Connections on vector bundles over super Riemann surfaces

Volume 220, number 4

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13 April 1989

C O N N E C T I O N S O N V E C T O R B U N D L E S OVER S U P E R R I E M A N N SURFACES Mark R A K O W S K I and George T H O M P S O N International Centre for Theoretical Physics, 1-34100 Trieste, Italy

Received 1 August 1988

We show that the globally inequivalent off-shell N= 1 super Yang-Mills theories in two dimensions classify the superholomorphic structures on vector bundles over super Riemann surfaces. More precisely, there is a one-to-one correspondence between superholomorphic structures on vector bundles over super Riemann surfaces and unitary connections satisfying certain curvature constraints. These curvature constraints are the canonical constraints used in superspace formulations of super Yang-Mills theories, but arise in our considerations as integrability requirements for the local existence of solutions to certain differential equations. Finally, we discuss the relationship of this work with some aspects of Witten's twistor-like transform.

1. Introduction In computing loop amplitudes in the Polyakov formulation of fermionic closed string theories [ 1 ], one considers an integral over the space of inequivalent two dimensional supergravity geometries [ 2 - 4 ]. In analogy to the bosonic string case [ 5 ], where the corresponding problem is equivalent to an integral over the space of conformally inequivalent compact Riemann surfaces [ 6 ] (or equivalently parametrized by moduli [ 7 ] ), this problem can be rephrased as an integral over inequivalent super Riemann surfaces (SRS's) [2-4,8-11 ]. In fact, the theory of super Riemann surfaces has evolved from precisely this problem. Understanding the way in which inequivalent SRS's can be parametrized - the supermoduli and super Teichmiiller spaces [ 2-4,8,10 ] - is therefore of prime importance. In a recent paper [12], Hitchin has developed a new way in which to characterize the moduli space of ordinary compact Riemann surfaces. Roughly speaking, he establishes a correspondence between solutions to the four-dimensional self-dual Yang-Mills equations dimensionally reduced to two dimensions, and the moduli space of compact Riemann surfaces. It would be interesting to extend this work to the context o f SRS's, starting from the self-dual super YangMills theory [ 13 ] in four dimensions. In this paper, we report on some progress in this direction. We es0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

tablish a one-to-one correspondence between superholomorphic structures on vector bundles over SRS's and unitary connections satisfying certain curvature constraints. These are the canonical constraints of super Yang-Mills theory, and have recently been considered in the context of line bundles over SRS's [ 14]. They arise in our considerations as integrability requirements for the local existence to certain differential equations. One can say that the globally inequivalent N = 1 super Yang-Mills theories classify the superholomorphic structures on vector bundles. A good review of the material reported in ref. [ 12 ] can be found in ref. [ 15 ]. In the next section, after quickly reviewing those aspects of super Riemann surface theory which are essential to our work, we prove the correspondence between superholomorphic structures and connections as stated above. Lastly, we discuss the relationship of this work with certain aspects of the twistorlike transform [ 16 ]. More complete accounts of SRS's and supermanifolds in general can be found in [ 911,17,18 ] and references contained therein.

2. Connections and superholomorphic structure We denote by C 1'1, the superspace o f one even and one odd complex coordinate (z, 0). The operators 557

Volume 220, number 4 D.=Oo+O0_-,

Dg=Oo+tTO:

PHYSICS LETTERS B (1)

13 April 1989

are the square roots of the derivatives with respect to z and z,

V such that noS= 1 ) of the vector bundle V over the SRS M, and gl (M, V) the space of one-form valued sections. A connection on V is a differential operator dA : go ( M, V ) ~ ~' (M, V ) such that

DoDo=O:,

dA (fs) = df®s+ fdA s,

D#D#=O:.

(2)

Our conventions for the complex conjugation of Grassmann variables is such that the order remains unchanged, e.g. ~ = 0~ We write the exterior differential operator as d=0+0,

O=qO:+dO Do,

(3)

where q = d z + 0 dO, and with similar expressions for the "barred", or conjugated quantities. A differentiable function f: C"t--,C"~ is called superholomorphic (or superanalytic) i f 0 f = 0; i.e. i f f i s only a function of z and 0, and not the complex conjugated variables. If we think of an invertible superholomorphic m a p p i n g f a s being a coordinate transformation in superspace,

f(z, O)= (2(z, 0), O(z, O)),

(4)

(D,,#)Dg+ (DOS- 0Do0)0--.

(5)

The mapping f i s said to be superconformal [19 ] if the second term in (5) proportional to 0~ vanishes. The stronger condition of superconformality is to be distinguished from the condition of being merely superholomorphic. A super Riemann surface (SRS) is defined to be a complex ( l, 1 ) supermanifold whose transition functions are superconformal [ 8-11 ]. We will consider vector bundles over compact SRS's whose fibre contains only even directions. By n-dimensional vector bundle, we will mean a vector bundle whose fibre is isomorphic to C "'°, and whose transition functions we will require to be smooth. If U~ and Uj are two trivializing chart of the base SRS which intersect, so that there exists maps h,: zr- ~( Ui ) ~ Ui × C "'°, then the transition function gij is defined by h, oh7 ~(p, v) = (p, gu(p)v).

(6)

A superholomorphic vector bundle will be a vector bundle with transition functions which are superholomorphic. Let go (M, V) denote the space of sections (a section s is a map from the base space M into the bundle 558

for all se go ( M, V ) and f~ d ( M ), the space of smooth functions. On a trivializing chart, dA takes the form dA = d + A = d + (flA. +dO Ao) + ( flAe +d6Ao)

= (O+qA. +dOAo) + (O+flA~+dOAa) = (~/~ +dO @0) + ( ~ . + d 0 g # ) .

(8)

As on ordinary Riemann surfaces, the space of oneforms g ' ( M ) decomposes into superholomorphic and antisuperholomorphic pieces, ~' (M) = ~ " ° ( M ) ~ # ° , ' (M). The curvature F of dA, and its curvature components are given by F= (d+A) (d+A)

= dO dO Foo + rldO F:o + dO dO Too + fldOF:o

then the differential operator Do transforms as Do=

(7)

+ rlflF-_~+ qdOF:# + fldO Feo + dO dOFoo,

(9 )

where Foo=-½{~o, go}+g~, F_-o= [g:, go], F._= [ ~ , ~ ] , F-_.a= [ ~ , g#], Fo,a= - { go, ga}, and similarly for the other components. Following the treatment of ordinary Riemann surfaces [12,15,20], we define a superholomorphic structure on a vector bundle as a differential operator d~ : go(M, V)--,go.' (M, V) satisfying the following conditions:

d'~ (fs) =~f®s+ fd'~;s, 0--d~ od~.

(10)

Similar to the case of a connection, we can locally write d~ as

d;~ = O+ flA ~+ dO Aa.

( 11 )

On a Riemann surface, the second condition in (10) is clearly satisfied automatically. The reason for this additional constraint will soon become apparent; for now we will just note that it is equivalent to the condition

Ae=DoAo,

(12)

and that for line bundles, flA~+dOAa is a 0-closed

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one-form. It is a straightforward exercise to check this equivalence, and we will omit the details. It would have perhaps been more natural to define a superholomorphic structure on a smooth vector bundle by choosing a collection of locally trivial charts from the atlas of the smooth structure such that the transition functions {g0} are superholomorphic. These two notions are in fact equivalent, as we now discuss. Given a set of transition functions {g,j} on some covering of M which are superholomorphic, it is easy to produce a superholomorphic structure in the differential operator sense. Take a frame of sections su over each trivializing open set U, so that su =guvSv on U c~ V; then define a matrix of one-forms Au by Osu = - A v s u . Since the {g,j} are superholomorphic, the various A U piece together to produce a global operator d;'~. To show that the superholomorphic structure (10) actually gives a trivialization with superholomorphic transition functions is not so simple. First, we need a lemma which extends a theorem of Atiyah and Bott [201 to the case of SRS's:

Lemma. If d;'~ is a superholomorphic structure, then there exists a frame of local sections s such that d;'js= 0 about each point of M. This is the origin o f the constraint d~ od~ = 0 in our definition (10), since (d~r~) 2 is not a differential operator, but a collection o f matrix valued two-forms. It is the integrability constraint for the local existence of solutions to d~'~s = 0.

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0 = (Cq.z"k-A2)So, O = ( Oz-I-A2 ) s l --}-A3so,

0=s2 +Aoso, 0=s3 +Aj So --AoSl.

(15)

Here we see that s 2 and s3 are determined algebraically in terms of the other components, while So and sj satisfy differential equations. The first equation for So is just the equation one would obtain on a Riemann surface, and the local existence of n-independent solutions is guaranteed by a theorem of Atiyah and Bott [20]. The linearity of the second equation suggests that we look for solutions which are of the form s~ =Aso, which reduces it to a simple equation forA:

OzA=A3.

(16)

Since the Dolbeault lemma [21] also applies to Grassmann functions, we are guaranteed that a function A exists, and this completes our proof o f lemma 1. Given a superholomorphic structure d~, it is now a simple matter to construct a collection {go} of superholomorphic transition functions. On each trivializing open set U, sufficiently small, we are now able to choose a frame o f local sections si satisfying d~jst = 0. If Ut c~Uj ~ ~J, then since st = g,jsj on the overlap of these coordinate neighborhoods,

0 = d~ s, = d] (gijsj) = Ogij@Sj "q-gij d"~Sj =Ogtj®Sj.

(17)

Proof To prove the lemma, we consider the most general 0-expansions for Aa and s:

Ao=Ao +OAt +OA2 -t-OOA3, S = S 0 "]- OS I "]- Os 2 "~- 00S3 ,

(13)

where the coefficient functions in these expressions are z and g dependent. Recalling that the Az component of the superholomorphic structure is determined by Ao via eq. (12), the lemma reduces to proving the existence of solutions to

( DcT+Ao)s=O.

(14)

In the most straightforward manner, one arrives at the following set o f component relations:

We have now obtained a trivialization in which the transition functions are superholomorphic. The correspondence which we really seek is between the connections on a vector bundle V, and the set of superholomorphic structures. We find it convenient to call a connection dA partially flat if the conditions

O=(dA)°"(dA)°" =(dA)"°(dA) ''°

(18)

are satisfied. This is equivalent to the vanishing of certain curvature components:

O=Foo=F-_o=Foa=Fza.

(19)

These constraints have been considered in ref. [ 14 ], 559

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but are essential from our point of view as they form the statement of a theorem:

Theorem. There is a one-to-one correspondence between the superholomorphic structures of a vector bundle, and the set of partially flat connections which are unitary with respect to some hermitian metric. Recall that a hermitian metric on a vector bundle V is a smooth function ( , ): VO3V--,C j'° such that the restriction to each fibre is a hermitian inner product: (v, w) = (w, v). Since the image of this map is in C ~.o, it does not make sense to impose a positive definite type condition. In this context, a connection d~ is unitary if it satisfies the metric compatibility condition

d(s, t) = (dAS, t) + (S, dAt)

(20)

for all sections s, te g°(M, V). We are now in a position to prove the theorem.

Proof Given a partially flat connection dA, w e can obtain a superholomorphic structure via d'~ - ( d A ) 0"1.

(21)

In the other direction, if we are given a superholomorphic structure d'/~, then with respect to some hermitian metric we can define a unitary connection by dA -= d.'.',+ 0 - q ~ -

d0.4o,

(22)

i.e., we take A : = - A : and Ao= --4o. This completes the verification of the theorem. It is noteworthy that the curvature of the connections we have considered is completely determined by the Foa component. Recalling that for a partially flat connection, 9090 = ~ and ~ o ~ = ~_~, it is trivial to verify the following relations: F : o = - ~oFoa,

F__:= ~ogaFoo.

(23)

From these equations, one can immediately see the structure of off-shell N = 1 super Yang-Mills theory; the Foa component o f curvature is simply a scalar superfield which generates the entire multiplet consisting of a scalar, a complex spinor, and a gauge field. 560

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In the case of line bundles, one should also note that F is both 0 and 0 closed. As a side remark, we note that if we had not imposed the condition that the connection be unitary, then the curvature would only vanish in the ~, t~ direction: F~7 = F-o = 0.

3. Concluding remarks A further remark that we would like to comment on concerns the relationship of this work with certain aspects of Witten's twistor-like transform [ 16 ]. In that paper, the canonical constraints of D = 10 super Yang-Mills theory were obtained as an integrability condition for the connection to be pure gauge along a certain light-like line in superspace. We can similarly consider this notion; the embedding superspace is a SRS, and the corresponding definition of a "lightlike" line is: z = ~o + z 2 - ~6o2, 0=tTo + M,

(24)

where E(r) is a one-parameter function, and 2 is a constant Grassmann number. It is noteworthy that these lines are in fact superconformal functions of (Zo, 0o). Requiring that the connection be both pure gauge when restricted to such a line in M, as well as unitary, leads to the curvature constraints ( 19 ) which we called partially fiat. As in the ten-dimensional case, these "superconformal lines" arise as solutions to the equations of motion for the corresponding two-dimensional superparticle. We also expect that the consistent coupling of the superparticle to a curved SRS background will lead to the standard torsion constraints of two-dimensional supergravity. A discussion of this will be treated in more detail elsewhere [221. In this paper, we have established a foundation for extending the work of Hitchin [ 12 ] to vector bundles over super Riemann surfaces. We have shown that the superholomorphic structures on vector bundles are in one-to-one correspondence with the globally inequivalent N = 1 super Yang-Mills theories. Ultimately, we hope that further work along the lines presented here will lead to a gauge field description

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of super moduli space. Much remains to be done, and a study of this problem is in progress.

Acknowledgement The authors would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

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[8 ] D. Friedan, in: Unified string theories, eds. M. Green and D. Gross (World Scientific, Singapore, 1986). [ 9 ] M.A. Baranov, I.V. Frolov and A.S. Shvarts, Teor. Mat. Fiz. 7O (1987) 92. [10]L. Crane and J.M. Rabin, Super Riemann surfaces: Uniformization and Teichmiiller theory, EFI preprint 8625 (1986). [ 11 ] A.A. Rosly, A.S. Schwarz and A.A. Voronov, Geometry of superconformal manifolds, ITEP preprint 87-107 ( 1987 ). [ 12] N.J. Hitchin, Proc. Loud. Math. Soc. (3)55 (1987) 59. [ 131 B. Zumino, Phys. Lett. B 69 ( 1977 ) 369. [ 14 ] S. Giddings and P. Nelson, Line bundles on super Riemann surfaces, Harvard preprint HUTP-87/A080 ( 1987 ). [ 15 ] N.J. Hitchin, Gauge theory on Riemann surfaces, lectures ICTP College on Riemann surfaces (Trieste, December 1987). [ 16] E. Witten, Nucl. Phys. B 266 (1986) 245. [ 17 ] B. DeWitt, Supermanifolds (Cambridge U.P., Cambridge, 1984). [18] A. Rogers, J. Math. Phys. 21 (1980) 1352. [19] R. Horsley, Nucl. Phys. B 138 (1978) 493. [20] M.F. Atiyah and R. Bott, Phil. Trans. R. Soc. Lond. A 308 (1982) 523. [21 ] O. Forster, Lectures on Riemann surfaces (Springer, Berlin, 1981). [ 22 ] M. Rakowski and G. Thompson, in preparation.

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