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I. A. Batalin

Publications and source records attributed to I. A. Batalin.

At least 19 recordsLinked to original sources
arXiv2021

Ideals generated by traces in the symplectic reflection algebra $H_{1,ν_1, ν_2}(I_2(2m))$. II

The associative algebra of symplectic reflections $\mathcal H:= H_{1,ν_1, ν_2}(I_2(2m))$ based on the group generated by the root system $I_2(2m)$ has two parameters, $ν_1$ and $ν_2$. For every value of these parameters, the algebra $\mathcal H$ has an $m$-dimensional space of traces. A given trace ${\rm tr}$ is called degenerate if the associated bilinear form $B_{\rm tr}(x,y)={\rm tr}(xy)$ is degenerate. Previously, there were found all values of $ν_1$ and $ν_2$ for which there are degenerate traces in the space of traces, and consequently the algebra $\mathcal H$ has a two-sided ideal. We proved earlier that any linear combination of degenerate traces is a degenerate trace. It turns out that for certain values of parameters $ν_1$ and $ν_2$, degenerate traces span a 2-dimensional space. We prove that non-zero traces in this $2d$ space generate three proper ideals of $\mathcal H$.

hep-th↗
arXiv2020

Ideals generated by traces or by supertraces in the symplectic reflection algebra $H_{1,ν}(I_2(2m+1))$ II

The algebra $\mathcal H:= H_{1,ν}(I_2(2m+1))$ of observables of the Calogero model based on the root system $I_2(2m+1)$ has an $m$-dimensional space of traces and an $(m+1)$-dimensional space of supertraces. In the preceding paper we found all values of the parameter $ν$ for which either the space of traces contains a~degenerate nonzero trace $tr_ν$ or the space of supertraces contains a~degenerate nonzero supertrace $str_ν$ and, as a~consequence, the algebra $\mathcal H$ has two-sided ideals: one consisting of all vectors in the kernel of the form $B_{tr_ν}(x,y)=tr_ν(xy)$ or another consisting of all vectors in the kernel of the form $B_{str_ν}(x,y)=str_ν(xy)$. We noticed that if $ν=\frac z {2m+1}$, where $z\in \mathbb Z \setminus (2m+1) \mathbb Z$, then there exist both a degenerate trace and a~degenerate supertrace on $\mathcal H$. Here we prove that the ideals determined by these degenerate forms coincide.

math.RT↗
arXiv2017

Superfield generating equation of field-antifield formalism

A simple quantum superfield generating equation of the field-antifield formalism is proposed. The Schroedinger equation with the Hamiltonian having $Δ$-exact form is derived. An $Sp(2)$ symmetric extension to the main construction, with specific features caused by the principal fact that all basic equations become $Sp(2)$ vector-valued ones, is presented. A principal role of quantum antibrackets in formulation of the Heisenberg equations of motion is shown.

hep-th↗
arXiv2007

Reducible Gauge Algebra of BRST-Invariant Constraints

We show that it is possible to formulate the most general first-class gauge algebra of the operator formalism by only using BRST-invariant constraints. In particular, we extend a previous construction for irreducible gauge algebras to the reducible case. The gauge algebra induces two nilpotent, Grassmann-odd, mutually anticommuting BRST operators that bear structural similarities with BRST/anti-BRST theories but with shifted ghost number assignments. In both cases we show how the extended BRST algebra can be encoded into an operator master equation. A unitarizing Hamiltonian that respects the two BRST symmetries is constructed with the help of a gauge-fixing Boson. Abelian reducible theories are shown explicitly in full detail, while non-Abelian theories are worked out for the lowest reducibility stages and ghost momentum ranks.

hep-th↗
arXiv2007

On Generalized Gauge-Fixing in the Field-Antifield Formalism

We consider the problem of covariant gauge-fixing in the most general setting of the field-antifield formalism, where the action W and the gauge-fixing part X enter symmetrically and both satisfy the Quantum Master Equation. Analogous to the gauge-generating algebra of the action W, we analyze the possibility of having a reducible gauge-fixing algebra of X. We treat a reducible gauge-fixing algebra of the so-called first-stage in full detail and generalize to arbitrary stages. The associated "square root" measure contributions are worked out from first principles, with or without the presence of antisymplectic second-class constraints. Finally, we consider an W-X alternating multi-level generalization.

hep-th↗
arXiv2005

Star Product for Second Class Constraint Systems from a BRST Theory

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class one and can also be understood as a far-going generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV-BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.

hep-th↗
arXiv2004

Hamiltonian Superfield Formalism with N Supercharges

An action principle that applies uniformly to any number N of supercharges is proposed. We perform the reduction to the N=0 partition function by integrating out superpartner fields. As a new feature for theories of extended supersymmetry, the canonical Pfaffian measure factor is a result of a Gaussian integration over a superpartner. This is mediated through an explicit choice of direction n^a in the θ-space, which the physical sector does not depend on. Also, we re-interpret the metric g^{ab} in the Susy algebra [D^a,D^b] = g^{ab}\partial_t as a symplectic structure on the fermionic θ-space. This leads to a superfield formulation with a general covariant θ-space sector.

hep-th↗
arXiv2003

Hamiltonian N=2 Superfield Quantization

We present a superfield construction of Hamiltonian quantization with N=2 supersymmetry generated by two fermionic charges Q^a. As a byproduct of the analysis we also derive a classically localized path integral from two fermionic objects Σ^a that can be viewed as ``square roots'' of the classical bosonic action under the product of a functional Poisson bracket.

hep-th↗
arXiv2003

BRST-anti-BRST Symmetric Conversion of Second-Class Constraints

A general method of the BRST--anti-BRST symmetric conversion of second-class constraints is presented. It yields a pair of commuting and nilpotent BRST-type charges that can be naturally regarded as BRST and anti-BRST ones. Interchanging the BRST and anti-BRST generators corresponds to a symmetry between the original second-class constraints and the conversion variables, which enter the formalism on equal footing.

hep-th↗
arXiv2003

Associativity and Operator Hamiltonian Quantization of Gauge Theories

We show that the associative algebra structure can be incorporated in the BRST quantization formalism for gauge theories such that extension from the corresponding Lie algebra to the associative algebra is achieved using operator quantization of reducible gauge theories. The BRST differential that encodes the associativity of the algebra multiplication is constructed as a second-order quadratic differential operator on the bar resolution.

math.QA↗
arXiv1998

Superfield Formulation of the Phase Space Path Integral

We give a superfield formulation of the path integral on an arbitrary curved phase space, with or without first class constraints. Canonical tranformations and BRST transformations enter in a unified manner. The superpartners of the original phase space variables precisely conspire to produce the correct path integral measure, as Pfaffian ghosts. When extended to the case of second-class constraints, the correct path integral measure is again reproduced after integrating over the superpartners. These results suggest that the superfield formulation is of first-principle nature.

hep-th↗
arXiv1998

Existence Theorem for Split Involution Constraint Algebra

Existence theorem is proven for the generating equations of the split involution constraint algebra. The structure of the general solution is established, and the characteristic arbitrariness in generating functions is described.

hep-th↗
arXiv1998

Superfield Quantization

We present a superfield formulation of the quantization program for theories with first class constraints. An exact operator formulation is given, and we show how to set up a phase-space path integral entirely in terms of superfields. BRST transformations and canonical transformations enter on equal footing, and they allow us to establish a superspace analog of the BFV theorem. We also present a formal derivation of the Lagrangian superfield analogue of the field-antifield formalism, by an integration over half of the phase-space variables.

hep-th↗
arXiv1997

General Solution of Quantum Master Equation in Finite-Dimensional Case

The general solution to the quantum master equation (and its $Sp(2)$ symmetric counterpart) is constructed explicitly in case of finite number of variables. It is shown that the finite-dimensional solution is physically trivial and thus can not be extended directly to cover the case of a local field theory. In this way we conclude that the locality condition plays an important role by making it possible to obtain nontrivial physical results when quantizing gauge field theories on the basis of field-antifield formalism.

hep-th↗
arXiv1997

Second Class Constraints in a Higher-Order Lagrangian Formalism

We consider the description of second-class constraints in a Lagrangian path integral associated with a higher-order $Δ$-operator. Based on two conjugate higher-order $Δ$-operators, we also propose a Lagrangian path integral with $Sp(2)$ symmetry, and describe the corresponding system in the presence of second-class constraints.

hep-th↗