DGLA Dg and BV formalism
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Springer
Received: September 1, 2020 Accepted: October 26, 2020 Published: December 11, 2020
Andrei Mikhailov Instituto de Física Teórica, Universidade Estadual Paulista, R. Dr. Bento Teobaldo Ferraz 271, Bloco II — Barra Funda, CEP:01140-070 — São Paulo, Brasil
E-mail: [email protected] Abstract: Differrential Graded Lie Algebra Dg was previously introduced in the context of current algebras. We show that under some conditions, the problem of constructing equivariantly closed form from closed invariant form is reduces to construction of a representation of Dg. This includes equivariant BV formalism. In particular, an analogue of intertwiner between Weil and Cartan models allows to clarify the general relation between integrated and unintegrated operators in string worldsheet theory. Keywords: BRST Quantization, Differential and Algebraic Geometry, Gauge Symmetry, Topological Strings ArXiv ePrint: 2007.02875
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP12(2020)077
JHEP12(2020)077
DGLA Dg and BV formalism
Contents 1 2 2 3 3 3 4
2 Notations
4
3 Dg 3.1 Definition of Dg 3.1.1 Supercommutative algebra and its dual coalgebra 3.1.2 Case of V = sg 3.1.3 Definition of Dg 3.2 Representation as vector fields 3.3 Simpler notations 3.4 Ghost number 3.5 Are F a Faddev-Popov ghosts?
4 4 4 5 6 8 8 9 9
4 D0 g
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5 Ansatz for equivariant form
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6 CDg 6.1 D0 g-differential modules 6.2 Pseudo-differential forms (PDF) 6.3 Special cocycles 6.4 A procedure for constructing r
11 11 11 12 13
7 Chevalley-Eilenberg complex of a differential module
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8 Integration measures from representations of Cg and Dg 8.1 PDFs from representations of Cg 8.1.1 Mapping to cochains 8.1.2 Mapping to PDFs 8.2 PDFs from representations of Dg
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9 BV 9.1 Half-densities as a representation of Ca 9.2 Correlation functions as a Lie superalgebra cocycle
17 17 17
–i–
JHEP12(2020)077
1 Introduction 1.1 The cone of Lie superalgebra 1.2 Dg 1.3 String measure 1.4 Equivariant string measure 1.5 Vertex operators 1.6 Previous work
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11 Integrating unintegrated vertices 11.1 Integration prescription using Cg 11.2 Integration using Dg 11.2.1 Deformations as a representation of Dg 11.2.2 Averaging procedure using Dg 11.3 Relation between two integration procedures 11.4 Is integration form base with respect to G0 ?
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A Nilpotence of dDg A.1 Commutator of Dg A.2 Nilpotence of dDg
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1
Introduction
BV formalism is a generalization of the BRST formalism, based on the mathematical theory of odd symplectic supermanifolds. In this formalism the path integral is interpreted as an integral of a density of weight 1/2 over a Lagrangian submanifold. It turns out that this “standard” formulation is not sufficient to describe string worldsheet theory. One has to also consider integration over families of Lagrangian submanifolds. Indeed, the idea of [1–3] was to interpret integration over the worldsheet metrics as a particular case of integration over the space of gauge f
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