Disk amplitudes in the topological A-model on Calabi-Yau spaces and Fano manifolds

We study topological A-model disk amplitudes with Calabi-Yau target spaces by mirror symmetries. This allows us to calculate holomorphic instantons of Riemann surfaces with boundaries that are mapped into SUSY cycles in Calabi-Yau d-folds. Also we analyse disk amplitudes in Fano manifold cases by considering fusion relations between A-model operators.     

The topological quantum field theory of Riemann’s theta functions

In this paper we prove the existence and uniqueness of a topological quantum field theory that incorporates, for all Riemann surfaces, the corresponding spaces of theta functions and the actions of the Heisenberg groups and modular groups on them.     

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.     

N=2 topological Yang-Mills theory on compact Kähler surfaces

We study a topological Yang-Mills theory withN=2 fermionic symmetry. Our formalism is a field theoretical interpretation of the Donaldson polynomial invariants on compact Kähler surfaces. We also study an analogous theory on compact oriented Riemann surfaces and briefly discuss a possible application of Witten's non-Abelian localization formula to the problems in the case of compact Kähler surfaces.This article was typeset by the author using Pjour1     

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus g > 1, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation of the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus g > 1. Received: 12 September 1996 / Accepted: 6 January 1997     

New Index Formulas as a Meromorphic Generalization of the Chern–Gauss–Bonnet Theorem

Laplace operators perturbed by meromorphic potential on the Riemann and separated-type Klein surfaces are constructed and their indices are calculated in two different ways. The topological expressions for the indices are obtained from the study of the spectral properties of the operators. Analytical expressions are provided by the heat kernel approach in terms of functional integrals. As a result, two formulae connecting characteristics of meromorphic (real meromorphic) functions and topological properties of Riemann (separated-type Klein) surfaces are derived.     

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.     

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.     

Geometry of Higgs and Toda fields on Riemann surfaces

We discuss geometrical aspects of Higgs systems and Toda field theory in the framework of the theory of vector bundles on Riemann surfaces of genus greater than one. We point out how Toda fields can be considered as equivalent to Higgs systems — a connection on a vector bundle E together with an End(E)-valued one form both in the standard and in the Conformal Affine case. We discuss how variations of Hodge structures can arise in such a framework and determine holomorphic embeddings of Riemann surfaces into locally homogeneous spaces, thus giving hints to possible realizations of W_n-geometries.     

Dimension on discrete spaces

In this paper we develop some combinatorial models for continuous spaces. We study the approximations of continuous spaces by graphs, molecular spaces, and coordinate matrices. We define the dimension on a discrete space by means of axioms based on an obvious geometrical background. This work presents some discrete models ofn-dimensional Euclidean spaces,n-dimensional spheres, a torus, and a projective plane. It explains how to construct new discrete spaces and describes in this connection several three-dimensional closed surfaces with some topological singularities. It also analyzes the topology of (3+1)-space-time. We are also discussing the question by R. Sorkin about how to derive the system of simplicial complexes from a system of open coverings of a topological space.     

Riemann�CHilbert Problems for Hurwitz Frobenius Manifolds

In this paper we give an overview of the solutions of Fuchsian and non-Fuchsian Riemann-Hilbert problems associated with Frobenius manifold structures on Hurwitz spaces found in recent works of the authors. We show that by an application of an appropriate Laplace transform, one can derive solutions of the non-Fuchsian Riemann-Hilbert problem from the more recent construction of solutions of the Fuchsian linear system written in terms of meromorphic differentials on Riemann surfaces.     

Dimensional Reduction Over the Quantum Sphere and Non-Abelian q-Vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a K?hler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU _q (2), and also of invariant gauge connections on these bundles. The reduction of Yang–Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-K?hler quotient construction.     

Generalized Weierstrass representation for surfaces in multi-dimensional Riemann spaces

Generalizations of the Weierstrass formulae to surface immersed into ⁴, ⁴ and into multidimensional Riemann spaces are proposed. Integrable deformations of surfaces in these spaces via the modified Veselov-Novikov equation are discussed.     

Coinvariants for Lattice VOAs and q-Supernomial Coefficients

 We propose an alternative definition of the $q$-supernomial coefficients as the characters of coinvariants for the one–dimensional lattice vertex operator algebras. This provides a new formula for the q-supernomial coefficients. We also prove that the spaces of the coinvariants form a bundle over the configuration space of complex points on Riemann surfaces (the configuration space includes the diagonals). Received: 13 August 2001 / Accepted: 23 January 2002 Published online: 6 August 2002     

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.