Abstract

Gauge fixing is interpreted in BV formalism as a choice of Lagrangian submanifold in an odd symplectic manifold (the BV phase space). A natural construction defines an integration procedure on families of Lagrangian submanifolds. In string perturbation theory, the moduli space integrals of higher genus amplitudes can be interpreted in this way. We discuss the role of gauge symmetries in this construction. We derive the conditions which should be imposed on gauge symmetries for the consistency of our integration procedure. We explain how these conditions behave under the deformations of the worldsheet theory. In particular, we show that integrated vertex operator is actually an inhomogeneous differential form on the space of Lagrangian submanifolds.

Highlights

  • We want to define some pseudo-differential form which can serve as string measure

  • (which associates a PDF on G to every Lagrangian submanifold)

  • A geometrically natural solution to these conditions can be found in the case of “traditional” BRST formalism where the fields φ are split into physical fields φ and ghosts c

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Summary

Introduction

We want to obtain BRST invariants by integrating over some closed cycles [1] This construction (and related) was used in [3,4], and on manifolds with a boundary in [4,5]. We want to extend the definition of to the space of all BRST-trivial deformations With all these generalizations, we still want to be a base form with respect to the worldsheet diffeomorphisms. The action of the group of diffeomorphisms on the BV phase space should come as part of the definition of the string worldsheet theory. This could be useful for the off-shell formulation of string theory [8]

Brief review of the BV formalism
Canonical transformations
Definition of can
The canonical operator is nilpotent
Moment map
Canonical operator is ill-defined in field-theoretic context
Summary
Definition of as a PDF on G
Algebraic interpretation
Descent to LAG
Use ghost number symmetry
Space of Legendrian submanifolds in M
Reduction to integration over single Lagrangian submanifold
Integration in two steps
Baranov–Schwarz transform
Special canonical transformations
Straightforward descent does not work
Equivariant half-densities
Summary: constraints on gauge symmetry
Deformations of equivariant half-density
Base form
Example
Example: an interpretation of Cartan complex
5.10. Equivariant effective action
Brief review of BRST formalism
Integration measure
Lift of symmetries to BRST configuration space
BV from BRST
Form in BRST formalism
Lifting the gauge symmetry to BRST formalism
Faddeev–Popov integration procedure
Conormal bundle to the constraint surface
Deformations of BV action
Deformations of F
Exact vertex operators
Relaxing Siegel gauge
Solution of Master Equation
Family of Lagrangian submanifolds
Integration over the family of Lagrangian submanifolds
BV action
Action of diffeomorphisms
10. Worldsheet with boundary
10.1. Variation of the wave function
10.2. Interpretation of as an intertwiner in the presence of a boundary
10.3. Base form
11.1. Definition of unintegrated vertex
11.2. Integration over the location of insertion points
11.2.2. Modified de Rham complex of H
11.2.4. Integration
11.5. Descent procedure
11.5.1. Integrated vertex and Lie algebra cohomology
11.5.3. Relation between integrated and unintegrated vertices
Contraction and Lie derivative
Symplectic structure and Poisson structure

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