Vilkovisky Formalism Research Articles

A homological approach to the Gaussian unitary ensemble

We study the Gaussian unitary ensemble (GUE) using noncommutative geometry and the Batalin–Vilkovisky (BV) formalism, and we show how this homological approach provides canonical relations between correlation functions in the GUE. As applications of this method, we obtain new ways to prove generalizations of Wigner’s semicircle law, to compute all the large N statistical correlations for multi-trace functions as random variables in the GUE, and to determine the leading (and subleading) order behavior of the correlation functions with respect to the rank N . Along the way, and illuminating the connection with prior work, we develop an explicit dictionary between this homological approach and the well-known combinatorics of ribbon graphs that lead to counting problems for the corresponding surfaces.

Renormalization of the Einstein–Cartan theory in first-order form

We examine the Einstein–Cartan (EC) theory in first-order form, which has a diffeomorphism as well as a local Lorentz invariance. We study the renormalizability of this theory in the framework of the Batalin–Vilkovisky formalism, which allows for a gauge invariant renormalization. Using the background field method, we discuss the gauge invariance of the background effective action and analyze the Ward identities which reflect the symmetries of the EC theory. As an application, we compute, in a general background gauge, the self-energy of the tetrad field at one-loop order.

Gravity with torsion as deformed BF theory *

We study a family of (possibly non topological) deformations of BF theory for the Lie algebra obtained by quadratic extension of so(3,1) by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin–Vilkovisky formalism. In particular, both Palatini–Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin–Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields.

A note on gluing via fiber products in the (classical) BV-BFV formalism

In classical field theory, gluing spacetime manifolds along boundary corresponds to taking a fiber product of the corresponding spaces of fields (as differential graded Fréchet manifolds) up to homotopy. We construct this homotopy explicitly in several examples in the setting of BV-BFV formalism (Batalin–Vilkovisky formalism with cutting–gluing).

Generalized Cohomological Field Theories in the Higher Order Formalism

In the classical Batalin--Vilkovisky formalism, the BV operator $\Delta$ is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by $\Delta$ is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin--Vilkovisky algebras where the operator $\Delta$ is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever $\Delta$ is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a "trinity" of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s.

Constraints in the BV formalism: Six-dimensional supersymmetry and its twists

We formulate the abelian six-dimensional N=(2,0) theory perturbatively, in a generalization of the Batalin–Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an L∞ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the N=(1,0) hypermultiplet, together with a degenerate holomorphic theory of holomorphic one-forms from the N=(1,0) tensor multiplet, which can be seen to describe the infinitesimal intermediate Jacobian variety. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira–Spencer theory. Matching our formalism to five-dimensional Yang–Mills theory after reduction leads to some issues related to electric–magnetic duality; we offer some speculation on a nonperturbative resolution.

Quantum Chern–Simons Theories on Cylinders: BV-BFV Partition Functions

We compute partition functions of Chern–Simons type theories for cylindrical spacetimes I times Sigma , with I an interval and dim Sigma = 4l+2, in the BV-BFV formalism (a refinement of the Batalin–Vilkovisky formalism adapted to manifolds with boundary and cutting–gluing). The case dim Sigma = 0 is considered as a toy example. We show that one can identify—for certain choices of residual fields—the “physical part” (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton–Jacobi action computed in the companion paper (Cattaneo et al., Constrained systems, generalized Hamilton–Jacobi actions, and quantization, arXiv:2012.13270), without any quantum corrections. This Hamilton–Jacobi action is the action functional of a conformal field theory on Sigma . For dim Sigma = 2, this implies a version of the CS-WZW correspondence. For dim Sigma = 6, using a particular polarization on one end of the cylinder, the Chern–Simons partition function is related to Kodaira–Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.

Perturbative path-integral of string fields and the A∞ structure of the BV master equation

Abstract The perturbative path-integral gives a morphism of the (quantum) A∞ structure intrinsic to each quantum field theory, which we show explicitly on the basis of the homological perturbation. As is known, in the Batalin–Vilkovisky (BV) formalism, any effective action also solves the BV master equation, which implies that the path-integral can be understood as a morphism of the BV differential. Since each solution of the BV master equation is in one-to-one correspondence with a quantum A∞ structure, the path-integral preserves this intrinsic A∞ structure of quantum field theory, where A∞ reduces to L∞ whenever multiplications of space-time fields are graded commutative. We apply these ideas to string-field theory and (re-)derive some quantities based on the perturbative path-integral, such as effective theories with finite α′, reduction of gauge and unphysical degrees, the S-matrix, and gauge-invariant observables.

The ghost fields and the BV extension for finite spectral triples

After arguing why the Batalin–Vilkovisky (BV) formalism is expected to find a natural description within the framework of noncommutative geometry, we explain how this relation takes form for gauge theories induced by finite spectral triples. In particular, we demonstrate how the two extension procedures appearing in the BV formalism, that is, the initial extension via the introduction of ghost/anti-ghost fields and the further extension with auxiliary fields, can be described in the language of noncommutative geometry using the notions of the BV spectral triple and total spectral triple, respectively. The construction is presented in detail for all U(2)-gauge theories induced by spectral triples on the algebra M2(C). Indications are given on how to extend the results to U(n)-gauge theories for n > 2.

L∞-algebras in Freedman–Townsend model and Batalin–Vilkovisky formalism

We present an [Formula: see text]-algebra description of the Freedman–Townsend tensor gauge theory with an emphasis on the Maurer–Cartan homotopy action form using cyclic inner product. It is explicitly shown that the gauge variations, the conservation law and the dynamics of the theory are all incorporated in the underlying [Formula: see text]-products. Also, following the more convenient method for reducible gauge theory, we propose the Batalin–Vilkovisky formalism of Freedman–Townsend model within the framework of [Formula: see text]-algebras.

Notes on Chern–Simons perturbation theory

We give a detailed introduction to the classical Chern–Simons gauge theory, including the mathematical preliminaries. We then explain the perturbative quantization of gauge theories via the Batalin–Vilkovisky (BV) formalism. We then define the perturbative Chern–Simons partition function at any (possibly non-acylic) reference flat connection using the BV formalism, using a Riemannian metric for gauge fixing. We show that it exhibits an anomaly known as the “framing anomaly” when the Riemannian metric is changed, that is, it fails to be gauge invariant. We explain how one can deal with this anomaly to obtain a topological invariant of framed manifolds.

Constrained systems, generalized Hamilton-Jacobi actions, and quantization

<p style='text-indent:20px;'>Mechanical systems (i.e., one-dimensional field theories) with constraints are the focus of this paper. In the classical theory, systems with infinite-dimensional targets are considered as well (this then encompasses also higher-dimensional field theories in the hamiltonian formalism). The properties of the Hamilton–Jacobi (HJ) action are described in details and several examples are explicitly computed (including nonabelian Chern–Simons theory, where the HJ action turns out to be the gauged Wess–Zumino–Witten action). Perturbative quantization, limited in this note to finite-dimensional targets, is performed in the framework of the Batalin–Vilkovisky (BV) formalism in the bulk and of the Batalin–Fradkin–Vilkovisky (BFV) formalism at the endpoints. As a sanity check of the method, it is proved that the semiclassical contribution of the physical part of the evolution operator is still given by the HJ action. Several examples are computed explicitly. In particular, it is shown that the toy model for nonabelian Chern–Simons theory and the toy model for 7D Chern–Simons theory with nonlinear Hitchin polarization do not have quantum corrections in the physical part (the extension of these results to the actual cases is discussed in the companion paper [<xref ref-type="bibr" rid="b21">21</xref>]). Background material for both the classical part (symplectic geometry, generalized generating functions, HJ actions, and the extension of these concepts to infinite-dimensional manifolds) and the quantum part (BV-BFV formalism) is provided.</p>

Spontaneous symmetry breaking: A view from derived geometry

We examine symmetry breaking in field theory within the framework of derived geometry, as applied to field theory via the Batalin–Vilkovisky formalism. Our emphasis is on the standard examples of Ginzburg–Landau and Yang–Mills–Higgs theories and is primarily interpretive. The rich, sophisticated language of derived geometry captures the physical story elegantly, allowing for sharp formulations of slogans (e.g., for the Higgs mechanism, that the unstable ghosts feed the Goldstone bosons to a hungry, massless gauge boson). Rewriting these results in the BV formalism provides, as one nice payoff, a reformulation of ’t Hooft’s family of gauge-fixing conditions for spontaneously broken gauge theory that behaves well in the ξ→∞ limit.

Notice of Retraction “U(1) gauged model of FJ-type chiral boson based on Batalin–Fradkin–Vilkovisky formalism”

This article has been retracted due to overlapping of its result with an earlier published article [Annals of Physics, Vol. 383 (2017) 497-510].

U(1) gauged model of FJ-type chiral boson based on Batalin–Fradkin–Vilkovisky formalism

The BRST quantization of the U(1) gauged model of FJ-type chiral boson for [Formula: see text] and [Formula: see text] are performed using the Batalin–Fradkin–Vilkovisky formalism. BFV formalism converts the second-class algebra into an effective first-class algebra with the help of auxiliary fields. Explicit expressions of the BRST charge, the involutive Hamiltonian, and the preserving BRST symmetry action are given and the full quantization has been carried through. For [Formula: see text], this Hamiltonian gives the gauge invariant Lagrangian including the well-known Wess–Zumino term, while for [Formula: see text] the corresponding Lagrangian has the additional new type of the Wess–Zumino term. The spectra in both cases have been analysed and the Wess–Zumino actions in terms of auxiliary fields are identified.

Equivariant Batalin–Vilkovisky formalism

We study an equivariant extension of the Batalin–Vilkovisky formalism for quantizing gauge theories. Namely, we introduce a general framework to encompass failures of the quantum master equation, and we apply it to the natural equivariant extension of AKSZ solutions of the classical master equation (CME). As examples of the construction, we recover the equivariant extension of supersymmetric Yang–Mills in 2d and of Donaldson–Witten theory.

The Batalin–Vilkovisky formalism description of self-dual Chern–Simons gauge theory coupled to matter fields in superspace

We construct the extended BRST and anti-BRST transformation by considering a shift symmetry both in Abelian and non-Abelian Chern–Simons theories coupled to scalar fields using the approach of Batalin–Vilkovisky quantization. The extended BRST invariant Lagrangian density is able to be addressed in the framework of superfield formalism with one fermionic coordinate. In the case of extended BRST–anti-BRST symmetry, it is shown that the Batalin–Vilkovisky action of the theory can be expressed in a covariant form with two fermionic coordinates in the superspace.

Renormalization for Holomorphic Field Theories

We introduce the concept of a holomorphic field theory on any complex manifold in the language of the Batalin–Vilkovisky formalism. When the complex dimension is one, this setting agrees with that of chiral conformal field theory. Our main result concerns the behavior of holomorphic theories under renormalization group flow. Namely, we show that holomorphic theories are one-loop finite. We use this to completely characterize holomorphic anomalies in any dimension. Throughout, we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories.

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

AbstractWe review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an ‐algebra and how quasi‐isomorphisms between ‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations.

‐Algebras, the BV Formalism, and Classical Fields

AbstractWe summarise some of our recent works on ‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of ‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of ‐quasi‐isomorphisms, and we propose a twistor space action.