Moduli of super Riemann surfaces

The basic properties of super Riemann surfaces are presented, and their supermoduli spaces are constructed, in a manner suitable for the application of algebro-geometric techniques to string theory.Supported in part by NSF Grant No. DMS-8704401Supported in part by NSF Grants No. DMS-8501783 and No. DMS-86107301(1)     

Line bundles on super Riemann surfaces

We give the elements of a theory of line bundles, their classification, and their connections on super Riemann surfaces. There are several salient departures from the classical case. For example, the dimension of the Picard group is not constant, and there is no natural hermitian form on Pic. Furthermore, the bundles with vanishing Chern number aren't necessarily flat, nor can every such bundle be represented by an antiholomorphic connection on the trivial bundle. Nevertheless the latter representation is still useful in investigating questions of holomorphic factorization. We also define a subclass of all connections, those which are compatible with the superconformal structure. The compatibility conditions turn out to be constraints on the curvature 2-form.     

Selberg supertrace formula for super Riemann surfaces

This paper is the third in a sequel to develop a super-analogue of the classical Selberg trace formula, the Selberg supertrace formula. It deals with bordered super Riemann surfaces. The theory of bordered super Riemann surfaces is outlined, and the corresponding Selberg supertrace formula is developed. The analytic properties of the Selberg super zeta-functions on bordered super Riemann surfaces are discussed, and super-determinants of Dirac-Laplace operators on bordered super Riemann surfaces are calculated in terms of Selberg super zeta-functions.Address from August 1993: II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany     

Connections on vector bundles over super Riemann surfaces

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.     

Reduction of 4-dim self-dual super Yang-Mills equations onto super Riemann surfaces

Recently, a self-dual super Yang-Mills equation over a super Reimann surface was obtained as the zero set of a moment map on the space of superconnections to the dual of the super Lie algebra of gauge transformations. We present a new formulation of the 4-dim Euclidean self-dual super Yang-Mills equations in terms of constraints on the supercurvature. By dimensional reduction, we obtain the same set of superconformal field equations which define self-dual connections on a super Rieman surface.     

Heat kernels and super determinants of Laplace operators on super Riemann surfaces

Heat kernels of Laplacians on superfields of arbitrary tensor weight on super Riemann surfaces are constructed, and are used to compute the super determinants of these operators in terms of the Selberg super zeta function.Work supported in part by the National Science Foundation grant, NSF-PHY-80-19754     

Automorphic functions on super Riemann surfaces and fermionic string

Using the theory of super Riemann surfaces, the light-cone gauge formulation of the fermionic string by Mandelstam is extended to the one-loop level. The one-loop N-massless vector particle amplitude is explicitly calculated. The higher-loop extension is also discussed. We also demonstrate the equivalence between the light-cone picture and the Polyakov picture at the one-loop level.     

On the problem of splitting deformations of super Riemann surfaces

Letters in Mathematical Physics - An odd deformation of a super Riemann surface $$\mathcal {S}$$ is a deformation of $$\mathcal {S}$$ by variables of odd parity. In this article, we study the...     

Superholomorphic structures,moment maps,and super self-dual Yang-Mills connections over super Riemann surfaces

We consider the space of superconnections with certain curvature constraints over super Riemann surfaces. We define a moment map over that space to the dual of the super Lie algebra of gauge transformations. The zero set of this moment map corresponds to the super self-dual Yang-Mills equations in two dimensions. This result generalizes the recently proposed scheme for the nonsupersymmetric case. The superfield equations also arise from super self-dual Yang-Mills equations in four dimensions by dimensional reduction.     

The geometry of the super KP flows

A supersymmetric generalization of the Krichever map is used to construct algebro-geometric solutions to the various super Kadomtsev-Petviashvili (SKP) hierarchies. The geometric data required consist of a suitable algebraic supercurve of genusg (generallynot a super Riemann surface) with a distinguished point and local coordinates (z, ) there, and a generic line bundle of degreeg–1 with a local trivialization near the point. The resulting solutions to the Manin-Radul SKP system describe coupled deformations of the line bundle and the supercurve itself, in contrast to the ordinary KP system which deforms line bundles but not curves. Two new SKP systems are introduced: an integrable Jacobian system whose solutions describe genuine Jacobian flows, deforming the bundle but not the curve; and a nonintegrable maximal system describing independent deformations of bundle and curve. The Kac-van de Leur SKP system describes the same deformations as the maximal system, but in a different parametrization.     

Selberg supertrace formula for super Riemann surfaces,analytic properties of Selberg super zeta-functions and multiloop contributions for the fermionic string

In this paper a complete derivation of the Selberg supertrace formula for super Riemann surfaces and a discussion of the analytic properties of the Selberg super zeta-functions is presented. The Selberg supertrace formula is based on Laplace-Dirac operators _m of weightm on super Riemann surfaces. The trace formula for allmZ is derived and it is shown that one must discriminate between even and oddm. Particularly the term in the trace formula proportional to the identity transformation is sensitive to this discrimination. The analytic properties of the two Selberg super zeta-functions are discussed in detail, first with, and the second without consideration of the spin structure. We find for the Selberg super zeta-functions similarities as well as differences in comparison to the ordinary Selberg zeta-function. Also functional equations for the two Selberg super zeta-functions are derived. The results are applied to discuss the spectrum of the Laplace-Dirac operators and to ccalculate their determinants. For the spectrum it is found that the nontrivial Eigenvalues are the same for _m and ₀ up to a constant depending onm, which is analogous to the bosonic case. The analytic properties of the determinants can be deduced from the analytic properties of the Selberg super zeta-functions, and it is shown that they are well-defined. Special cases (m=0,2) for the determinants are important in the Polyakov approach for the fermionic string. With these results it is deduced that the fermionic string integrand of the Polyakov functional integral is well-defined.     

Quantum Riemann surfaces I. The unit disc

We construct a non-commutative ^*-algebra which is a quantum deformation of the algebra of continuous functions on the closed unit disc . is generated by the Toeplitz operators on a suitable Hilbert space of holomorphic functions onU.Supported in part by the National Science Foundation under grant DMS/PHY 88-16214     

Noncommutative coordinates on Riemann surfaces

We show that the definition of a projective coordinate frame within a Laguerre–Forsyth scheme, leads to the extension of the factorized diffeomorphism algebra. The quantum improvement of this symmetry can be performed only if these coordinates switch, at the quantum level, into a noncommutative regime.     

Quantum Riemann surfaces: II. The discrete series

We continue our study of noncommutative deformations of two-dimensional hyperbolic manifolds which we initiated in Part I. We construct a sequence of ^*-algebras which are quantizations of a compact Riemann surface of genus g corresponding to special values of the Planck constant. These algebras are direct integrals of finite-dimensional ^*-algebras.Supported by DOE under Grant DE-FG02-88ER25065.