Current Algebra on the Torus

We derive the N-point one-loop correlation functions for the currents of an arbitrary affine Kac-Moody algebra. The one-loop amplitudes, which are elliptic functions defined on the torus Riemann surface, are specified by group invariant tensors and certain constant tau-dependent functions. We compute the elliptic functions via a generating function, and explicitly construct the invariant tensor functions recursively in terms of Young tableaux. The lowest tensors are related to the character formula of the representation of the affine algebra. These general current algebra loop amplitudes provide a building block for open twistor string theory, among other applications.     

Spectral functions,special functions and the Selberg zeta function

The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.Member of CNRS     

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.     

Liouville Correlation Functions from Four-Dimensional Gauge Theories

We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2{\mathcal{N}=2} SCFTs recently defined by one of the authors. We conduct extensive tests of the conjecture at genus 0, 1.     

Algebraic structure of topological superconformal field theory on Riemann surfaces

The algebraic structure of a topological superconformal field theory on a compact Riemann surface is investigated. The Krichever-Novikov [K-N] global operator formalism is used to obtain anN=4 super K-N algebra on a Riemann surface. Subsequently thisN=4 algebra is shown to posses anN=3 K-N subalgebra. TheN=3 subalgebra is then twisted to derive a topological version of the Krichever-Novikov algebra with a residualN=2 superconformal structure. The BRST charge of the associated topological field theory on the Riemann surface is shown to be genus dependent in this formalism and the global generalization of the BRST derivative conditions are obtained. The complete BRST structure of the theory is explicitly elucidated.     

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     

Non-linearσ-models on compact Riemann surfaces

The classicalO(3) non-linear -model is generalised to a theory of fields defined on a compact Riemann surfaceM with values in a compact Kähler manifoldV. The dimension of the space of self-dual fields fromM to the complex projective space ^N is calculated and the classifying space for the inequivalent quantisations of the theory is also calculated.work supported by the Science and Engineering Research Council     

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.     

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     

On the Schwinger model on Riemann surfaces

In this paper the Schwinger model or two-dimensional quantum electrodynamics is exactly solved on a Riemann surface providing the explicit expression of the partition function and of the generating functional of the amplitudes between the fermionic currents. This offers one of the few examples in which it is possible to integrate in an explicit way a gauge field theory interacting with matter on a Riemann surface.     

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.     

Graded Riemann surfaces and Krichever-Novikov algebras

Following the work of Krichever and Novikov, Bonora, Martellini, Rinaldi and Russo defined a superalgebra associated to each compact Riemann surface with spin structure. Noting that this data determines a graded Riemann surface, we find a natural interpretation of the BMRR-algebra in terms of the geometry of graded Riemann surfaces. We also discuss the central extensions of these algebras (correcting the form of the central extension given by Bonoraet al.). It is hoped that this work will be the first step towards defining Krichever-Novikov algebras for (the more general) super-Riemann surfaces; in particular we emphasise the importance ofgraded conformal vectorfields.     

Krichever-Novikov formulation of topological conformal field theory

The Krichever-Novikov (KN) global operator formalism is applied to construct a topological conformal field theory on a compact Riemann surface from an N=2 super-conformal field theory. The topological version of the KN algebra is derived and the BRST charge is shown to be genus-dependent in this formulation. This leads to an interesting cohomology structure for the physical subspace of the Hilbert space.     

Non-archimedean string dynamics

Explicit formulas for the N-point tree amplitudes of the non-archimedean open string are derived. These amplitudes can be generated from a simple non-local lagrangian involving a single scalar field (the tachyon) in ambient space-time. This lagrangian is studied and is found to possess a tachyon free vacuum with no “particles” but with soliton solutions. The question of generalizing the adelic product formular to N-point amplitudes is taken up. The infinite product of 5-point amplitudes is shown to converge in a suitably chosen kinematic region whence it can be analytically continued. Though the precise form of the product formula for the 5-point (and N-point)amplitudes is not found, it is shown that the product is not equal to one as it is for the 4-point amplitudes but rather involves the famous zeros of the Riemann zeta function. Chan-Paton rules for non-archimedean open strings are given. A string over the (global) field of rational numbers is constructed. Other problems that are addressed are the introduction of supersymmetry, the nature of a p-adic string lagrangian, and the possibility of strings over other locally compact fields.     

Determinants of Laplacians on surfaces of finite volume

Determinants of the Laplace and other elliptic operators on compact manifolds have been an object of study for many years (see [MP, RS, Vor]). Up until now, however, the theory of determinants has not been extended to non-compact situations, since these typically involve a mixture of discrete and continuous spectra. Recent advances in this theory, which are partially motivated by developments in mathematical physics, have led to a connection, in the compact Riemann surface case, between determinants of Laplacians on spinors and the Selberg zeta function of the underlying surface (see [DP, Kie, Sar, Vor]).Our purpose in this paper is to introduce a notion of determinants on non-compact (finite volume) Riemann surfaces. These will be associated to the Laplacian shifted by a parameters(1–s), and will be defined in terms of a Dirichlet series (w, s) which is a sum that represents the discrete as well as the continuous spectrum. It will be seen to be regular atw=0, and our main theorem (see Sect. 1) will express exp as the Selberg zeta function of the surface times the appropriate -factor.A Sloan Fellow and partially supported by NSF grant DMS-8701865     

String theory beyond the Planck scale

The high-energy behavior of string scattering amplitudes is studied to all orders in perturbation theory, with the aim of exploring the short-distance structure of string theory. It is shown that the sum over all Riemann surfaces is dominated by a saddle point. Consequently, the high-energy limit is universal and simple to calculate. In this limit, furthermore, the amplitudes fall off in a stringy way - much faster than that allowed by field theory. The dominant saddle points are identified as coming from world sheets which are Z_G+1 symmetric algebraic curves, and their contribution to the scattering amplitude is evaluated. An interesting spacetime picture of the high-energy limit emerges. The issue of summing the perturbation expansion is addressed.     

From Euclidean to Minkowski space with the Cauchy–Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from Green’s functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy–Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson–Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy–Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation.     

The Topological Vertex

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.Acknowledgement We would like to thank D.-E.Diaconescu, R. Dijkgraaf, J. Gomis, A. Grassi, A. Iqbal, A. Kapustin, S. Katz, V. Kazakov, I. Kostov, C-C. Liu, H. Ooguri, J. Schwarz, S. Shenker and E. Zaslow for valuable discussions (and the cap!). The research of MA and CV was supported in part by NSF grants PHY-9802709 and DMS-0074329. In addition, CV thanks the hospitality of the theory group at Caltech, where he is a Gordon Moore Distinguished Scholar. M.A. is grateful to the Caltech theory group for hospitality during part of this work. A.K. is supported in part by the DFG grant KL-1070/2-1.     

P-adic theta functions and solutions of the KP hierarchy

Based on Schottky uniformization theory of Riemann surfaces, we construct a universal power series for (Riemann) theta function solutions of the KP hierarchy. Specializing this power series to the coordinates associated with Schottky groups overp-adic fields, we show that thep-adic theta functions of Mumford curves give solutions of the KP hierarchy.