Volume 213, number l PHYSICS LETTERS B 13 October 1988

CONTOUR INTEGRATION ON S U P E R - R I E M A N N S U R F A C E S

Alice R O G E R S Department of Mathematics, King's College, Strand, London WC2R 2LS, UK

Received 3 June 1988

Super contours are defined to be superconformal maps of real ( 1,1 )-dimensional superspace into complex ( 1,1 )-dimensional superspace. A method for integration along supercontours is defined in terms of a modified Berezin integral on ( 1,1 )-dimensional real superspace. The natural transformation properties of this modified Berezin integral are inherited by the contour integral, so that the contour integral of a half form on a general super-Riemann surface is defined.

Super -Riemann surfaces are complex supermani- folds which play the same role in the Polyakov quan- t isat ion of the spinning string that convent ional R iemann surfaces play in the Polyakov quant isa t ion of the bosonic string. Their geometric structure orig- inates in Howe's formula t ion of two-dimensional su- perconformal supergravi ty in (2 ,2 ) -d imens iona l real superspace [1 ]. Baranov and Schwarz [2] and Fr iedan [ 3 ] showed that the geometry of two-dimen- sional superconformal theories can natural ly be ex- pressed in terms of (1 ,1 ) -d imens iona l complex supergeometry. A detai led account of the geometry o f super -Riemann surfaces has been given by Crane and Rabin [4 ], and the extension to super -Riemann surfaces of many aspects of classical R iemann sur- faces has been considered in a number of papers, such as those by Batchelor and Bryant, Giddings and Nelson, Hodgkin, Lebrun and Rothstein, and by Teofil lato [5] , while many recent papers on super- string quant isa t ion use and develop the mathemat ics of super-Riemann surfaces ~. However, there is as yet no unambiguous answer obta ined to the Polyakov quant isa t ion o f the spinning string, and many un- solved problems remain in the theory of super -R iemann surfaces.

Contour integrat ion is a powerful tool in the study

Research supported by the Science and Engineering Research Council of Great Britain.

"~ A selection, unevitably incomplete, of papers in this field are listed in ref. [6].

of classical R iemann surfaces; it is impor tant , for in- stance, in the construct ion of Jacobi variet ies and theta functions. The purpose of this letter is to de- scribe what appears to be the appropr ia te generalisa- t ion of contour integration for super -Riemann surfaces, so that analogous construct ions can be car- r ied out on these surfaces.

A super -Riemann surface is a (1 ,1 ) -d imens iona l complex supermanifold whose local coordinates Z = (z, ~) change on overlapping coordinate regions by superconformal t ransformations, that is transfor- mat ions Z = (z, ~ ) ~ Z ' = (z ' , ~' ) where

z' =f (z )+¢g~(z) fx/f T ~ ,

{'= ~(z)+{x/f ' (z)+ TJ(z) 7u' (~, (I)

w i t h f a n d ~ b o t h analytic functions of the even vari- able z. The reason for this restr ict ion is that the dif- ferential opera tor

D: = c3/8~+ {Olaz (2)

then t ransforms covariantly, according to the rule

D=, = ( D : , ~ ) D = . (3)

Fr iedan [ 3 ] has defined a generalised contour inte- gral on such a supermanifold for closed contours by combining the Berezin integral [7] with s tandard contour integration. However this approach does not give coordinate invar iant results when integrating over open contours, because Berezin integrat ion does not lead to coordinate invar iant integrals when inte-

0370-2693 /88 /$ 03.50 © Elsevier Science Publishers B.V. ( Nor th -Hol land Physics Publishing Divis ion )

37

Volume 213, n u m b e r 1 P H Y S I C S L E T T E R S B 13 O c t o b e r 1988

grating over a finite region if the integrand does not z~ Z vanish at the boundary. Also, defining fz2f( ) dz d(

as a function of the endpoints, as in ref. [ 3 ], can lead to ambiguities.

Instead, in this letter, a full, unambiguous theory of integration over supercontours, which are maps of real (1,1)-dimensional superspace into a super- Riemann surface, is constructed. The approach uses a modified Berezin integral, which effectively allows the introduction of odd as well as even limits of in- tegration, and thus overcomes the difficulties with the boundary points. Just as classical contour integrals are defined as integrals on the real line, their super- generalisation will be defined as integrals in ( 1,1 )- dimensional real superspace, and we begin by defin- ing modified Berezin integrals in this space, extend- ing ideas first introduced in ref. [ 8 ] in the context of superspace path integration.

Real (1,1)-dimensional superspace will be de- noted RI'J and complex ( 1,1 )-dimensional super- space C ~'~. Thus ~'~ consists of pairs (t, r) , where t is an even element of A (~L) and z an odd element, and C 1, ~ is defined similarly in terms ofA ( C L ). (The value of L, the number of generators of the Grassmann al- gebras, is unimportant provided that it is greater than two. More detailed descriptions of these spaces may be found in ref. [ 9 ]. ) Lower case Roman and Greek letters will be used to denote even and odd variables respectively, while a capital letter will denote a pair of variables, one even and one odd.

Suppose that f is a function mapping ~'~ into A(~C), and that (a, a ) and (b, fl) are points in ~L~. Then the integral of f between limits (a, a ) and (b, fl) is defined to be

i iDTf(t ,Q: f dr(a+f"f(t,r) dt). (4) fl t, b + rfl

Here the integration on the right-hand side with re- spect to the odd variable z is a standard Berezin in- tegral, while the integral with respect to the even variable t needs further discussion because the end- points are not simply real numbers, but even ele- ments of the Grassmann algebra. Such integrals may be defined according to the following rule: h 0

ff(t) dt= ff(a+y(b-a))(b-a)dy, (5) a 1

where y is an ordinary real variable. This approach was first introduced by De Witt [ 10 ], and developed more fully in ref. [ 11 ]. Nilpotent shifts of limits of integration were introduced in the physics literature by Restuccia and Taylor [ 12 ]. The rule (5) allows one to express the even integral in (4) as

b+rf l a

f f(t) dt= ff(t)dt+rflf(b)-zaf(a). (6) a+ ra b

Two useful properties of the modified Berezin in- tegral (4) can be established. First there is the fun- damental theorem of ( 1,1 )-dimensional calculus,

iiDTf(t,z) DT=f(b, fl)-f(a,a), (7) f l b

where

D T = O / O Z + " c O l O r . ( 8 )

Also there is the transformation rule for change of variables in ~ ~'1. Suppose that a mapping

K : ~ l ' l - - - ~ 1"1 ,

(t, z)-* (k(t, r), x(t, r) )

is such that the even and odd components have the related form

k(t, z) = p ( t ) +rn(t) ~ (t) , x(t, z) = n ( t ) + z~/p' (t) + n ( t ) n' (t) , (9)

with p and n piecewise differentiable. (Such a map will be called real superconformal, by analogy with the condition ( 1 ). ) Then, if

K(a,a)=(c,?), K(b, fl)=(d,~), ~ S d f i b

y c oz a

(10)

This result is a consequence of (7), together with the following chain rule for super derivatives DT: sup- pose that f: ~ 1.~ ~A (R L) is differentiable and that K is the real superconformal map defined in (9). Then

DT (f(K(t, z) ) ) = jr(k(t, z), x(t, z) ) DT(X(t, Z) ) , (11)

38

Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

where

f( k, ~c)=DKf( k, t¢) . (12)

(A similar chain rule is valid when K maps ~1.] into R ~,1 and f m a p s ~L, into A(CL), when K maps ~1,~ into C ~,l a n d f m a p s C l'~ into A(CL), and also when K maps C ~'' into C 1,, a n d f m a p s C 1'1 into A (C L), pro- vided that in each case K is superconformal in the appropriate sense. This last result is crucial to the ex- istence of non-trivial super-Riemann surfaces, be- cause it ensures that the combinat ion of two superconformal transformations is also superconfor- mal. )

Equipped with this definition of integration on ~Ll, there is now a natural definition of contour integra- tion on C 1,1. Suppose that C :~ 1'1~C1"1 is real super- conformal, so that

C(t, z) = (c(t, z), y(t, z) ) ,

c(t, ~) = s ( t ) + w ( t ) ~ ( t ) ,

y(t, t) = a ( t ) + ~x/s' (t) +a(t) a' (t) . (13)

Also suppose that C(a, o~) = (p, ~r) and C(b, 13) = (r, p), and that f : C ~' J ~ A (C L). Then the integral o f f over the supercontour C is defined to be

f f(z, () DZ ¢'[p,r<r.l.,]

.8 t,

dcf

It is easily seen that this integral has two essential transformation properties. First, it is invariant under superconformal reparametrisation o f the curve C. That is, i fK i s an invertible real superconformal map ofN j'~ into N ~'1, then

f f ( z ,~ )DZ= f f ( z ,~ )DZ . (15) C[p,~;r,p] CoK[p,~;r,p]

This result follows from the change of variable rule (10) together with the chain rule ( 11 ) in the case where K maps D ~'1 into R I'1 and f maps D ~'1 into a(ct-).

Secondly, the integral has the following transfor- mation rule for change of coordinate in CLI: Suppose F: C ~'~ ~ C 1,1 is superconformal with

F(z , ( )=( f ( z ,¢ ) , tP(z,() ) . (16)

Then

f g(Z' ) DZ' FoC[p ' ,n ' ; r ' , p ' ]

= t- g ( F ( Z ) ) D : T D Z . (17) C[p,mr ,p]

This result follows from the chain rule ( l 1 ) in the case where K maps C ~'~ into C I'1 and f m a p s C 1'~ into A(CL).

On a super-Riemann surface one may define a half form as an object W which in local coordinates is written W( Z) DZ, with

W ( Z ) = W ( Z ' ) D ~ T , (18)

i fZ ' = F ( Z ) as in (16). The transformation rule (17) shows that a method of integrating half forms over supercontours on a super-Riemann surface is ob- tained naturally from the supercontour integral on C 1,1 defined in (14).

This contour integral obeys the following Cauchy theorem. Suppose that C is a closed supercontour in C LI and that Co is the projection of C onto the com- plex plane C under the natural projection ofC 1'1 onto C. Also suppose that f : C 1,1 ~ A (C L ) is analytic inside C (that is, at all points in C 1'1 which project down onto the interior of Co). Then

f f ( Z ) = O . (19) C

A simple integrable representation is possible for analytic functions o f C 1,1. First suppose that C is a closed supercontour in C 1'1 and (r ,p) a point in C 1,1, whose projection in C does not lie on the projection o f C in C. Then the winding number of C about the point (r,p) is defined to be

1 f ( ( -p) DZ (20) n(C;r,p)= ~ i ( z - r ) " C

A simple application of the Cauchy theorem then leads to

2hi f f ( z , ( ) ( ~ - P ) D Z (21) f(r,p) - n(C; r,p~) ~ - ~

C

i f f i s an analytic function o fC I'l. The key definition (14) o f contour integration in

C ~'1, together with the modified Berezin integral (4),

39

Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

thus appears to lead to a natural approach to contour integration on super Riemann surfaces. Investigation of the application of this form of integration to the theory of super Riemann surfaces is currently in progress.

References

[1] P.S. Howe, J. Phys. A 12 (1979) 393. [2] M. Baranov and A. Schwarz, Pis'ma Zh. Eksp. Teor. Fiz. 42

(1985) 340 [Soy. Phys. JETP Lett. 42 (1986) 419]; M. Baranov, I. Frolov and A. Schwarz, Teor. Mat. Fiz. 70 (1987) 92 [Theor. Math. Phys. 70 (1987) 64].

[3] D. Friedan, Notes on string theory and two-dimensional conformal field theory, in: Proc. Workshop on Unified string theories, eds. D. Gross and M. Green (World Scientific, Singapore, 1986).

[4] L. Crane and J. Rabin, Super Riemann surfaces: uniformi- zation and Teichm/Jller theory, Commun. Math. Phys., to be published.

[5] M. Batchelor and P. Bryant, Graded Riemann surfaces, Commun. Math. Phys., to be published; S.B. Giddings and P. Nelson, The geometry of super Rie- mann surfaces, Boston and Harvard preprint BUHEP-87- 31, HUTP-87/A070;

L. Hodgkin, Problems of fields on super Riemann surfaces, King's College London preprint ( 1988 ); C. Lebrun and M. Rothstein, Moduli of super Riemann sur- faces, IAS Princeton preprint; P. Teofillato, Line bundles and divisors on a super Riemann surface, King's College London preprint (1987).

[6] J.J. Attick, G. Moore and A. Sen, Catoptric tadpoles, pre- print SLAC-pub-4500, IASSNS-HEP-87/66; J.J. Attick, J. Rabin and A. Sen, Nucl. Phys. B 299 ( 1988 ) 279; M.A. Baranov and A. Schwarz, Intern. J. Mod. Phys. A 2 (1987) 1773; E. D'Hoker and D. Phong, Nucl. Phys. B 278 ( 1986 ) 225; G. Moore, P. Nelson and J. Polchinski, Phys. Len. B 169 (1986) 47; E. Verlinde and H. Verlinde, Phys. Lett. B 192 (1987) 95.

[7] F.A. Berezin, The method of second quantization (Aca- demic Press, New York, 1966).

[ 8 ] A. Rogers, Supersymmetric path integration, Phys. Lett. B 193 (1987) 48.

[9] A. Rogers, Commun. Math. Phys. 105 (1986) 375. [10]B.S. DeWitt, Supermanifolds (Cambridge U. P., Cam-

bridge, 1984). [ 11]V.S. Vladimirov and I.V. Volovicb, Teor. Mat. Fiz. 60

(1984) 169 [Theor. Math. Phys. 60 (1985) 743]; A. Rogers, J. Math. Phys. 26 (1985) 385.

[ 12] J. Hassoun, A. Restuccia and J.G. Taylor, J. Math. Phys. 26 (1985) 345.

40