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.     

Grassmannian and String Theory

The infinite-dimensional Grassmannian manifold contains moduli spaces of Riemann surfaces of all genera. This well known fact leads to a conjecture that non-perturbative string theory can be formulated in terms of the Grassmannian. We present new facts supporting this hypothesis. In particular, it is shown that Grassmannians can be considered as generalized moduli spaces; this statement permits us to define corresponding “string amplitudes” (at least formally). One can conjecture that it is possible to explain the relation between non-perturbative and perturbative string theory by means of localization theorems for equivariant cohomology; this conjecture is based on the characterization of moduli spaces, relevant to string theory, as sets consisting of points with large stabilizers in certain groups acting on the Grassmannian. We describe an involution on the Grassmannian that could be related to S-duality in string theory. Received: 19 December 1996 / Accepted: 27 March 1998     

String theory and algebraic geometry of moduli spaces

It is shown how the algebraic geometry of the moduli space of Riemann surfaces entirely determines the partition function of Polyakov's string theory. This is done by using elements of Arakelov's intersection theory applied to determinants of families of differential operators parametrized by moduli space. As a result we write the partition function in terms of exponentials of Arakelov's Green functions and Faltings' invariant on Riemann surfaces. Generalizing to arithmetic surfaces, i.e. surfaces which are associated to an algebraic number fieldK, we establish a connection between string theory and the infinite primes ofK. As a result we conjecture that the usual partition function is a special case of a new partition function on the moduli space defined overK.     

Explicit parametrizations of degenerate surfaces from the KP equation and closed bosonic string amplitudes

The p-loop amplitude of closed bosonic string theory involves the integration over the moduli space. We seek an explicit parametrization of Riemann matrices in terms of 3p - 3 complex variables by solving the Kadomcev-Petviasvili (KP) equation. We find explicit solutions of this problem (Schottky problem) for certain types of degenerate surfaces. For these classes of surfaces, we obtain closed bosonic string amplitudes from the Belavin-Knizhnik theorem using our parametrizations. We show in what precise way they are related to the correlation functions on the Riemann surfaces.     

Conformal geometry and string field theory

The covariant perturbation theory rules that should arise from any gauge invariant string field theory, such as those proposed on the basis of BRST formalism, are set forth. The resulting path integral expressions naturally produce coordinate invariant densities on the moduli space of Riemann surfaces; these include the Koba-Nielsen amplitudes. The connection between string field theory and modular invariance is discussed, and it is proposed that future explorations in string field theory focus on coordinate invariant quantities on moduli space.     

Summations over equilaterally triangulated surfaces and the critical string measure

We propose a new approach to the summation over dynamically triangulated Riemann surfaces which does not rely on properties of the potential in a matrix model. Instead, we formulate a purely algebraic discretization of critical string path integral. This is combined with a technique which assigns to each equilateral triangulation of a two-dimensional surface a Riemann surface defined over a certain finite extension of the field of rational numbers, i.e. an arthmetic surface. Thus we establish a new formulation in which the sum over randomly triangulated surfaces defines an invariant measure on the moduli space of arithmetic surfaces. It is shown that because of this it is far from obvious that this measure for large genera approximates the measure defined by the continuum theory, i.e. Liouville theory or critical string theory. In low genus this subtlety does not exist. In the case of critical string theory we explicity compute the volume of the moduli space of arithmetic surfaces in terms of the modular height function and show that for low genus it approximates correctly the continuum measure. We also discuss a continuum limit which bears some resemblance with a double scaling limit in matrix models.This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098Supported in part by NSF grant PHY85-15857     

Homotopy algebra morphism and geometry of classical string field theories

We discuss general properties of classical string field theories with symmetric vertices in the context of deformation theory. For a given conformal background, there are many string field theories corresponding to different decomposition of moduli space of Riemann surfaces. It is shown that any classical open string field theories on a fixed conformal background are A_∞-quasi-isomorphic to each other. This indicates that they have isomorphic moduli space of classical solutions. The minimal model theorem in A_∞-algebras plays a key role in these results. Its natural and geometric realization on formal supermanifolds is also given. The same results hold for classical closed string field theories, whose algebraic structures are governed by L_∞-algebras.     

The Hurwitz enumeration problem of branched covers and Hodge integrals

We use algebraic methods to compute the simple Hurwitz numbers for arbitrary source and target Riemann surfaces. For an elliptic curve target, we reproduce the results previously obtained by string theorists. Motivated by the Gromov–Witten potentials, we find a general generating function for the simple Hurwitz numbers in terms of the representation theory of the symmetric group S_n. We also find a generating function for Hodge integrals on the moduli space of Riemann surfaces with two marked points, similar to that found by Faber and Pandharipande for the case of one marked point.     

A proof that Witten's open string theory gives a single cover of moduli space

We show that Witten's open string diagrams are surfaces with metrics of minimal area under the condition that all nontrivial open Jordan curves be longer or equal to . The minimal area property is used together with a mini-max problem to establish a new existence and uniqueness theorem for quadratic differentials in open Riemann surfaces with or without punctures on the boundaries. This theorem implies that the Feynman rules of open string theory give a single cover of the moduli of open Riemann surfaces.Supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069     

Quadratic differentials and closed string vertices

Existence theorems for quadratic differentials on Riemann surfaces are used to explore the possible structure of local vertices in closed string field theory. The cell decomposition of the moduli space of the three-holed sphere is studied in some detail, and the problem of obtaining a single covering of moduli space is investigated.     

An ambiguity in fermionic string perturbation theory

Recent investigation by Verlinde and Verlinde has shown that the fermionic string loop amplitudes change by a total derivative term in the moduli space under a change of basis of the supermoduli. This ambiguity is addressed in the context of the heterotic string theory, and shown to be a consequence of an inherent ambiguity in defining integration over the variables of a Grassmann algebra—in this case the Grassmann-valued coordinates of the supermoduli space. A resolution of this ambiguity in genus-two within this formalism is also presented.     

Factorization of bosonic string amplitudes

We consider the bosonic string path integral over degenerating Riemann surfaces. We first review the factorization of conformal field theory on a degenerating surface. A careful treatment of the degeneration of the measure for moduli leads to a modification of the usual ghost insertions so as to assure covariance under a change of conformal frame. More generally, amplitudes with BRST invariant but conformally non-invariant operators are well defined with the covariant ghost insertions. As a detailed application we study the string modifications to the background field equations. We find to first order in the tadpole and all orders in string coupling that the ratio of the graviton source, dilaton source, and zero-point amplitude agrees with that found from general covariance and the soft-dilaton theorem in the low-energy field theory. We also discuss the unitarity of the bosonic string theory,     

Intersection Numbers on Moduli Spaces and Symmetries of a Verlinde Formula

We investigate the geometry and topology of a standard moduli space of stable bundles on a Riemann surface, and use a generalization of the Verlinde formula to derive results on intersection pairings. Received: 5 April 1996 / Accepted: 6 February 1997     

Geometric realization of conformal field theory on Riemann surfaces

Conformal field theory on a family of Riemann surfaces is formulated. We derive equations of motion of vacua which are parametrized by moduli of Riemann surfaces and show that these vacua are characterized uniquely by these equations. Our theory has a deep connection with Sato's theory of KP equations.     

Beta functions and S-matrix in string theory

The relationship between sigma model β-functions and string theory scattering amplitudes is proved. We derive our results for the closed bosonic string using the weak field expansion around the flat space. The equations of motion for all the background fields, including the heavy fields are considered. We show that the effective equations for the light fields are obtained by integrating out the heavy fields. It is shown that the contributions to the β-functions come from the boundary of moduli space on a punctured Riemann surface. String loop corrections to the equations of motion are also studied.     

Roaming moduli space using dynamical triangulations

In critical as well as in non-critical string theory the partition function reduces to an integral over moduli space after integration over matter fields. For non-critical string theory this moduli integrand is known for genus one surfaces. The formalism of dynamical triangulations provides us with a regularization of non-critical string theory. We show how to assign in a simple and geometrical way a moduli parameter to each triangulation. After integrating over possible matter fields we can thus construct the moduli integrand. We show numerically for c=0 and c=−2 non-critical strings that the moduli integrand converges to the known continuum expression when the number of triangles goes to infinity.     

Off-shell closed string amplitudes: Towards a computation of the tachyon potential

We derive an explicit formula for the evaluation of the classical closed string action for any off-shell string field, and for the calculation of arbitrary off-shell amplitudes. The formulae require a parametrization, in terms of some moduli space coordinates, of the family of local coordinates needed to insert the off-shell states on Riemann surfaces. We discuss in detail the evaluation of the tachyon potential as a power series in the tachyon field. The expansion coefficients in this series are shown to be geometrical invariants of Strebel quadratic differentials whose variational properties imply that closed string polyhedra, among all possible choices of string vertices, yield a tachyon potential which is as small as possible order by order in the string coupling constant. Our discussion emphasizes the geometrical meaning of off-shell amplitudes.     

Open-Closed Moduli Spaces and Related Algebraic Structures

We set up a Batalin–Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered J-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L _∞ and A _∞ algebraic structures associated to it.