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Line integrals on super-Riemann surfaces
Volume 206, number 2
PHYSICS LETTERS B
19 May 1988
LINE INTEGRALS ON SUPER-RIEMANN SURFACES I.N. M c A R T H U R Institut fiir Theoretische Physik, Universitdt Karlsruhe, D-7500 Karsruhe 1, Fed. Rep. Germany
Received 29 January 1988
A procedure for computing line integrals on super-Riemann surfaces, based on an analogue of the sheaf theoretic prescription of line integration on an ordinary Riemann surface, is presented.
1. Introduction There has been much progress recently in the analysis of two-dimensional field theories defined on a Riemann surface (or in the presence of a nontrivial gravitational background). If the theory is Weyl invariant, then it realizes the conformal structure of the underlying Riemann surface, and this is sufficient to allow the computation of exact Green functions (see, for example, ref. [ 1 ] ). The ability to do explicit calculations in string theory is one important application. It is therefore natural to attempt to make full use of the superconformal structure on super-Riemann surfaces, which is realized by super-Weyl invariant theories defined on a super-Riemann surface (or propagating in a non-trivial two-dimensional supergravity background). An important ingredient in the exploitation of the conformal structure on a Riemann surface is the ability to take line integrals of holomorphic (1, 0) forms (locally of the form ~o=(z) dz), and the existence of a corresponding property for super-Riemann surfaces is likely to be important in the use of the superconformal structure. A local definition of line integration on super-Riemann surfaces has been given by Friedan [ 2 ], and in this paper, this local construction is generalized to take into account the global structure of the surface. It is the analogue of the approach to integration on Riemann surfaces using sheaf cohomology. Although the paper could be abbreviated by drawing upon notation and assuming concepts from sheaf theory, an 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
attempt has been made to make it self-contained. Above all, it is aimed at giving an answer to a question a physicist might ask - how does one compute a number from a line integral around a closed path in a nontrivial supergravity background? In section 2, super-Riemann surfaces and their superconformal structure are reviewed. A formulation of cohomology on these surfaces is presented in section 3, and use is made of this in section 4 to define line integration. The conclusion discusses avenues for the application of these results to superconformal field theories.
2. Super-Riemann surfaces and superconformal structure For convenience, we consider ( 1, 0) supergravity [ 3 ], which is relevant to the superfield formulation o f the heterotic string, although the results are readily formulated in ( 1, 1 ) superspace. Superspace conventions in the appendix o f ref. [4] are used here. The local geometric structure of a super-Riemann surface is determined by the assignment of complexified frames {EA~i) } ( A = + , --, o~) on superspace coordinate patches Ui, together with a connection form co ~) for the local SO (2) action satisfying appropriate torsion constraints [3]. Note that o~ is a single spinor index. The coordinate covering is chosen such that all finite intersections of the U, are simply connected. The global structure of the super-Riemann surface is 221
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determined by the S O ( 2 ) valued transition functions on the overlaps o f charts: E (i) = exp [ + ifl(6) IE~) , E~,) = exp [ i f l ( 6 ) / 2 ]E~j) , where the fl(ij) are real-valued superfields defined on
U~c~Uj. Integrability of the subspaces ofsuperspace spanned by ~ ;~(i) + , ~p(i) } and {E(_') } is ensured by the torsion constraints [ 5 ], so that local coordinates z Mu) = (z (~), ~(i), 0(i)) may be introduced with {0~), 0(0~)} and {8~i) } spanning the respective subspaces (0 is a single component spinor coordinate, see ref. [ 4 ] ). The torsion constraints are solved if the frame {EA(i) } is related to the flat (1, 0) superspace frame {8~i), 8~(°, D(oi) -- 3(oi) + 0(')3~(~) } by a super-Weyl transformation
[6]:
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3. Cohomology on the super-Riemann surface In the next section, line integrals will be defined for tensor superfield 4(~i) defined on the charts Uj, with V~) 4(d ) = 0 and with• w~ -(i) E+(i)E(ff ) well-defined on overlaps o f coordinate charts. This means that 4(~° transforms under S 0 ( 2 ) and super-Weyl transformations as E(~i) does. Because V(j ) 4(ff ) = e x p [ ½ifl(ij) ]V__) 4 ~ ) on Uic~ Uj, the condition V(_i) 4(d ) = 0 is also well-defined on overlaps of charts. Additionally, we require that neither of the component fields 4 (/) [o=o or V(° 4(d ) [o=o be singular on Uz for a frame {EA(i) } which varies smoothly on Ui. In terms o f local superconformal coordinates z M(~) on Ui, V~ ) 4(d ) = exp (3-q (i))O~u) e x p ( - ~ ( ' ) )4(d ) , so 0 ( i ) f /"~-"t~ (0i ) = 0 , g where 4(i)=exp(-~J(i)4(~i)). Thus 4O(i) has the component expansion
E ~ ) = exp(2g2 u))O~i) + 2 exp (2~2 u) ) (D(oi) ~2(°)Do(') ,
4(0 ~) (z, 0 ) = h (o")(z('))+O(i)h(j ) ( z " ) ) .
E(,) _ exp ( ~ ( , ) ) D (oi)
The one-form h(zi)(z (i)) c a n be written as O~(°fu) (z (i)) on U~ (recall that the Ui are simply connected and h(zi) is non-singular), and thus locally 415(i) = D(oi) 4 (i) with
E _(i) =exp(mgJ (i) )0e(i) ,
(1)
where gju) is a real-valued superfield. There is freedom involved in the choice of these superconformal coordinates, and the flat frame datermined by an alternative choice z' MU) is related to {E(A° } by a combination of S O ( 2 ) and super-Weyl transformations. Since E (_') and E u) transform linearly under these transformations, Do(!) =exp(A(i))D(o i) ,
ve,a (i) =exp(2A(i)0 (i)
(2)
for a complex-valued superfield A (s). This in turn implies the following conditions for a change o f coordinates for the integrable subspaces [ 2 ] : z' ( ~ ) - z ' ( ' ) ( z , O) , O'(i)=O'")(z,O)
,
e' (~)-e' (i)(z) , Do(S)z, (,)=2 0' (,)DO(,)0, (,),
A(O=D(o*)O, (i)
(3)
This will be called a superconformal transformation. In particular, the transformation which relates local superconformal coordinates z M(o on U~ and Z M (j) on Uj is of this type. The coordinate transition functions z M(i) (z Mu) ) describe the superconformal structure of the super-Riemann surface. 222
4 (i)(Z, 0) = f ( i ) ( Z (i)) +OU)h(j) (z(i)) . In terms of the frame, this translates to 4(~u)-V(i) 4 °) with V(j ) 4 (o = 0. Note that this is consistent with V(')4(~° = 0 because the torsion components TAd vanish. Although the tensor superfield 4 a with the properties above can be written locally as V , 4 with V_ 4 = 0, in general this is not possible globally for a well-defined scalar superfield (it). The equivalence classes of superfields 4~ under the equivalence relation tlt),~ 4 a + V a 4 for globally defined 4 are what could be termed the cohomology classes of tensor superfields 4 , . An alternative formulation of this concept is possible. Given a tensor superafield 4 , , we choose on each Ui a superfield 4 u) with V(_')4(i)=0 (or, equivalently, with 0z(°4(i)=0) such that 4 ( i ) = V ( ° 4 u). Nonsingular scalar superfields A satisfying V_A = 0 will be termed holomorphic, and they form an abelian group under pointwise addition. Since V~)A =exp(ifl(ij) ) , V(J)A, on Uic~Uj, the concept of holomorphy is indepen-
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dent of coordinate chart. Denoting the abelian group ofholomorphic superfields on U, by ¢(Ui), the choice ofsuperfields 4 (z) above assigns an element of (9(U~) to each Ug. In the language of sheaf theory, the 4 (') are determined by a cochain 4 ~ C (°) (M, (9), where M is the super-Riemann surface (see ref. [ 7 ] for descriptions of sheaf theory readily accessible to physicists, or ref. [ 8 ] for a more mathematical treatment; an element 4 ~ C ( ~ ) ( M , (9) assigns a superfield 4 ( g ° ' " ) ~ (9(U~oC~... ~U~,,) to each nonempty ( n + 1 )-fold intersection of charts ). On n o n e m p t y intersections U#~Uj, there are superfields 4 ¢o and 4 0.) with 4o¢~)=D(o~)4 (~), 4o°) = D~ ) 4 ¢j). However, since the coordinates z M(~) and z M(j) are related by a superconformal transformation, 4(oi) = ( D(oi) 0 u) ) 4o°) = D(0° 4 o ) where (2) and (3) have been used. Thus D(o i) ( 4 0 ) __ 4 (i)) = 0 .
(4)
The superfield 4 ° ) - 4 (~) is holomorphic on U~ ~ U j, and is thus determined by a cochain in C (~) (M, (9). This cochain is denoted 6~°)4 and is defined by (8¢°)4) (~'J)= 4 °) - 4 °). It is an example of the map 8(n):c(n)----~c(nq-l) with the property 8(n+l)8(n)=0 which is the basis of sheaf cohomology (see refs. [ 7 ] and [ 8 ] ). For the purpose of this work, only 8 ¢°) defined above and 8(1) :C(I)--,C (~), defined by (8(,)A ) ¢i.j.k) =AO.k) _AO.k) + A(Z.S) , for A ~ C ('), are required; it is trivial to check that 8(')8(o)=0. The condition (4), together with 0~~) ( 4 ( " - 4 (") = 0 from holomorphy, determines that 8(°)4('.J) is a constant superfield, 8 ¢ ° ) 4 ° J ) = c ~,s), c ° J ) ~ C (the concept of constant superfield is independent of local superconformal coordinates, as is easily checked). So c "J) can be considered to be determined by a cochain c ~ C ( ' ) ( M , C) which assigns an element c"J)~C (U~c~ U s) to each U~ c~Us, where C (U~ ~ U s) is the additive abelian group o f constant functions on U, c~U s (with C (') (M, C) similarly defined). Also, since 8(~)8(°)=0 and c = 8 ( ° ) 4 , 8 ( ' ) c = 0 . This is denoted c~Z (') (M, C), where Z (~) (M, C) is the group of 1cocycles, or the subspace o f C (1) (M, C) which is the
19 May 1988
kernel of 8 (1). Thus c determines an element [c] in the cohomology group H ¢') (M, C), where H (1) (M, C ) is the set of equivalence classes in Z (~) (M, C ) under the equivalence relation c ~ c + 8 ( ° ) d for d in C (°) (M, C). Note that [c] is not the trivial class despite the fact that c = 8 ¢ ° ) 4 , because 4 e C (°) (M, (9) and not C (°) (M, C). A choice was made in defining [c], namely the choice of 4 ~ C (°) (M, (9), or the assignment o f h o l o morphic superfields 4 (i) to the Ui. Suppose another assignment 4 '(i), with O~°4'(g)=O and 4(o~)= D~/) 4 ' ('), is made. Then it follows that D(oi) ( 4 ' (i) - 4 ( ' ) ) = 0 , so that qr ( i ) - 4 (z) is a constant superfield d ( ' ) e C ( U , ) . Thus 4 ' - 4 = d with d~C (°) (M, C), and c ' = 8 ( ° ) 4 ' = c + 8 ( ° ) d , so [ c ' ] = [ c ] . This shows that the cohomology class [c] in H ('~ (M, C) is independent o f the choice 4 ~ C (°) (M, (9). In this sense, the cohomology class is associated with the tensor superfield 4 , we began with. Further, if 4 , can be written as V,A for a globally defined holomorphic superfield A, it is easy to check that the class [c] associated with 4 , is the trivial class, so this agrees with the notion of triviality under the equivalence relation 4 , ~ 4 , + V,A. Line integrals of 4~ will be defined making use of the correspondence between 4 , and the cohomology class [c]. (This is the analogue of the correspondence between holomorphic ( 1, 0) forms and elements of H (~) (M, C) on ordinary Riemann surfaces, and line integrals will be defined using an analogue of the duality between HI ( M , Y ) a n d H ( I ) ( M , C ) ) .
4. Line integrals In ref. [2 ], Friedan has proposed a local definition of line integration on a super-Riemann surface, which will be adopted here within coordinate charts. However, to allow a definition of the line integral for paths which are not contained in a single coordinate patch, it will be necessary to make use of the cohomology class in H (') (M, C) associated with the tensor superfield 4 , . Given a simply connected subspace U of superspace with local coordinates z M, a path y in U with endpoints z ~ and z ~ (that is, a map 7: [ 0,1 ] ~ U with y ( 0 ) = ( Z o , Zo, 0o) and y ( 1 ) = ( Z l , 2,, G ) ) , and a superfield 4o with 0_~4o= 0 and nonsingular compo223
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definition of the line integral in the coordinate patch that
nent fields, then the line integral f~dz q)o(Z, O) defined in ref. [2] is characterized by the following property: if cl)o(z, O) = DoCI)(z, O) on U,
f d~o(Z,
~/+1 f n-t-I Z dz'qb0u>= E [ q b ( O ( p , + , ) _ q S U ) ( p , ) ] . i=0 i=0 7i
O ) = ( ~ ( Z l , 01 ) - - ~ ( Z o , 00) •
7
As already noted, this is independent o f the representatives (it) (i) chosen. The sum may be rewritten as
The nature of the measure dz need not concern us as the above property is sufficient to characterize the integral. Note that the integral is independent o f the choice of holomorphic superfield ~ ( z , 0), as any other choice differs from it by a constant superfield. Also, the line integral is a functional of the endpoints of the path, and is independent of the path in superspace between the points. The behaviour of the functional under an infinitesimal deformation o f an endpoint of the path described by the tangent vector Do to superspace is given by
Do,
(~)(n+l)(z', 0 ' ) +
-~(°)(z,O)
j dz ~o(Z, O) =Cl)o(Zl, O1 ) •
• ('+l)(z', 0')+
It is also clear that if 71 is a path from (Zo, 0o) to (z', O' ) and 72 is a path from (z', O' ) to (zb O~), then =
N o w consider a path y on the super-Riemann surface which does not lie in one coordinate chart. Let 7(0) be the point p in a coordinate chart U, and 7( 1 ) the point p' in a chart U ' . We wish to define the line integral along 7 of a tensor superfield ~ , o f the type considered in section 3. Suppose 7 passes sequentially through charts UI, ..., U,, and let U = U o , U ' = U , + ~. Choose points Pl, -.., Pn+l such that p~ lies in U~_ 1c~ Us (see fig. 1 ) and split yas y = XT+~l 7~, with 7s the portion o f y between Ps and p,+~, and p o = p , p , + 2 = p '. Then on U,, ~b, restricts to the superfield q¢o~) in local superconformal coordinates, and we define the line integral o f q~, along y to be 7_-+Jf~,dz (')~ o(') . Choosing holomorphic superfields qb (i) such that ~(oi) = D ( o ° ~ (°, it follows from the U i-2
@
f ...
[(TI)(i-1)(pi)--(i)(i)(pi)]
,
~ c(~,'-l)-~(°)(z, 0). i=l
In the case that 7 is a closed loop, p = p ' eUoc~U,+ l, and the line integral gives ~" c ua-l) where U _ t U , + t. When the class [c] in H (l) (M, C) determined by ~ , is trivial, this sum vanishes because c = ~d for some dEC (°) (M, C), so c U ' ~ - l ) = d U - l ) - d ( n . Thus the line integral of a cohomologically trivial form ~ , vanishes. So the above prescription assigns a complex number to the line integral of q b around a closed loop. Actually, this should be thought of as a functional o f the supergravity background which defines the superRiemann surface (or equivalently, of the superconformal structure it defines). In particular, it is a functional of the even and odd parameters which describe the supermoduli space of the super-Riemann surface. The prescription for line integration is also invariant under deformations of the path, providing the
L,, + L,~ L,.
Uo=U
L i=l
where (z, 0) and (z', 0' ) are the coordinates o f p and p'. However, if the ~ u ) are thought of as being determined by a class ~ e C (°) (M, to) as in section 3, then since 8 q b = c e C (l) (M, C), it follows that qb(i-l) (pi) - ~(') (p~) = c u J- t)eC. Thus the line integral is
7
Ui-1
Ui
Ui+ 1
..... >
-->~-~
k Pn+2=P Pi-1
Pi
Fig. 1. 224
19 May 1988
Pi+ I
~ - - ~ "
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deformed path can be reached continuously from the original (or the region enclosed by the difference between the original and deformed paths is contractible). It therefore suffices to consider a deformed path y' of the type shown in fig. 2. A simple computation shows that the difference between the line integral along y and 7' is [OU)(qj+ l ) - O U ) ( q j ) ] + [ • ( ; ) ( p ; ) - O ( i ) ( q j + 1)] + [ 0 (i- l)(qj)-- 0 (;- l)(p;) ] :C(t,j) __c(i--
l,j)
dff c( i-- 1,i)
.
However, this is (5c) (;- 1,;j), which vanishes because 5c=0, as seen in section 3. This argument also shows that the line integral of ~ , around a contractible loop vanishes.
5. Conclusion
In this paper, a prescription for computing line integrals on nontrivial super-Riemann surfaces has been given, based on the sheaf theoretic prescription for line integration on an ordinary Riemann surface. However, the question of the existence of nontrivial tensor superfields O , of the type described in section 3 (that is, determining a nontrivial class in H ¢~) (M, C) ) has not been addressed. In general, the presence of nontrivial fermionic components in the superconformal coordinate transition functions (nontrivial in that they cannot be removed by changes of superconformal coordinates on the charts) means that they do not reduce simply to holomorphic ( 1, 0) forms on an ordinary Riemann surface. Nontrivial ~ , would provide connections for flat line bundles on the superPi
~
; ~//l
q/+l
19 May 1988
Riemann surface, and as such are relevant to the study of superconformal field theories. The jacobian variety of Riemann surface is a mathematical construction helpful in the treatment of twodimensional conformal field theories, particularly in relation to the computation of Green functions. The ability to compute exact Green functions on higher genus Riemann surfaces has been responsible for much progress in string theory. The Riemann surface can be embedded in its jacobian variety via line integrals of the abelian differentials, and the invariant interval for Green functions is essentially a straight line on the jacobian variety. As yet, the computation of exact Green functions on super-Riemann surfaces making full use of the superconformal structure of the surface has not been achieved. This is a problem of importance in the computation of superstring amplitudes, and the naive expectation is that its solution will rely on a suitable generalization of the jacobian variety to the case of a super-Riemann surface via line integrals of the superfields O~ which are the analogues of abelian differentials. For example, line integrals of these superfields around noncontractible loops give complex numbers which, suitably normalized, are the period matrices of the super-Riemann surface. However, any generalization of the jacobian variety by this means will be nontrivial, as can be seen by considering the case where the super-Riemann surface has no nontrivial fermionic structure in its transition functions, and reduces to an ordinary Riemann surface with a choice of spin structure [ 2 ]. Although the Riemann surface and its canonical bundle (or forms on the surface) embed naturally in the jacobian variety via the jacobian map (essentially line integrals of the abelian differentials), the spin bundle does not. Spinors inherit square root cuts at the images of the zeroes of the abelian differentials, and only make sense on a double cover of the jacobian with ramification points at the images of the zeroes of the abelian differentials. The exception is the torus, where the abelian differential has no zero, and approaches based on uniformization suggest that in this case a straightforward analogue of the jacobian exists [9].
Vi
F i g . 2.
225
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N o t e added S h e a f t h e o r e t i c m e t h o d s h a v e also b e e n a p p l i e d t o super-Riemann surfaces in papers by Giddings and N e l s o n [ 10 ] c o p i e s o f w h i c h I r e c e i v e d a f t e r t h i s w o r k was completed.
References [ 1 ] H. Sonoda, Phys. Lett. B 178 (1986) 390. [2] D. Friedan, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986). [3 ] R. Brooks, F. Muhammad and S.J. Gates, Nucl. Phys. B 268 (1986) 599;
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G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 ( 1986 ) 47; M. Evans and B. Ovrut, Phys. Lett. B 171 (1986) 177. [4] l.N. McArthur, Phys. Lett. B 185 (1987) 358. [ 5 ] S.B. Giddings and P. Nelson, Torsion constraints and superRiemann surfaces, preprint BUHEP-87-28, HUTP-87/ A062. [6] E. Martinec, Phys. Rev. D 28 (1983) 2604. [7] O. Alvarez, Commun. Math. Phys. 100 (1985) 279; E. Witten and J. Bagger, Phys. Lett. B 115 (1982) 202. [8] R. Bott and L.W. Tu, Differential forms in algebraic topology (Springer, Berlin, 1982). [ 9 ] L. Crane and J.M. Rabin, Commun. Math. Phys. 113 ( 1988 ) 601. [10] S.B. Giddings and P. Nelson, preprints BUHEP-87-31, HUTP-87/A070; BUHEP-87-48, HUTP-87/A080.
PHYSICS LETTERS B
19 May 1988
LINE INTEGRALS ON SUPER-RIEMANN SURFACES I.N. M c A R T H U R Institut fiir Theoretische Physik, Universitdt Karlsruhe, D-7500 Karsruhe 1, Fed. Rep. Germany
Received 29 January 1988
A procedure for computing line integrals on super-Riemann surfaces, based on an analogue of the sheaf theoretic prescription of line integration on an ordinary Riemann surface, is presented.
1. Introduction There has been much progress recently in the analysis of two-dimensional field theories defined on a Riemann surface (or in the presence of a nontrivial gravitational background). If the theory is Weyl invariant, then it realizes the conformal structure of the underlying Riemann surface, and this is sufficient to allow the computation of exact Green functions (see, for example, ref. [ 1 ] ). The ability to do explicit calculations in string theory is one important application. It is therefore natural to attempt to make full use of the superconformal structure on super-Riemann surfaces, which is realized by super-Weyl invariant theories defined on a super-Riemann surface (or propagating in a non-trivial two-dimensional supergravity background). An important ingredient in the exploitation of the conformal structure on a Riemann surface is the ability to take line integrals of holomorphic (1, 0) forms (locally of the form ~o=(z) dz), and the existence of a corresponding property for super-Riemann surfaces is likely to be important in the use of the superconformal structure. A local definition of line integration on super-Riemann surfaces has been given by Friedan [ 2 ], and in this paper, this local construction is generalized to take into account the global structure of the surface. It is the analogue of the approach to integration on Riemann surfaces using sheaf cohomology. Although the paper could be abbreviated by drawing upon notation and assuming concepts from sheaf theory, an 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
attempt has been made to make it self-contained. Above all, it is aimed at giving an answer to a question a physicist might ask - how does one compute a number from a line integral around a closed path in a nontrivial supergravity background? In section 2, super-Riemann surfaces and their superconformal structure are reviewed. A formulation of cohomology on these surfaces is presented in section 3, and use is made of this in section 4 to define line integration. The conclusion discusses avenues for the application of these results to superconformal field theories.
2. Super-Riemann surfaces and superconformal structure For convenience, we consider ( 1, 0) supergravity [ 3 ], which is relevant to the superfield formulation o f the heterotic string, although the results are readily formulated in ( 1, 1 ) superspace. Superspace conventions in the appendix o f ref. [4] are used here. The local geometric structure of a super-Riemann surface is determined by the assignment of complexified frames {EA~i) } ( A = + , --, o~) on superspace coordinate patches Ui, together with a connection form co ~) for the local SO (2) action satisfying appropriate torsion constraints [3]. Note that o~ is a single spinor index. The coordinate covering is chosen such that all finite intersections of the U, are simply connected. The global structure of the super-Riemann surface is 221
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determined by the S O ( 2 ) valued transition functions on the overlaps o f charts: E (i) = exp [ + ifl(6) IE~) , E~,) = exp [ i f l ( 6 ) / 2 ]E~j) , where the fl(ij) are real-valued superfields defined on
U~c~Uj. Integrability of the subspaces ofsuperspace spanned by ~ ;~(i) + , ~p(i) } and {E(_') } is ensured by the torsion constraints [ 5 ], so that local coordinates z Mu) = (z (~), ~(i), 0(i)) may be introduced with {0~), 0(0~)} and {8~i) } spanning the respective subspaces (0 is a single component spinor coordinate, see ref. [ 4 ] ). The torsion constraints are solved if the frame {EA(i) } is related to the flat (1, 0) superspace frame {8~i), 8~(°, D(oi) -- 3(oi) + 0(')3~(~) } by a super-Weyl transformation
[6]:
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3. Cohomology on the super-Riemann surface In the next section, line integrals will be defined for tensor superfield 4(~i) defined on the charts Uj, with V~) 4(d ) = 0 and with• w~ -(i) E+(i)E(ff ) well-defined on overlaps o f coordinate charts. This means that 4(~° transforms under S 0 ( 2 ) and super-Weyl transformations as E(~i) does. Because V(j ) 4(ff ) = e x p [ ½ifl(ij) ]V__) 4 ~ ) on Uic~ Uj, the condition V(_i) 4(d ) = 0 is also well-defined on overlaps of charts. Additionally, we require that neither of the component fields 4 (/) [o=o or V(° 4(d ) [o=o be singular on Uz for a frame {EA(i) } which varies smoothly on Ui. In terms o f local superconformal coordinates z M(~) on Ui, V~ ) 4(d ) = exp (3-q (i))O~u) e x p ( - ~ ( ' ) )4(d ) , so 0 ( i ) f /"~-"t~ (0i ) = 0 , g where 4(i)=exp(-~J(i)4(~i)). Thus 4O(i) has the component expansion
E ~ ) = exp(2g2 u))O~i) + 2 exp (2~2 u) ) (D(oi) ~2(°)Do(') ,
4(0 ~) (z, 0 ) = h (o")(z('))+O(i)h(j ) ( z " ) ) .
E(,) _ exp ( ~ ( , ) ) D (oi)
The one-form h(zi)(z (i)) c a n be written as O~(°fu) (z (i)) on U~ (recall that the Ui are simply connected and h(zi) is non-singular), and thus locally 415(i) = D(oi) 4 (i) with
E _(i) =exp(mgJ (i) )0e(i) ,
(1)
where gju) is a real-valued superfield. There is freedom involved in the choice of these superconformal coordinates, and the flat frame datermined by an alternative choice z' MU) is related to {E(A° } by a combination of S O ( 2 ) and super-Weyl transformations. Since E (_') and E u) transform linearly under these transformations, Do(!) =exp(A(i))D(o i) ,
ve,a (i) =exp(2A(i)0 (i)
(2)
for a complex-valued superfield A (s). This in turn implies the following conditions for a change o f coordinates for the integrable subspaces [ 2 ] : z' ( ~ ) - z ' ( ' ) ( z , O) , O'(i)=O'")(z,O)
,
e' (~)-e' (i)(z) , Do(S)z, (,)=2 0' (,)DO(,)0, (,),
A(O=D(o*)O, (i)
(3)
This will be called a superconformal transformation. In particular, the transformation which relates local superconformal coordinates z M(o on U~ and Z M (j) on Uj is of this type. The coordinate transition functions z M(i) (z Mu) ) describe the superconformal structure of the super-Riemann surface. 222
4 (i)(Z, 0) = f ( i ) ( Z (i)) +OU)h(j) (z(i)) . In terms of the frame, this translates to 4(~u)-V(i) 4 °) with V(j ) 4 (o = 0. Note that this is consistent with V(')4(~° = 0 because the torsion components TAd vanish. Although the tensor superfield 4 a with the properties above can be written locally as V , 4 with V_ 4 = 0, in general this is not possible globally for a well-defined scalar superfield (it). The equivalence classes of superfields 4~ under the equivalence relation tlt),~ 4 a + V a 4 for globally defined 4 are what could be termed the cohomology classes of tensor superfields 4 , . An alternative formulation of this concept is possible. Given a tensor superafield 4 , , we choose on each Ui a superfield 4 u) with V(_')4(i)=0 (or, equivalently, with 0z(°4(i)=0) such that 4 ( i ) = V ( ° 4 u). Nonsingular scalar superfields A satisfying V_A = 0 will be termed holomorphic, and they form an abelian group under pointwise addition. Since V~)A =exp(ifl(ij) ) , V(J)A, on Uic~Uj, the concept of holomorphy is indepen-
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dent of coordinate chart. Denoting the abelian group ofholomorphic superfields on U, by ¢(Ui), the choice ofsuperfields 4 (z) above assigns an element of (9(U~) to each Ug. In the language of sheaf theory, the 4 (') are determined by a cochain 4 ~ C (°) (M, (9), where M is the super-Riemann surface (see ref. [ 7 ] for descriptions of sheaf theory readily accessible to physicists, or ref. [ 8 ] for a more mathematical treatment; an element 4 ~ C ( ~ ) ( M , (9) assigns a superfield 4 ( g ° ' " ) ~ (9(U~oC~... ~U~,,) to each nonempty ( n + 1 )-fold intersection of charts ). On n o n e m p t y intersections U#~Uj, there are superfields 4 ¢o and 4 0.) with 4o¢~)=D(o~)4 (~), 4o°) = D~ ) 4 ¢j). However, since the coordinates z M(~) and z M(j) are related by a superconformal transformation, 4(oi) = ( D(oi) 0 u) ) 4o°) = D(0° 4 o ) where (2) and (3) have been used. Thus D(o i) ( 4 0 ) __ 4 (i)) = 0 .
(4)
The superfield 4 ° ) - 4 (~) is holomorphic on U~ ~ U j, and is thus determined by a cochain in C (~) (M, (9). This cochain is denoted 6~°)4 and is defined by (8¢°)4) (~'J)= 4 °) - 4 °). It is an example of the map 8(n):c(n)----~c(nq-l) with the property 8(n+l)8(n)=0 which is the basis of sheaf cohomology (see refs. [ 7 ] and [ 8 ] ). For the purpose of this work, only 8 ¢°) defined above and 8(1) :C(I)--,C (~), defined by (8(,)A ) ¢i.j.k) =AO.k) _AO.k) + A(Z.S) , for A ~ C ('), are required; it is trivial to check that 8(')8(o)=0. The condition (4), together with 0~~) ( 4 ( " - 4 (") = 0 from holomorphy, determines that 8(°)4('.J) is a constant superfield, 8 ¢ ° ) 4 ° J ) = c ~,s), c ° J ) ~ C (the concept of constant superfield is independent of local superconformal coordinates, as is easily checked). So c "J) can be considered to be determined by a cochain c ~ C ( ' ) ( M , C) which assigns an element c"J)~C (U~c~ U s) to each U~ c~Us, where C (U~ ~ U s) is the additive abelian group o f constant functions on U, c~U s (with C (') (M, C) similarly defined). Also, since 8(~)8(°)=0 and c = 8 ( ° ) 4 , 8 ( ' ) c = 0 . This is denoted c~Z (') (M, C), where Z (~) (M, C) is the group of 1cocycles, or the subspace o f C (1) (M, C) which is the
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kernel of 8 (1). Thus c determines an element [c] in the cohomology group H ¢') (M, C), where H (1) (M, C ) is the set of equivalence classes in Z (~) (M, C ) under the equivalence relation c ~ c + 8 ( ° ) d for d in C (°) (M, C). Note that [c] is not the trivial class despite the fact that c = 8 ¢ ° ) 4 , because 4 e C (°) (M, (9) and not C (°) (M, C). A choice was made in defining [c], namely the choice of 4 ~ C (°) (M, (9), or the assignment o f h o l o morphic superfields 4 (i) to the Ui. Suppose another assignment 4 '(i), with O~°4'(g)=O and 4(o~)= D~/) 4 ' ('), is made. Then it follows that D(oi) ( 4 ' (i) - 4 ( ' ) ) = 0 , so that qr ( i ) - 4 (z) is a constant superfield d ( ' ) e C ( U , ) . Thus 4 ' - 4 = d with d~C (°) (M, C), and c ' = 8 ( ° ) 4 ' = c + 8 ( ° ) d , so [ c ' ] = [ c ] . This shows that the cohomology class [c] in H ('~ (M, C) is independent o f the choice 4 ~ C (°) (M, (9). In this sense, the cohomology class is associated with the tensor superfield 4 , we began with. Further, if 4 , can be written as V,A for a globally defined holomorphic superfield A, it is easy to check that the class [c] associated with 4 , is the trivial class, so this agrees with the notion of triviality under the equivalence relation 4 , ~ 4 , + V,A. Line integrals of 4~ will be defined making use of the correspondence between 4 , and the cohomology class [c]. (This is the analogue of the correspondence between holomorphic ( 1, 0) forms and elements of H (~) (M, C) on ordinary Riemann surfaces, and line integrals will be defined using an analogue of the duality between HI ( M , Y ) a n d H ( I ) ( M , C ) ) .
4. Line integrals In ref. [2 ], Friedan has proposed a local definition of line integration on a super-Riemann surface, which will be adopted here within coordinate charts. However, to allow a definition of the line integral for paths which are not contained in a single coordinate patch, it will be necessary to make use of the cohomology class in H (') (M, C) associated with the tensor superfield 4 , . Given a simply connected subspace U of superspace with local coordinates z M, a path y in U with endpoints z ~ and z ~ (that is, a map 7: [ 0,1 ] ~ U with y ( 0 ) = ( Z o , Zo, 0o) and y ( 1 ) = ( Z l , 2,, G ) ) , and a superfield 4o with 0_~4o= 0 and nonsingular compo223
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definition of the line integral in the coordinate patch that
nent fields, then the line integral f~dz q)o(Z, O) defined in ref. [2] is characterized by the following property: if cl)o(z, O) = DoCI)(z, O) on U,
f d~o(Z,
~/+1 f n-t-I Z dz'qb0u>= E [ q b ( O ( p , + , ) _ q S U ) ( p , ) ] . i=0 i=0 7i
O ) = ( ~ ( Z l , 01 ) - - ~ ( Z o , 00) •
7
As already noted, this is independent o f the representatives (it) (i) chosen. The sum may be rewritten as
The nature of the measure dz need not concern us as the above property is sufficient to characterize the integral. Note that the integral is independent o f the choice of holomorphic superfield ~ ( z , 0), as any other choice differs from it by a constant superfield. Also, the line integral is a functional of the endpoints of the path, and is independent of the path in superspace between the points. The behaviour of the functional under an infinitesimal deformation o f an endpoint of the path described by the tangent vector Do to superspace is given by
Do,
(~)(n+l)(z', 0 ' ) +
-~(°)(z,O)
j dz ~o(Z, O) =Cl)o(Zl, O1 ) •
• ('+l)(z', 0')+
It is also clear that if 71 is a path from (Zo, 0o) to (z', O' ) and 72 is a path from (z', O' ) to (zb O~), then =
N o w consider a path y on the super-Riemann surface which does not lie in one coordinate chart. Let 7(0) be the point p in a coordinate chart U, and 7( 1 ) the point p' in a chart U ' . We wish to define the line integral along 7 of a tensor superfield ~ , o f the type considered in section 3. Suppose 7 passes sequentially through charts UI, ..., U,, and let U = U o , U ' = U , + ~. Choose points Pl, -.., Pn+l such that p~ lies in U~_ 1c~ Us (see fig. 1 ) and split yas y = XT+~l 7~, with 7s the portion o f y between Ps and p,+~, and p o = p , p , + 2 = p '. Then on U,, ~b, restricts to the superfield q¢o~) in local superconformal coordinates, and we define the line integral o f q~, along y to be 7_-+Jf~,dz (')~ o(') . Choosing holomorphic superfields qb (i) such that ~(oi) = D ( o ° ~ (°, it follows from the U i-2
@
f ...
[(TI)(i-1)(pi)--(i)(i)(pi)]
,
~ c(~,'-l)-~(°)(z, 0). i=l
In the case that 7 is a closed loop, p = p ' eUoc~U,+ l, and the line integral gives ~" c ua-l) where U _ t U , + t. When the class [c] in H (l) (M, C) determined by ~ , is trivial, this sum vanishes because c = ~d for some dEC (°) (M, C), so c U ' ~ - l ) = d U - l ) - d ( n . Thus the line integral of a cohomologically trivial form ~ , vanishes. So the above prescription assigns a complex number to the line integral of q b around a closed loop. Actually, this should be thought of as a functional o f the supergravity background which defines the superRiemann surface (or equivalently, of the superconformal structure it defines). In particular, it is a functional of the even and odd parameters which describe the supermoduli space of the super-Riemann surface. The prescription for line integration is also invariant under deformations of the path, providing the
L,, + L,~ L,.
Uo=U
L i=l
where (z, 0) and (z', 0' ) are the coordinates o f p and p'. However, if the ~ u ) are thought of as being determined by a class ~ e C (°) (M, to) as in section 3, then since 8 q b = c e C (l) (M, C), it follows that qb(i-l) (pi) - ~(') (p~) = c u J- t)eC. Thus the line integral is
7
Ui-1
Ui
Ui+ 1
..... >
-->~-~
k Pn+2=P Pi-1
Pi
Fig. 1. 224
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Pi+ I
~ - - ~ "
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deformed path can be reached continuously from the original (or the region enclosed by the difference between the original and deformed paths is contractible). It therefore suffices to consider a deformed path y' of the type shown in fig. 2. A simple computation shows that the difference between the line integral along y and 7' is [OU)(qj+ l ) - O U ) ( q j ) ] + [ • ( ; ) ( p ; ) - O ( i ) ( q j + 1)] + [ 0 (i- l)(qj)-- 0 (;- l)(p;) ] :C(t,j) __c(i--
l,j)
dff c( i-- 1,i)
.
However, this is (5c) (;- 1,;j), which vanishes because 5c=0, as seen in section 3. This argument also shows that the line integral of ~ , around a contractible loop vanishes.
5. Conclusion
In this paper, a prescription for computing line integrals on nontrivial super-Riemann surfaces has been given, based on the sheaf theoretic prescription for line integration on an ordinary Riemann surface. However, the question of the existence of nontrivial tensor superfields O , of the type described in section 3 (that is, determining a nontrivial class in H ¢~) (M, C) ) has not been addressed. In general, the presence of nontrivial fermionic components in the superconformal coordinate transition functions (nontrivial in that they cannot be removed by changes of superconformal coordinates on the charts) means that they do not reduce simply to holomorphic ( 1, 0) forms on an ordinary Riemann surface. Nontrivial ~ , would provide connections for flat line bundles on the superPi
~
; ~//l
q/+l
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Riemann surface, and as such are relevant to the study of superconformal field theories. The jacobian variety of Riemann surface is a mathematical construction helpful in the treatment of twodimensional conformal field theories, particularly in relation to the computation of Green functions. The ability to compute exact Green functions on higher genus Riemann surfaces has been responsible for much progress in string theory. The Riemann surface can be embedded in its jacobian variety via line integrals of the abelian differentials, and the invariant interval for Green functions is essentially a straight line on the jacobian variety. As yet, the computation of exact Green functions on super-Riemann surfaces making full use of the superconformal structure of the surface has not been achieved. This is a problem of importance in the computation of superstring amplitudes, and the naive expectation is that its solution will rely on a suitable generalization of the jacobian variety to the case of a super-Riemann surface via line integrals of the superfields O~ which are the analogues of abelian differentials. For example, line integrals of these superfields around noncontractible loops give complex numbers which, suitably normalized, are the period matrices of the super-Riemann surface. However, any generalization of the jacobian variety by this means will be nontrivial, as can be seen by considering the case where the super-Riemann surface has no nontrivial fermionic structure in its transition functions, and reduces to an ordinary Riemann surface with a choice of spin structure [ 2 ]. Although the Riemann surface and its canonical bundle (or forms on the surface) embed naturally in the jacobian variety via the jacobian map (essentially line integrals of the abelian differentials), the spin bundle does not. Spinors inherit square root cuts at the images of the zeroes of the abelian differentials, and only make sense on a double cover of the jacobian with ramification points at the images of the zeroes of the abelian differentials. The exception is the torus, where the abelian differential has no zero, and approaches based on uniformization suggest that in this case a straightforward analogue of the jacobian exists [9].
Vi
F i g . 2.
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N o t e added S h e a f t h e o r e t i c m e t h o d s h a v e also b e e n a p p l i e d t o super-Riemann surfaces in papers by Giddings and N e l s o n [ 10 ] c o p i e s o f w h i c h I r e c e i v e d a f t e r t h i s w o r k was completed.
References [ 1 ] H. Sonoda, Phys. Lett. B 178 (1986) 390. [2] D. Friedan, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986). [3 ] R. Brooks, F. Muhammad and S.J. Gates, Nucl. Phys. B 268 (1986) 599;
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G. Moore, P. Nelson and J. Polchinski, Phys. Lett. B 169 ( 1986 ) 47; M. Evans and B. Ovrut, Phys. Lett. B 171 (1986) 177. [4] l.N. McArthur, Phys. Lett. B 185 (1987) 358. [ 5 ] S.B. Giddings and P. Nelson, Torsion constraints and superRiemann surfaces, preprint BUHEP-87-28, HUTP-87/ A062. [6] E. Martinec, Phys. Rev. D 28 (1983) 2604. [7] O. Alvarez, Commun. Math. Phys. 100 (1985) 279; E. Witten and J. Bagger, Phys. Lett. B 115 (1982) 202. [8] R. Bott and L.W. Tu, Differential forms in algebraic topology (Springer, Berlin, 1982). [ 9 ] L. Crane and J.M. Rabin, Commun. Math. Phys. 113 ( 1988 ) 601. [10] S.B. Giddings and P. Nelson, preprints BUHEP-87-31, HUTP-87/A070; BUHEP-87-48, HUTP-87/A080.