Operadic formulation of topological vertex algebras and Gerstenhaber or Batalin-Vilkovisky algebras

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad).The author is supported in part by NSF grant DMS-9104519     

Dyons in topological field theories

We examine a class of topological field theories defined by Lagrangians that under certain conditions can be written as the sum of two characteristic numbers or winding numbers. Therefore, the action or the energy is a topological invariant and stable under perturbations. The sufficient conditions required for stability take the form of first-order field equations, analogous to the self-duality and Bogomol'nyi equations in Yang-Mills(-Higgs) theory. Solutions to the first-order equations automatically satisfy the full field equations. We show the existence of nontrivial, nonsingular, minimum energy spherically symmetric dyon solutions and that they are stable. We also discuss evidence for a dual field theory to Yang-Mills-Higgs in topological field theory. The existence of dual field theories and electric monopoles is predicted by Montonen and Olive.     

On topological field theories and duality

Topologically non-trivial effects appearing in the discussion of duality transformations in higher genus manifolds are discussed in a simple example. Their relation with the properties of Topological Field Theories is established.     

KAC-moody algebras,Yang-Baxter-Zamolodchikov-Faddeev algebras and integrable field theories

A large class of integrable two-dimensional field theories exhibit Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and Kac-Moody (KM) algebras. Examples of them are chiral fermionic models, sigma models and Wess-Zumino-Witten sigma models. With their help an explicit link is found between representations of YBZF and KM algebras.     

On the identification of finite operator algebras in two-dimensional conformally invariant field theories

We investigate the structure of the linear differential operators whose solutions determine the four-point correlations of primary operators in the d = 2 conformally invariant SU(2) α-model with Wess-Sumino term and the d = 2 critical statistical systems with central Virasoro charge smaller than one. Factorisation properties of the differential operators are related to a finite closure of the operator algebras. We recover the selection and fusion rules of Fateev, Zamolodchikov and Gepner, Witten for the SU(2) α-model. It is outlined how the results of the SU(2) model can be used for the identification of closed operator algebras in the statistical model.     

Multi-instantons, supersymmetry and topological field theories

In this letter we argue that instanton-dominated Green's functions in N=2 Super Yang–Mills theories can be equivalently computed either using the so-called constrained instanton method or making reference to the topological twisted version of the theory. Defining an appropriate BRST operator (as a supersymmetry plus a gauge variation), we also show that the expansion coefficients of the Seiberg–Witten effective action for the low-energy degrees of freedom can be written as integrals of total derivatives over the moduli space of self-dual gauge connections.     

Electric monopoles in topological field theories

A tensor product generalization ofB ∧F theories is proposed that has a Bogomol'nyi structure. Nonsingular, stable, finite-energy particle-like solutions to the Bogomol'nyi equations are studied. Unlike Yang-Mills(-Higgs) theory, the Bogomol'nyi structure does not appear as a perfect square in the Lagrangian. Consequently, the Bogomol'nyi energy can be obtained in more than one way. The added flexibility permits electric monopole solutions.     

Linking observables in perturbed topological field theories

The topological antisymmetric tensor field theory in n-dimensions is perturbed by the introduction of local metric dependent interaction terms in the curvatures. The correlator describing the linking number between two surfaces in n-dimensions is shown to be not affected by the quantum corrections.     

New observables in topological instantonic field theories

Instantonic theories are quantum field theories where all correlators are determined by integrals over the finite-dimensional space (space of generalized instantons). We consider novel geometrical observables in instantonic topological quantum mechanics that are strikingly different from standard evaluation observables. These observables allow jumps of special type for the trajectory (at the point of insertion of such observables). They do not (anti)commute with evaluation observables and raise the dimension of the space of allowed configurations, while the evaluation observables lower this dimension. We study these observables in geometric and operator formalisms. Simple examples are explicitly computed; they depend on the linking of points.     

Morse theory interpretation of topological quantum field theories

Topological quantum field theories are interpreted as a generalized form of Morse theory. This interpretation is applied to formulate the simplest topological quantum field theory: topological quantum mechanics. The only non-trivial topological invariant corresponding to this theory is computed and identified with the Euler characteristic. Using field theoretical methods this topological invariant is calculated in different ways and in the process a proof of the Gauss-Bonnet-Chern-Avez formula as well as some results of degenerate Morse theory are obtained.     

Hidden fermionic symmetry in conformal topological field theories

We point out that bosonic conformal coset modelsG _l ×G _k /G _l+k for all semi-simple Lie algebrasG have a hidden fermionic symmetry atl=0 (the central charge=0) and may be interpreted as twisted versions of some superconformal theories.     

A discrete approach to topological quantum field theories

First, we describe a rather general scheme for constructing three-dimensional euclidean topological quantum field theories, whose basic building blocks are provided by the representation theory of a certain class of (bi-)algebras. Secondly, we discuss in some detail examples, where the algebra is either the function algebra of a finite group, the group algebra of a finite group or a deformation of the enveloping algebra of a classical simple Lie group.     

Strings on pp-waves and massive two-dimensional field theories

We find a general class of pp-wave solutions of type IIB string theory such that the light cone gauge worldsheet Lagrangian is that of an interacting massive field theory. When the light cone Lagrangian has (2,2) supersymmetry we can find backgrounds that lead to arbitrary superpotentials on the worldsheet. We consider situations with both flat and curved transverse spaces. We describe in some detail the background giving rise to the N=2 sine Gordon theory on the worldsheet. Massive mirror symmetry relates it to the deformed CP¹ model (or sausage model) which seems to elude a purely supergravity target space interpretation. To cite this article: J. Maldacena, L. Maoz, C. R. Physique 4 (2003).     

The quenched approximation in some two-dimensional field theories

We investigate the effect of suppressing closed fermion loops (the quenched approximation) in some (1 + 1)-dimensional field theories. In the Schwinger and Thirring models we find that effects of fermion loops on bound state masses can be absorbed in a rescaling of the coupling constant. In the Schwinger model an extra (decoupled) massless ghost appears and the order parameter $\text{〈}\text{ψ}\text{ψ〉}$ becomes infrared divergent. In an extension of the Schwinger model we compute effective lagrangians for the bound state spectrum in the quenched and unquenched case that look rather different.     

Quasi-quantum groups as internal symmetries of topological quantum field theories

We show that every topological quantum field theory (understood as a functor) has an associated quasi-quantum group of internal symmetries.