2D Field Theories Research Articles

The structure of 2D semi-simple field theories

I classify the cohomological 2D field theories based on a semi-simple complex Frobenius algebra A. They are controlled by a linear combination of κ-classes and by an extension datum to the Deligne–Mumford boundary. Their effect on the Gromov–Witten potential is described by Givental’s Fock space formulae. This leads to the reconstruction of Gromov–Witten (ancestor) invariants from the quantum cup-product at a single semi-simple point and the first Chern class of the manifold, confirming Givental’s higher-genus reconstruction conjecture. This in turn implies the Virasoro conjecture for manifolds with semi-simple quantum cohomology. The classification uses the Mumford conjecture, proved by Madsen and Weiss (European Congress of Mathematics, pp. 283–303, 2005).

(de)Tails of Toda CFT

The relation between the partition function of N=2 gauge theories in 4d and conformal Toda field theory in 2d is explained for the case where the 4d theory is a linear quiver with "quiver tails". That is when the 4d theory has gauge groups of different rank. We propose an identification of a subset of the states of Toda CFT which represent the Coulomb-branch parameters of the different rank gauge multiplets and study their three-point functions and descendants.

2D and 3D topological field theories for generalized complex geometry

Using the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) prescription we construct 2D and 3D topological field theories associated to generalized complex manifolds. These models can be thought of ...

Finite volume spectrum of 2D field theories from Hirota dynamics

We propose, using the example of the O(4) sigma model, a general method for solving integrable two dimensional relativistic sigma models in a finite size periodic box. Our starting point is the so-called Y-system, which is equivalent to the thermodynamic Bethe ansatz equations of Yang and Yang. It is derived from the Zamolodchikov scattering theory in the cross channel, for virtual particles along the non-compact direction of the space-time cylinder. The method is based on the integrable Hirota dynamics that follows from the Y-system. The outcome is a nonlinear integral equation for a single complex function, valid for an arbitrary quantum state and accompanied by the finite size analogue of Bethe equations. It is close in spirit to the Destri-deVega (DdV) equation. We present the numerical data for the energy of various states as a function of the size, and derive the general Luscher-type formulas for the finite size corrections. We also re-derive by our method the DdV equation for the SU(2) chiral Gross-Neveu model.

Equivalent Hamiltonians forPT-symmetric versions of dual 2D field theories

In the context of non-Hermitian but PT-symmetric Hamiltonians, we explore the systematics of the coefficients of the multiple commutators that appear in the perturbative expansions for Q, the exponent of the metric operator, and h, the equivalent Hermitian Hamiltonian. We find exact expressions for both Q and h for PT-symmetric versions of the dual two-dimensional quantum field theories, the sine-Gordon model and the massive Thirring model, and elucidate how these solutions arise from the structure of the multiple commutator coefficients.

Non-constant non-commutativity in 2d field theories and a new look at fuzzy monopoles

We write down scalar field theory and gauge theory on two-dimensional non-commutative spaces M with non-vanishing curvature and non-constant non-commutativity. Usual dynamics results upon taking the limit of M going to (i) a commutative manifold M 0 having non-vanishing curvature and (ii) the non-commutative plane. Our procedure does not require introducing singular algebraic maps or frame fields. Rather, we exploit the Kähler structure in the limit (i) and identify the symplectic two-form with the volume two-form. As an example, we take M to be the stereographically projected fuzzy sphere, and find magnetic monopole solutions to the non-commutative Maxwell equations. Although the magnetic charges are conserved, the classical theory does not require that they be quantized. The non-commutative gauge field strength transforms in the usual manner, but the same is not, in general, true for the associated potentials. We develop a perturbation scheme to obtain the expression for gauge transformations about limits (i) and (ii). We also obtain the lowest order Seiberg–Witten map to write down corrections to the commutative field equations and show that solutions to Maxwell theory on M 0 are stable under inclusion of lowest order non-commutative corrections. The results are applied to the example of non-commutative AdS 2.

String topology: background and present state

(from [?]) The data of a “2D field theory with a closed string compactification” is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following assumptions. It is assumed the punctures are labeled and divided into nonempty sets of inputs and outputs. The inputs are marked by a tangent direction and the outputs are weighted by nonnegative real numbers adding to unity. It is assumed the gluing of inputs to outputs lands on the pseudomanifold boundary of the cell decomposition and the entire pseudomanifold boundary is decomposed into pieces by all such factorings. It is further assumed that the action is equivariant with respect to the toroidal action of rotating the markings. A main result of compactified string topology is the Theorem 1. (closed strings) Each oriented smooth manifold has a 2D field theory with a closed string compactification on the equivariant chains of its free loop space mod constant loops. The sum over all surface types of the top pseudomanifold chain yields a chain X satisfying the master equation dX + X ∗ X = 0 where ∗ is the sum over all gluings. This structure is well defined up to homotopy*. The genus zero parts yields an infinity Lie bialgebra on the equivariant chains of the free loop space mod constant loops. The higher genus terms provide further elements of algebraic structure* called a “quantum Lie bialgebra” partially resolving the involutive identity. There is also a compactified discussion and a Theorem 2 for open strings as the first step to a more complete theory. We note a second step for knots. *See the Appendix “Homotopy theory of the master equation” for more explanation.

Nonperturbative Superpotentials and Compactification to Three Dimensions

We consider four-dimensional N=2 supersymmetric gauge theories with gauge group U(N) on R^3 x S^1, in the presence of a classical superpotential. The low-energy quantum superpotential is obtained by simply replacing the adjoint scalar superfield in the classical superpotential by the Lax matrix of the integrable system that underlies the 4d field theory. We verify in a number of examples that the vacuum structure obtained in this way matches precisely that in 4d, although the degrees of freedom that appear are quite distinct. Several features of 4d field theories, such as the possibility of lifting vacua from U(N) to U(tN), become particularly simple in this framework. It turns out that supersymmetric vacua give rise to a reduction of the integrable system which contains information about the field theory but also about the Dijkgraaf-Vafa matrix model. The relation between the matrix model and the quantum superpotential on R^3 x S^1 appears to involve a novel kind of mirror symmetry.

The cyclic universe: An informal introduction

The Cyclic Model is a radical, new cosmological scenario which proposes that the Universe undergoes an endless sequence of epochs which begin with a ‘big bang’ and end in a ‘big crunch.’ When the Universes bounces from contraction to re-expansion, the temperature and density remain finite. The model does not include a period of rapid inflation, yet it reproduces all of the successful predictions of standard big bang and inflationary cosmology. We point out numerous novel elements that have not been used previously which may open the door to further alternative cosmologies. Although the model is motivated by M-theory, branes and extra dimensions, here we show that the scenario can be described almost entirely in terms of conventional 4d field theory and 4d cosmology.

Geometric dualities in 4d field theories and their 5d interpretation

We study four-dimensional N=1 gauge theories arising on D3-branes probing toric singularities. Toric dualities and flows between theories corresponding to different singularities are analyzed by encoding the geometric information into (p,q) webs. A new method for identifying global symmetries of the four-dimensional theories using the brane webs is developed. Five-dimensional theories are associated to the theories on the D3-branes by using (p,q) webs. This leads to a novel interpretation of Seiberg duality, as crossing curves of marginal stability in five dimensions.

Exact Solution of a Charge-Asymmetric Two-Dimensional Coulomb Gas

The model under consideration is an asymmetric two-dimensional Coulomb gas of positively (q1=+1) and negatively (q2=−1/2) charged pointlike particles, interacting via a logarithmic potential. This continuous system is stable against collapse of positive-negative pairs of charges for the dimensionless coupling constant (inverse temperature) β<4. The mapping of the Coulomb gas is made onto the complex Bullough–Dodd model, and recent results about that integrable 2D field theory are used. The mapping provides the full thermodynamics (the free energy, the internal energy, the specific heat) and the large-distance asymptotics of the particle correlation functions, in the whole stability regime of the plasma. The results are checked by a small-β expansion and close to the collapse β=4 point. The comparison is made with the exactly solvable symmetric version of the model (q1=+1,q2=−1), and some fundamental changes in statistics caused by the charge asymmetry are pointed out.

Prospects and problems of tachyon matter cosmology

We consider the evolution of FRW cosmological models and linear perturbations of tachyon matter rolling towards a minimum of its potential. The tachyon coupled to gravity is described by an effective 4d field theory of string theory tachyon. In the model where a tachyon potential V( T) has a quadratic minimum at finite value of the tachyon field T 0 and V( T 0)=0, the tachyon condensate oscillates around its minimum with a decreasing amplitude. It is shown that its effective equation of state is p=− ε/3. However, linear inhomogeneous tachyon fluctuations coupled to the oscillating background condensate are exponentially unstable due to the effect of parametric resonance. In another interesting model, where tachyon potential exponentially approaches zero at infinity of T, rolling tachyon condensate in an expanding Universe behaves as pressureless fluid. Its linear fluctuations coupled with small metric perturbations evolve similar to these in a pressureless fluid. However, this linear stage changes to a strongly non-linear one very early, so that the usual quasi-linear stage observed at sufficiently large scales in the present Universe may not be realized in the absence of the usual particle-like cold dark matter.

Problems with Tachyon Inflation

We consider cosmological consequences of string theory tachyon condensation. We show that it is very difficult to obtain inflation in the simplest versions of this theory. Typically, inflation in these theories could occur only at super-Planckian densities, where the effective 4D field theory is inapplicable. Reheating and creation of matter in models where the tachyon potential V(T) has a minimum at infinitely large T is problematic because the tachyon field in such theories does not oscillate. If the universe after inflation is dominated by the energy density of the tachyon condensate, it will always remain dominated by the tachyons. It might happen that string condensation is responsible for a short stage of inflation at a nearly Planckian density, but one would need to have a second stage of inflation after that. This would imply that the tachyon played no role in the post-inflationary universe until the very late stages of its evolution. These problems do not appear in the recently proposed models of hybrid inflation where the complex tachyon field has a minimum at T << M_p.

Hierarchic trees with branching number close to one: Noiseless Kardar-Parisi-Zhang equation with additional linear term for imitating two-dimensional and three-dimensional phase transitions.

An imitation of two-dimensional (2D) field theory is formulated by means of a model on the hierarchic tree (with branching number close to one) with the same potential and the free correlators identical to 2D correlator ones. Such a model carries on some features of the original model for certain scale invariant theories. The renormalization group equation for the free energy is the noiseless Kardar-Parisi-Zhang equation with an additional linear term.

Type II NS Five-Branes: Non Critical Strings and their Topological Sectors

The near-horizon geometry of parallel NS5-branes is described by an exact superconformal 2d field theory based on K^10 = W(k)^4 x M^6, with M^6 a flat 5+1 space time and W(k)^4 =U(1) x SU(2)(k), a four dimensional background with non-trivial dilaton. The ten-dimensional ``BULK'' spectrum of excitations of K^10 can be derived combining unitary representations of the N=4 superconformal theory of W(k)^4 and M^6 in a modular-invariant way. All Bulk states are massive and belong to the long representations of the N=2 six dimensional space-time supersymmetry. The NS5-brane states, propagating on M^6, belong to the short representations of N=2. Both the bulk and the brane spectrum is derived using the powerful worldsheet technics of the N=4 superconformal theories. We claim that both bulk and brane states are necessary to well define the theory. The non abelian U(n) or SO(2n) structure of the 5-brane fields follows from the fusion coefficients appearing in the correlation functions involving SU(2)(k) conformal fields.

Comments on the Holographic Picture of the Randall-Sundrum Model

We discuss some issues about the holographic interpretation of the compact Randall-Sundrum model, which is conjectured to be dual to a 4d field theory with non-linearly realized conformal symmetry. We make several checks of this conjecture. In particular, we show that the radion couples conformally to a background 4d metric. We also discuss the interpretation of the Goldberger-Wise mechanism for stabilizing the radion. We consider situations where the electroweak breaking stabilizes the radion and we discuss the issue of natural conservation of flavor quantum numbers.

Vacuum polarization in the Schwarzschild spacetime and dimensional reduction

A massless scalar field minimally coupled to gravity and propagating in Schwarzschild spacetime is considered. After dimensional reduction under spherical symmetry the resulting 2D field theory is canonically quantized and the renormalized expectation values 〈Tab〉 of the relevant energy-momentum tensor operator are investigated. Asymptotic behaviors and analytical approximations are given for 〈Tab〉 in the Boulware, Unruh and Hartle-Hawking states. Special attention is devoted to the black-hole horizon region where the WKB approximation breaks down.

BPS spectrum of 5 dimensional field theories, (p,q) webs and curve counting

We study the BPS spectrum of supersymmetric 5 dimensional field theories and their representations as string webs. It is found that a state of given charges exists when it has a representation as an irreducible string web. Its spin is determined by the string web. The number of fermionic zero modes is 8g+4b, where g is the number of internal faces and b is the number of boundaries. In the lift to M theory of 4d field theories such states are described by membranes ending on the 5-brane, breaking SUSY from 8 to 4, and g becomes the genus of the membrane. Mathematically, we obtain a diagrammatic method to find the spectrum of curves on a toric complex surface, and the number of their moduli.

On thermodynamic approaches to conformal field theory

We present the thermodynamic Bethe ansatz as a way to factorize the partition function of a 2d field theory, in particular, a conformal field theory and we compare it with another approach to factorization due to Schoutens which consists of diagonalizing matrix recursion relations between the partition functions at consecutive levels. We prove that both are equivalent, taking as examples the SU(2) spinons and the 3-state Potts model. In the latter case we see that there are two different thermodynamic Bethe ansatz equation systems with the same physical content, of which the second is new and corresponds to a one-quasiparticle representation, as opposed to the usual two-quasiparticle representation. This new thermodynamic Bethe ansatz system leads to a new dilogarithmic formula for the central charge of that model.

The field theory of non-supersymmetric brane configurations

We identify the 4d field theories living on the world-volume of D4-branes in non-supersymmetric type IIA string theory constructions. They are softly broken N = 2 SQCD with the breakings introduced through vevs of the auxiliary fields in the spurion coupling field. Exact solutions of these theories for perturbing soft breakings exist in the literature. We calculate the ratios of string tensions in softly broken N = 2 SU( N) gauge theory testing the recently proposed M-theory prediction. The semi-classical result of M-theory is renormalized in the non-supersymmetric models.