~nmg’~’+ EnmEab)i3aAlt3b],

(A.7)

e2~. e~1~o)[ ~nmV’

Here p*nap

=

(~V~,, —~‘V~p~,),

(A.8)

and we have assumed that the scalar product in the space of ~ is defined with an extra function M 0(z) (ci. (20)). Since Q is a 2d scalar operator the dependence of ZQ on the conformal factor p of the 2-metric is determined by the (Weyl) anomaly. In view of the “product” structure of Q the same is effectively true also for the dependence on M0 and M1. Computing the variations of det Q with respect to M, and p we get (cf. (24), (25)) ~ log ZQ=fd2z[(i5p+oAO)b2(zIQ) Here the operator

Q

—6A1 b2(zIQ)].

(A.9)

or, equivalently,

=

Q =M1PM~P”,

(A.10)

is a laplacian acting on vectors, —M1 VCM

Qab

gaiY

e2I~o)[

=

~

VaMi~’Vb + Ra1’ + 2(g~~gC~i + Eab’~)dCAOVd].

(A.11)

Note that det Q det Q and that the structure of the operator Q is different from the structure of ~ in (21). The Seeley coefficients b2(z I Q) and b2(z I ~) are found using (26), (27) (ci. (30), (31), (34)) =

2(A

b2(zIQ)

=

~—~I~[~R+ ~V

1 —A0) _2V2A1J,

b2(zj~)~_v~[kR_R+~V2(Ai_Ao)+2V2Ao].

(A.12)

(A.13)

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Integrating (A.9) we get (cf. (37)) log ZQ(A0, A1)

=

~Id2V~[_~Rv_2R+RAl 2(A —

log Z1

~-R(A1

A0)

—

=

log ZQ(AO, A1)

=

~fd2zv~{RAi

—

—

~-(A0 A1) V

2A 0

—

—

A1)

—

2A0V

1J, (A.14)

—

A1)

—

2A0V2A1].

log ZQ(A0, 0) 2(2A ~RA1 + 4A1V

—

0

(A.15)

In the special case when M0 M1 M (i.e. A0 A1 A), which is relevant for the discussion of the duality transformation in the path integral, one has =

=

=

2(8~~g~th + EnmEal3)ôaAô1’,

Qnm

=

~nm172

Qab

=

—gabV

log ZQIA.A

=

~Id2[~21~1~A

—

=

+ Rab +

2(ga1’g~+ abE~’)t9CAVd, +28aA oaAJ.

(A.16) (A.17) (A.18)

It is possible to check the presence of the aaA 3a)~ term in log ZQ by an explicit perturbative calculation. Changing the variables from in (A.4) to ~“

=

for which the scalar product will be A-independent, one can represent ZQ as a path integral with the following action (ci. (A.16)): 3aUn + 2fnmEab 3AU~ ôaUm

1=

~Id2zV~1a~~uP1 + aaA 9aA)UnUI

or

1=

4Id2zV~Dau12Du~—Fu”u~),

DaUn=8aUn+A~mUm,

A~m=EflmEababA,

F

E~thEnmF~,m= —V2A.

AS. Schwarz, A.A. Tseytlin / Dilaton shift

705

Computing the 0(A2) term in the effective action on a flat background using dimensional regularisation one finds the finite Schwinger-type term *

log ZQ

1

2

__fd2z(_Tr(A~~)

=

+

(A±is the transverse part of A,,). This result is in agreement with (A.18). 2. The operator equivalent to Q with M 02z%/~Rlog M1 M was considered in refs.of[1,3] M in the logarithm its and the presence of the term (1/8~)f d determinant was established by computing the conformal anomaly. We have found that the complete expression for log det Q (A.18) contains also the additional M-dependent term =

~fd2z~~a~(log

=

M)0a(log M).

(A.19)

By formally doing the duality transformation in the path integral one could expect [1,3] that the ratio (5), (6) should be equal to Z 1 (A.1) or to ZQ. Comparing (37), (43) with (A.14), (A.15), (A.18) we conclude that in fact this cannot be true since the term (A.19) in (A.18) cannot be present in the logarithm of (5) (the latter must change sign under M—sM~). This apparent contradiction is resolved by discovering that the actual relation between the torsion (5), (23) and ZQ contains also a contribution of another determinant of second order elliptic operator. To derive the exact form of this relation let us repeat the standard steps corresponding to the duality transformation at the path integral level Let us start with the following path integral: ~

1=f[dPa][d91

(A.20)

exp(_fd2z~(~M(z)gabpap1’+ieabpad1’9)).

1”2

Integrating Pa integrates we get thefirst path integral the dual Z—(det ~0Ythat (cf. (5), (6)).over If one over 9 one of obtains the theory s-function implying Pa 8aY’ i.e. the path integral becomes that of the original theory, i.e. Z (det 1/2 There is, however, an additional determinant which appears from =

~

*

It is easy to check that UV and IR divergences cancel separately so that one may set the massless tadpole equal to zero and use the standard integrals ,.

d~Jpp~p~,

J

**

1 =—

(2ir)~~p2(p — q)2

4(d

1)

d’~p

J=1

(q2ôb—dqqb)J ‘

J

i =

(

2

~r(d —2)

2~.)d~2(~ — q) as well as E”E*d = It should be noted that the discussion which follows is based on operations with ill-defined integrals and thus is not rigorous.

706

A.S. Schwarz, A.A. Tseytlin

/

Dilaton shift

the s-function. To give a precise sense to the above relations let us first change the variables as in (A.4), Pa

(A.21)

3aY + Cab i3bY,

exp(

Z=Jf[dy][d7][d91

_Id2Z~[~M(Z)(3ay

+ ~,,b ~)2

+

id~ ôa91). (A.22)

Here J is the jacobian corresponding to (A.21) (see (A.5)). In the context of the sigma model duality transformation we should assume that the scalar products in the spaces of y and 7 are defined with the function M 0 M while the scalar product in the space of 9 contains M’ (cf. (41), (42)). Integrating first over 7 and y and then over 9 we get (ci. (A.5)) =

*

Integrating first over multiplied by

9

2(det A 2. (A.23) 2=J(det Q)~’ 0)’~ we get the s-function factor which can be represented as

~(j3)

ZH=

I[d7][d9]

aa7 8a9)

exp(_iId2zv~

(det Hy’~2.

(A.24)

Then 2

=

J(det H) — 1”2(det 4)~ 1/2

(A.25)

Comparing (A.23) with (A.25) we find that the following relation must be true: det

Q

=

det H det 4 0(det

A0)

1,

(A.26)

i.e. Z

ZQ =

(A.27)

—.

ZH

Z and ZQ defined in (6), (23) and (A.5) (with M, M) were already computed in (36) and (A.18). According to (A.27) 2R+2aaA 3aA). (A.28) log ZH ~Id2zv~L~~? V =

*

To compute the integral over 53 and y one is to make a (non-local) shift of the fields which can be determined from the corresponding shift of p,, in (A.20).

AS. Schwarz, A.A. Tseytlin / Dilaton shift

707

3. It may be useful to give another (equivalent) definition of Q which makes the analogy with the discussion in sects. 2,3 more transparent. Given the DeRham elliptic complex (E,, T,), i 0, 1, 2, one can define the operator =

P:E0~E2—E1, P(to0,w2)=T0w0+T~w2, which maps a scalar and a 2-form into a vector. Then P~w1 (T~’to1,T1to1),

pt : E1 —*E0 ~E2,

=

PtP:E0~E2~_sEo~E2, PtP(w0, w2)=(T0*Tow0,TiT~w2), ppt

=

=

TOTO* + T~T1.

Since T1T0 0, T~’ T~ 0 the operator PtP is diagonal in E0 ~ E2 and det P~P det ppt det ~ One can obtain a new non-diagonal elliptic second order operator on E0 ~ E2 by introducing a “twist” in the definition of P, =

=

=

=

P(w0,

to2)

=

T0to0 +A1T~A2w2,

(A.29)

where A, are self-adjoint operators in E, (e.g. multiplication by a function). In this case Q~ptp, Q

(we, to2)

=

(To*Tocoo + T0*A1T~A2w2,A2T1A~T~A2w2 +A2T1A1T0w0). (A.30)

Using the relation (17) between T, and the exterior differentials T0 =d0,

T1 =d1,

T~=M~’ d~’M1, T~=Mj’ d”M2,

and comparing (A.4), (A.7) with (A.29), (A.30) we conclude that they are equivalent if 1, M A1=M1, A2=M~ 0=M2, and if

to0 is

identified with ~ and the 2-form

to2

1d M~

with the scalar

~2

Explicitly,

1d(~’M 1~’M1d0 M1~

M~ d1M1d0

1d~’

M~d1M1d~’

,

(A.31)

708

AS. Schwarz, A.A. Tseytlin

/

Dilaton shift

and Q=M

1PPtM1I=M1

d~’M~d1+M1 d0M1~ld0*~.

(A.32)

Eq. (A.32) is equivalent to (A.11).

References [1] T.H. Buscher, Phys. Lett. B194 (1987) 59; B201 (1988) 466 [2] 5. Cecotti, S. Ferrara and L. Girardello, Nucl. Phys. B308 (1988) 436; A. Giveon, E. Rabinovici and G. Veneziano, NucI. Phys. B322 (1989) 167; G. Veneziano, Phys. Lett. B265 (1991) 287 [3] A.A. Tseytlin, Mod. Phys. Lett. A6 (1991) 1721 [4] M. Ro~ekand E. Verlinde, NucI. Phys. B373 (1992) 630; A. Giveon and M. Ro~ek,NucI. Phys. B380 (1992) 128 [51B.E. Fridling and A. Jevicki, Phys. Lett. B134 (1984) 70; E.S. Fradkin and A.A. Tseytlin, Ann. Phys. 162 (1985) 48 [61 P. Ginsparg and C. Vafa, NucI. Phys. B289 (1987) 414; T. Banks, M. Dine, H. Dijkstra and W. Fischler, Phys. Lett. B212 (1988) 466; E. Smith and J. Polchinski, Phys. Lett. B263 (1991) 59 [7] P. Gilkey, The index theorem and the heat equation (Publish or Perish, Boston, MA, 1974) [8] AS. Schwarz, Commun. Math. Phys. 67 (1979) 1 [9] A.S. Schwarz, Lett. Math. Phys. 2 (1978) 201, 247 [10] P.K. Townsend, Phys. Lett. B88 (1979) 97; W. Siegel, Phys. Lett. B93 (1979) 259; B138 (1980) 107; M.J. Duff and P. van Nieuwenhuizen, Phys. Lett. B94 (1980) 179 [11] AS. Schwarz and Yu.S. Tyupkin, NucI. Phys. B242 (1984) 436 [12] R.T. Seeley, Proc. Symp. Pure Math. 10 (1967) 288; P. Gilkey, Adv. Math. 10 (1973) 344; Proc. Symp. Pure Math. 27 (1975) 265; Compos. Math. 38 (1979) 201