Stochastic and parastochastic aspects of supersymmetric functional measures: A new non-perturbative approach to supersymmetry

We study the properties of supersymmetric models having a local Nicolai mapping. In these cases the Nicolai mapping can be interpreted as a stochastic differential equation, and hence we can use all the standard stochastic techniques to extract physical information from the theory. The corresponding Langevin equation does not describe, in general, a system approaching (asymptotically) a thermal equilibrium. We construct explicitly and nonperturbatively the Nicolai mapping for a large class of two dimensional models. In particular, this is the first non-perturbative proof of the existence of the mapping. The properties of the mapping agree with the expectations from general arguments. We show how the Nicolai mapping can be used to eliminate completely the fermions from the perturbative expansion, leaving a simpler set of diagrammatric rules involving only scalars. Finally, we argue that the present approach may be very powerful for studying finiteness properties of extended supersymmetric theories.     

Parent Field Theory and Unfolding in BRST First-Quantized Terms

For free-field theories associated with BRST first-quantized gauge systems, we identify generalized auxiliary fields and pure gauge variables already at the first-quantized level as the fields associated with algebraically contractible pairs for the BRST operator. Locality of the field theory is taken into account by separating the space–time degrees of freedom from the internal ones. A standard extension of the first-quantized system, originally developed to study quantization on curved manifolds, is used here for the construction of a first-order parent field theory that has a remarkable property: by elimination of generalized auxiliary fields, it can be reduced both to the field theory corresponding to the original system and to its unfolded formulation. As an application, we consider the free higher-spin gauge theories of Fronsdal.Senior Research Associate of the National Fund for Scientific Research (Belgium).Postdoctoral Visitor of the National Fund for Scientific Research (Belgium).     

Geometric quantization of the BRST charge

In the first half of this paper (Sects. 1–4) we generalise the standard geometric quantization procedure to symplectic supermanifolds. In the second half (Sects. 5, 6) we apply this to two examples that exhibit classical BRST symmetry, i.e., we quantize the BRST charge and the ghost number. More precisely, in the first example we consider the reduced symplectic manifold obtained by symplectic reduction from a free group action with Ad^*-equivariant moment map; in the second example we consider a foliated configuration space, whose cotangent bundle admits the construction of a BRST charge associated to this foliation. We show that the classical BRST symmetry can be described in terms of a hamiltonian supergroup action on the extended phase space, and that geometric quantization gives us a super-unitary representation of this supergroup. Finally we point out how these results are related to reduction at the quantum level, as compared with the reduction at the classical level.Research supported by the Dutch Organization for Scientific Research (NWO)     

Semi-infinite cohomology in conformal field theory and 2D gravity

We discuss various techniques for computing the semi-infinite cohomology of highest weight modules which arise in the BRST quantization of two dimensional field theories. In particular, we concentrate on two such theories - the G/H coset models and 2D gravity coupled to c ≤ 1 conformal matter.     

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.     

Gauge fixing and renormalization in topological quantum field theory

We show how Witten's topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero.     

The coupling of Chern–Simons theory to topological gravity

We couple Chern–Simons gauge theory to 3-dimensional topological gravity with the aim of investigating its quantum topological invariance. We derive the relevant BRST rules and Batalin–Vilkovisky action. Standard BRST transformations of the gauge field are modified by terms involving both its anti-field and the super-ghost of topological gravity. Beyond the obvious couplings to the metric and the gravitino, the BV action includes hitherto neglected couplings to the super-ghost. We use this result to determine the topological anomalies of certain higher ghost deformations of SU(N)$\mathit{SU}(N)$ Chern–Simons theory, introduced years ago by Witten. In the context of topological strings these anomalies, which generalize the familiar framing anomaly, are expected to be cancelled by couplings of the closed string sector. We show that such couplings are obtained by dressing the closed string field with topological gravity observables.     

Stochastic quantization,supersymmetry and the Nicolai map

We present a formulation of the method of stochastic quantization of Parisi and W that reveals its intimate connection with supersymmetry. The crucial ingredient of this analysis is the Nicolai map. By using supersymmetric Ward identities, we derive relations between two Fokker-Planck-type Hamiltonians which arise naturally in this formalism.     

Langevin equation in field theory

A new approach to quantum field theory is developed based on the Langevin equation (stochastic quantization). Applications to conventional and gauge theories are discussed, as well as various extensions; the Langevin difference equation, the complex Langevin equation in Minkowski space, etc.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 66–76, March, 1986.     

Topological Field Theories Associated with Three-Dimensional Seiberg–Witten Monopoles

Three-dimensional topological field theoriesassociated with the three-dimensional version of Abelianand non-Abelian Seiberg–Witten monopoles arepresented. These three-dimensional monopole equationsare obtained by a dimensional reduction of thefour-dimensional ones. The starting actions to beconsidered are Gaussian types with random auxiliaryfields. As the local gauge symmetries with topologicalshifts are found to be first-stage-reducible, theBatalin–Vilkovisky algorithm is suitable forquantization. Then the BRST transformation rules areautomatically obtained. Nontrivial observablesassociated with Chern classes are obtained from the geometricsector and are found to correspond to those of thetopological field theory of Bogomol'nyimonopoles.     

An interacting geometry model and induced gravity

We propose the theory of quantum gravity with interactions introduced by topological principle. The fundamental property of such a theory is that its energy-momentum tensor is a BRST anticommutator. Physical states are elements of the BRST cohomology group. The model with only topological excitations, introduced recently by Witten, is discussed from the point of view of the induced gravity program. We find that the mass scale is induced dynamically by gravitational instantons. The low-energy effective theory has gravitons, which occur as the collective excitations of geometry, when the metric becomes dynamical. Applications of cobordism theory to quantum gravity are discussed.This essay received the second award from the Gravity Research Foundation for the year 1988.—Ed.     

BRST quantization in the Siegel gauge

From a formal generalization to N copies of the free open string field theory BRST-quantized in the Siegel gauge we reproduce the BRST quantization of the free closed bosonic string field theory and obtain the one of massless higher spin field theories.     

Basic Brackets of a 2D Model for the Hodge Theory Without its Canonical Conjugate Momenta

We deduce the canonical brackets for a two (1+1)-dimensional (2D) free Abelian 1-form gauge theory by exploiting the beauty and strength of the continuous symmetries of a Becchi-Rouet-Stora-Tyutin (BRST) invariant Lagrangian density that respects, in totality, six continuous symmetries. These symmetries entail upon this model to become a field theoretic example of Hodge theory. Taken together, these symmetries enforce the existence of exactly the same canonical brackets amongst the creation and annihilation operators that are found to exist within the standard canonical quantization scheme. These creation and annihilation operators appear in the normal mode expansion of the basic fields of this theory. In other words, we provide an alternative to the canonical method of quantization for our present model of Hodge theory where the continuous internal symmetries play a decisive role. We conjecture that our method of quantization is valid for a class of field theories that are tractable physical examples for the Hodge theory. This statement is true in any arbitrary dimension of spacetime.     

Moyal Deformation,Seiberg–Witten Maps,and Integrable Models

A covariant formalism for Moyal deformations of gauge theory and differential equations which determine Seiberg–Witten maps is presented. Replacing the ordinary product of functions by the noncommutative Moyal product, noncommutative versions of integrable models can be constructed. We explore how a Seiberg–Witten map acts in such a framework. As a specific example, we consider a noncommutative extension of the principal chiral model.     

BRST quantization of the string model with extrinsic curvature

The string model proposed by Polyakov is investigated as a two-dimensional field theory with higher-order derivatives. We reduce the model lagrangian to a simple useful form and achieve the BRST quantization. We show the nilpotence of the BRST charge under certain conditions, and discuss the unitarity of the theory.     

Characteristic classes of star products on Marsden–Weinstein reduced symplectic manifolds

In this note we consider a quantum reduction scheme in deformation quantization on symplectic manifolds proposed by Bordemann, Herbig and Waldmann based on BRST cohomology. We explicitly construct the induced map on equivalence classes of star products which will turn out to be an analogue to the Kirwan map in the Cartan model of equivariant cohomology. As a byproduct, we shall see that every star product on a (suitable) reduced manifold is equivalent to a reduced star product.     

Group Field Theory: An Overview

We give a brief overview of the properties of a higher-dimensional generalization of matrix model which arise naturally in the context of a background approach to quantum gravity, the so-called group field theory. We show in which sense this theory provides a third quantization point-of-view on quantum gravity. Prepared for the proceedings of Peyresq Physics 9Meeting: Micro and Macro Structureof Spacetime, Peyresq, France, 19-26 June 2004. 5 Except in 2 + 1 dimension (Witten, 1988).     

A Geometric Approach to Boundaries and Surface Defects in Dijkgraaf–Witten Theories

Dijkgraaf–Witten theories are extended three-dimensional topological field theories of Turaev–Viro type. They can be constructed geometrically from categories of bundles via linearization. Boundaries and surface defects or interfaces in quantum field theories are of interest in various applications and provide structural insight. We perform a geometric study of boundary conditions and surface defects in Dijkgraaf–Witten theories. A crucial tool is the linearization of categories of relative bundles. We present the categories of generalized Wilson lines produced by such a linearization procedure. We establish that they agree with the Wilson line categories that are predicted by the general formalism for boundary conditions and surface defects in three-dimensional topological field theories that has been developed in Fuchs et al. (Commun Math Phys 321:543–575, 2013)     

Anticommutativity equation in topological quantum mechanics

We consider topological quantum mechanics as an example of topological field theory and show that its special properties lead to numerous interesting relations for topological correlators in this theory. We prove that the generating function ? for these correlators satisfies the anticommutativity equation (D?F)². We show that the commutativity equation [dB, dB]=0 can be considered as a special case of the anticommutativity equation.