Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy

Based on the dispersionless KP (dKP) theory, we study a topological Landau-Ginzburg (LG) theory characterized by a rational potential. Writing the dKP hierarchy in a general form treating all the primaries in an equal basis, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having a finite number of primaries. Several flat solutions of the topological LG theory are obtained in this formulation, and are identified with those discussed by Dubrovin. We explicitly construct gravitational descendants for all the primary fields. Giving a residue formulae for the 3-point functions of the fields, we show that these 3-point functions satisfy the topological recursion relation. The string equation is obtained as the generalized hodograph solutions of the dKP hierarchy, which show that all the gravitational effects to the constitutive equations (2-point functions) can be renormalized into the coupling constants in the small phase space. Supported in part by NSF grant DMS-9403597.     

Operators with Large R-Charge in N=4 Yang-Mills Theory

It has been recently proposed that string theory in the background of a plane wave corresponds to a certain subsector of the N=4 supersymmetric Yang-Mills theory. This correspondence follows as a limit of the AdS/CFT duality. As a particular case of the AdS/CFT correspondence, it is a priori a strong-weak coupling duality. However, the predictions for the anomalous dimensions which follow from this particular limit are analytic functions of the 't Hooft coupling constant λ and have a well-defined expansion in the weak coupling regime. This allows one to conjecture that the correspondence between the strings on the plane wave background and the Yang-Mills theory works at the level of perturbative expansions. In our paper we perform perturbative computations in the Yang-Mills theory that confirm this conjecture. We calculate the anomalous dimension of the operator corresponding to the elementary string excitation. We verify at the two-loop level that the anomalous dimension has a finite limit when the R-charge J→∞ keeps λ/J² finite. We conjecture that this is true at higher orders of perturbation theory. We show, by summing an infinite subset of Feynman diagrams, under the above assumption, that the anomalous dimensions arising from the Yang-Mills perturbation theory are in agreement with the anomalous dimensions following from the string worldsheet sigma-model.     

BV Quantization of Topological Open Membranes

We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids.Dept. of Particle Physics, Weizmann Institute, Rehovot, IsraelMathematics Graduate Center, CUNY, New York, USA     

Large N 2D Yang-Mills Theory and Topological String Theory

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A= 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's “Ω^-1 points ”. We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach. Received: 3 March 1994 / Accepted: 2 February 1995     

A fermionic loop wave equation for quantum chromodynamics at Nc = + ∞

A fermionic loop wave equation for euclidean QCD in the 't Hooft topological limit is considered. Arguments are given that this equation leads to a fermionic (supersymmetric) string representation for the above theory.     

High-energy QCD as a topological field theory

We propose an identification of the conformal field theory underlying Lipatov's spin-chain model of high-energy scattering in perturbative QCD. It is a twisted N = 2 supersymmetric topological field theory, which arises as the limiting case of the SL(2,R)/U(1) non-linear model that also plays a role in describing the Quantum Hall effect and black holes in string theory. The doubly-infinite set of non-trivial integrals of motion of the high-energy spin-chain model displayed by Faddeev and Korchemsky are identified as the Cartan subalgebra of a bosonic sub-symmetry possessed by this topological theory. The renormalization group and an analysis of instanton perturbations yield some understanding why this particular topological spin-chain model emerges in the high-energy limit, and provide a new estimate of the asymptotic behaviour of multi-Reggeized-gluon exchange. Received: 31 August 1998 / Published online: 11 March 1999     

Test of the anti-de Sitter-space/conformal-field-theory correspondence using high-spin operators

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the N=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resumes the perturbative series. In a confirmation of the anti-de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in AdS5. We also make a numerical prediction for the third term in the strong coupling series.     

Stokes phenomena and non-perturbative completion in the multi-cut two-matrix models

The Stokes multipliers in the matrix models are invariants in the string-theory moduli space and related to the D-instanton chemical potentials. They not only represent non-perturbative information but also play an important role in connecting various perturbative string theories in the moduli space. They are a key concept to the non-perturbative completion of string theory and also expected to imply some remnant of strong coupling dynamics in M theory. In this paper, we investigate the non-perturbative completion problem consisting of two constraints on the Stokes multipliers. As the first constraint, Stokes phenomena which realize the multi-cut geometry are studied in the Zk symmetric critical points of the multi-cut two-matrix models. Sequence of solutions to the constraints are obtained in general k-cut critical points. A discrete set of solutions and a continuum set of solutions are explicitly shown, and they can be classified by several constrained configurations of the Young diagram. As the second constraint, we discuss non-perturbative stability of backgrounds in terms of the Riemann-Hilbert problem. In particular, our procedure in the 2-cut (1,2) case (pure-supergravity case) completely fixes the D-instanton chemical potentials and results in the Hastings-McLeod solution to the Painlevé II equation. It is also stressed that the Riemann-Hilbert approach realizes an off-shell background independent formulation of non-critical string theory.     

Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

We formulate the spin foam perturbation theory for three-dimensional Euclidean Quantum Gravity with a cosmological constant. We analyse the perturbative expansion of the partition function in the dilute-gas limit and we argue that the Baez conjecture stating that the number of possible distinct topological classes of perturbative configurations is finite for the set of all triangulations of a manifold is not true. However, the conjecture is true for a special class of triangulations which are based on subdivisions of certain 3-manifold cubulations. In this case we calculate the partition function and show that the dilute-gas correction vanishes for the simplest choice of the volume operator. By slightly modifying the dilute-gas limit, we obtain a nonvanishing correction which is related to the second order perturbative correction. By assuming that the dilute-gas limit coupling constant is a function of the cosmological constant, we obtain a value for the partition function which is independent of the choice of the volume operator. Member of the Mathematical Physics Group, University of Lisbon.     

Matrix string theory

Via compactification on a circle, the matrix mode] of M-theory proposed by Banks et a]. suggests a concrete identification between the large N limit of two-dimensional N = 8 supersymmetric Yang-Mills theory and type IIA string theory. In this paper we collect evidence that supports this identification. We explicitly identify the perturbative string states and their interactions, and describe the appearance of D-particle and D-membrane states.     

All conjugate-maximal-helicity-violating amplitudes from topological open string theory in twistor space

It has recently been proposed that the D-instanton expansion of the open topological B model on P(3|4) is equivalent to the perturbative expansion of the maximally supersymmetric Yang-Mills theory in four dimensions. In this letter we show how to construct the gauge theory results for all n-point conjugate-maximal-helicity-violating amplitudes by computing the integral over the moduli space of curves of degree n-3 in P(3|4), providing strong support to the string theory construction.     

Broken supersymmetry in the matrix model on a circle

We consider the discretization of aD=2 surface using polygons. We map the surface onto superspace and integrate over surfaces of arbitrary genus, obtaining a discretized version of the Green-Schwarz string inD=1. Taking an unusual critical limit of the supersymmetric matrix model involved, we construct exact solutions, to all perturbative orders, for the discretized superstring in one dimension, both when the target space is a real line and when the theory is represented in terms of matrix variables on a circle of finite radius. We comment on the behavior of the compactfied perturbative expansion under duality transformations.BITNET: BELLUCCI at IRMLNF     

Extended holomorphic anomaly and loop amplitudes in open topological string

Open topological string amplitudes on compact Calabi–Yau threefolds are shown to satisfy an extension of the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. The total topological charge of the D-brane configuration must vanish in order to satisfy tadpole cancellation. The boundary state of such D-branes is holomorphically captured by a Hodge theoretic normal function. Its Griffiths' infinitesimal invariant is the analogue of the closed string Yukawa coupling and plays the role of the terminator in a Feynman diagram expansion for the topological string with D-branes. The holomorphic anomaly equation is solved and the holomorphic ambiguity is fixed for some representative worldsheets of low genus and with few boundaries on the real quintic.     

Superstring on pp-wave orbifold from large-N quiver gauge theory

We extend the proposal of Berenstein, Maldacena and Nastase to the Type IIB superstring propagating on a pp-wave over the R ⁴/Z _k orbifold. We show that first-quantized free string theory is described correctly by the large-N, fixed gauge coupling limit of [U(N)] ^k quiver gauge theory. We propose a precise map between gauge theory operators and string states for both untwisted and twisted sectors. We also compute leading-order perturbative correction to the anomalous dimensions of these operators. The result is in agreement with the value deduced from the string energy spectrum, thus substantiating our proposed operator-state map. Received: 14 March 2002 / Published online: 5 July 2002     

Variational perturbation theory for Fokker-Planck equation with nonlinear drift

We develop a recursive method for perturbative solutions of the Fokker-Planck equation with nonlinear drift. The series expansion of the time-dependent probability density in terms of powers of the coupling constant is obtained by solving a set of first-order linear ordinary differential equations. Resumming the series in the spirit of variational perturbation theory we are able to determine the probability density for all values of the coupling constant. Comparison with numerical results shows exponential convergence with increasing order.     

Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local $${\mathbb{C}\mathbb{P}^2}$$

The holomorphic anomaly equations describe B-model closed topological strings in Calabi–Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local \({\mathbb{C}\mathbb{P}^2}\) toric Calabi–Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by \({\mathbb{Z}_3}\) symmetry, alongside another action related to the Kähler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.     

Saturation and universality in QCD at small x

We find approximate solutions to the renormalization group equation which governs the quantum evolution of the effective theory for the Color Glass Condensate. This is a functional Fokker–Planck equation which generates in particular the non-linear evolution equations previously derived by Balitsky and Kovchegov within perturbative QCD. In the limit where the transverse momentum of the external probe is large compared to the saturation momentum, our approximations yield the Gaussian ansatz for the effective action of the McLerran–Venugopalan model. In the opposite limit, of a small external momentum, we find that the effective theory is governed by a scale-invariant universal action which has the correct properties to describe gluon saturation.     

Generalised Scalar-tensor Theory in the Bianchi Type I Model

We use a conformal transformation to find solutions to the generalised scalar-tensor theory, with a coupling constant dependent on a scalar field, in an empty Bianchi type I model. We describe the dynamical behaviour of the metric functions for three different couplings: two exact solutions to the field equations and a qualitative one are found. They exhibit non-singular behaviours and kinetic inflation. Two of them admit both General Relativity and string theory in the low-energy limit as asymptotic cases.     

A Path Integral Approach¶to the Kontsevich Quantization Formula

We give a quantum field theory interpretation of Kontsevich's deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra. Received: 10 March 1999 / Accepted: 30 January 2000