Global structures for the moduli of (punctured) super riemann surfaces

A fine moduli superspace for algebraic super Riemann surfaces with a level-n structure is constructed as a quotient of the split superscheme of local spin-gravitivo fields by an étale equivalence relation. This object is not a superscheme, but still has an interesting structure: it is an algebraic superspace, that is, an analytic superspace with sufficiently many meromorphic functions. The moduli of super Riemann surfaces with punctures (fixed points in the supersurface) is also constructed as an algebraic superspace. Moreover, when one only considers ordinary punctures (fixed points in the underlying ordinary curve), it turns out that the moduli is a true superscheme. We prove furthermore that this moduli superscheme is split.     

Super Riemann surfaces: Uniformization and Teichmüller theory

Teichmüller theory for super Riemann surfaces is rigorously developed using the supermanifold theory of Rogers. In the case of trivial topology in the soul directions, relevant for superstring applications, the following results are proven. The super Teichmüller space is a complex super-orbifold whose body is the ordinary Teichmüller space of the associated Riemann surfaces with spin structure. For genusg>1 it has 3g-3 complex even and 2g-2 complex odd dimensions. The super modular group which reduces super Teichmüller space to super moduli space is the ordinary modular group; there are no new discrete modular transformations in the odd directions. The boundary of super Teichmüller space contains not only super Riemann surfaces with pinched bodies, but Rogers supermanifolds having nontrivial topology in the odd dimensions as well. We also prove the uniformization theorem for super Riemann surfaces and discuss their representation by discrete supergroups of Fuchsian and Schottky type and by Beltrami differentials. Finally we present partial results for the more difficult problem of classifying super Riemann surfaces of arbitrary topology.Enrico Fermi Fellow. Research supported by the NSF (PHY 83-01221) and DOE (DE-AC02-82-ER-40073).     

New moduli spaces from string background independence consistency conditions

In string field theory an infinitesimal background deformation is implemented as a canonical transformation whose hamiltonian function is defined by moduli spaces of punctured Riemann surfaces having one special puncture. We show that the consistency conditions associated to the commutator of two deformations are implemented by virtue of the existence of moduli spaces of punctured surfaces with two special punctures. The spaces are antisymmetric under the exchange of the special punctures, and satisfy recursion relations relating them to moduli spaces with one special puncture and to string vertices. We develop the theory of moduli spaces of surfaces with arbitrary number of special punctures and indicate their relevance to the construction of a string field theory that makes no reference to a conformal background. Our results also imply a partial antibracket cohomology theorem for the string action.     

Conformal Lie superalgebras and moduli spaces

A conformal Lie superalgebra is a superextension of the centerless Virasoro algebra W—the Lie algebra of complex vector fields on the circle. The algebras of Ramond and Neveu-Schwarz are not the only examples of such superalgebras. All known superconformal algebras can be obtained as comlexifications of Lie superalgebras of vector fields on a supercircle with an additional structure. For every such superalgebra

a class of geometric objects—complex

— is defined. For the superalgebras of Neveu-Schwarz and Ramond they are super Riemann surfaces with punctures of different kinds. We construct moduli superspaces for compact

, and show that the superalgebra

acts infinitesimally on the corresponding moduli space.     

Neveu-Schwarz- and Ramond-type superalgebras on genus-g Riemann surfaces

We extend the work of Krichever and Novikov on Virasoro type algebras by constructing Neveu-Schwarz and Ramond type superalgebras on genus-g Riemann surfaces. We realize these superalgebras and their central extension in the framework of superstring theory. We construct the corresponding BRST operators, which turn out to be nilpotent in ten dimensions.     

Higher genus superstring amplitudes from the geometry of moduli space

We show that the higher genus 4-point superstring amplitude is strongly constrained by the geometry of moduli space of Riemann surfaces. A detailed analysis leads to a natural proposal which satisfies several conditions. The result is based on the recently derived Siegel induced metric on the moduli space of Riemann surfaces and on combinatorial products of determinants of holomorphic Abelian differentials.     

Construction of constant curvature punctured Riemann surfaces with particle-scattering interpretation

A class of punctured constant curvature Riemann surfaces, with boundary conditions similar to those of the Poincaré half plane, is constructed. It is shown to describe the scattering of particle-like objects in two Euclidian dimensions. The associated time delays and classical phase shifts are introduced and connected to the behaviour of the surfaces at their punctures. For each such surface, we conjecture that the time delays are partial derivatives of the phase shift. This type of relationship, already known to be correct in other scattering problems, leads to a general integrability condition concerning the behaviour of the metric in the neighbourhood of the punctures. The time delays are explicitly computed for three punctures, and the conjecture is verified. The result, reexpressed as a product of Riemann zeta-functions, exhibits an intringuing number-theoretic structure: a p-adic product formula holds and one of Ramanujan's identities applies. An ansatz is given for the corresponding exact quantum S-matrix. It is such that the integrability condition is replaced by a finite difference relation only involving the exact spectrum already derived, in the associated Liouville field theory, by Gervais and Neveu.     

Two-loop superstrings VII. Cohomology of chiral amplitudes

The relation between superholomorphicity and holomorphicity of chiral superstring N-point amplitudes for NS bosons on a genus 2 Riemann surface is shown to be encoded in a hybrid cohomology theory, incorporating elements of both de Rham and Dolbeault cohomologies. A constructive algorithm is provided which shows that, for arbitrary N and for each fixed even spin structure, the hybrid cohomology classes of the chiral amplitudes of the N-point function on a surface of genus 2 always admit a holomorphic representative. Three key ingredients in the derivation are a classification of all kinematic invariants for the N-point function, a new type of 3-point Green's function, and a recursive construction by monodromies of certain sections of vector bundles over the moduli space of Riemann surfaces, holomorphic in all but exactly one or two insertion points.     

Super Teichmüller spaces: Punctures and elliptic points

A model for the super Teichmüller space is described and calculated when the super Riemann surfaces no longer have a compact nonsingular body manifold. The bosonic and fermionic dimensions turn out to be the dimensions of the appropriate spaces of automorphic forms, in the case where the body has a finite number of punctures and/or elliptic points.     

A local index theorem for families of\bar \partial -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

We prove a local index theorem for families of \(\bar \partial \) -operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmüller spaceT _g,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.     

The Selberg zeta function and a new Kähler metric on the moduli space of punctured Riemann surfaces

A local index theorem for families of -operators on Riemann surfaces with functures is proved. A new Kähler metric on the moduli space of punctured surfaces is described in terms of the Eisenstein-Maass series.     

How covariant closed string theory solves a minimal area problem

For a given genusg Riemann surface withn0 punctures (n3 forg=0) we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2. The minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures. These metrics are induced by Jenkins-Strebel quadratic differentials. They arise from the string diagrams corresponding to restricted Feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra.Supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069     

Conformal theories and punctured surfaces

We define conformal theories as realizations of certain operations involving punctured Riemann surfaces (with coordinates chosen at the punctures) in a Hilbert space. We describe the connections of our formalism with other formulations of conformal theories.     

Matrix theory interpretation of discrete light cone quantization string worldsheets

We study the null compactification of type-IIA string perturbation theory at finite temperature. We prove a theorem about Riemann surfaces establishing that the moduli spaces of infinite-momentum-frame superstring worldsheets are identical to those of branched-cover instantons in the matrix-string model conjectured to describe M theory. This means that the identification of string degrees of freedom in the matrix model proposed by Dijkgraaf, Verlinde, and Verlinde is correct and that its natural generalization produces the moduli space of Riemann surfaces at all orders in the genus expansion.     

Consistency of open superstring theories

The infinities and the anomalies in the G≠SO(32) open superstring arise from the same limit of the one-loop Riemann surfaces. The dilaton tadpole infinity is removable by an appropriate choice of background, as in the work of Fischler and Susskind. However, we find that the anomalies arise from the tadpole of an unphysical field, and cannot be removed by any choice of background. Thus, the spacetime anomalies are due to a world-sheet superconformal anomaly of topological origin. We also give a general discussion of the relation between anomalies and surface terms in moduli space, and point out several new applications.     

Estimates of Weil–Petersson Volumes¶via Effective Divisors

We study the asymptotics of the Weil–Petersson volumes of the moduli spaces of compact Riemann surfaces of genus g with n punctures, for fixed n as g→∞. Received: 18 May 2000 / Accepted: 1 April 2001