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Families of gauge conditions in BV formalism
Revised: May
Families of gauge conditions in BV formalism
S~ao Paulo
Brazil
Davis
U.S.A.
Andrei Mikhailov 0 3
Albert Schwarz 2
0 On leave from Institute for Theoretical and Experimental Physics , Moscow , Russia
1 sica Teorica, Universidade Estadual Paulista
2 Department of Mathematics, University of California , USA
3 Instituto de Fi
In BV formalism we can consider a Lagrangian submanifold as a gauge condition. Starting with the BV action functional we construct a closed form on the space of Lagrangian submanifolds. If the action functional is invariant with respect to some
BRST Quantization; Gauge Symmetry; Topological Strings
-
group H and
is an H-invariant family of Lagrangian submanifold then under certain
conditions we construct a form on
that descends to a closed form on
=H. Integrating
the latter form over a cycle in
=H we obtain numbers that can have interesting physical
meaning. We show that one can get string amplitudes this way. Applying this construction
to topological quantum
eld theories one obtains topological invariants.
1 Introduction
4 Gauge symmetries
5 From BRST to BV
2 Families of equivalent action functionals
3 Families of Lagrangian submanifolds in BV phase space
6 Topological quantum
eld theories. Bosonic strings
7 String amplitudes
7.1
String amplitudes for critical string
8 Pure spinor superstring
A Some useful formulas
B De nition of using marked points
C Central extension of the group of canonical transformations
C.1 De nition of b C.2 How to build a form on LAG starting from b
family of physically equivalent action functionals. As was noticed in [1] this is wrong. The
consideration of a family of equivalent action functionals or family of gauge conditions
labeled by points of (super) manifold
leads to a construction of a closed di erential form
{ 1 {
(a closed pseudodi erential form if
is a supermanifold). If our action functionals
are invariant with respect to some group H then the form
is H-invariant, but it does not
necessarily descend to
=H. Under some conditions we construct a closed H-equivariant
form
H and show that this equivariant form is homologous to a form descending to
=H.
This allows us to obtain interesting physical quantities integrating over cycles in
=H.
For example, we can start with topological quantum
eld theory on some manifold .
One can apply our results to the family of equivalent action functionals labeled by metrics
on
. We obtain topological invariants of
this way; it would be interesting to calculate
them and compare with known invariants.
This machinery can be applied to string amplitudes. The worldsheet of bosonic string
HJEP07(21)63
can be considered as two-dimensional topological quantum
eld theory. Considering
as a
space of metrics and H as a group generated by di eomorphisms and Weyl transformations
we get formulas for string amplitudes; for appropriate choice of Lagrangian submanifolds
these formulas coincide with the standard ones. Similar constructions work for other types
of strings.
Some remarks about terminology and notations.
We are saying \manifold" instead
of \supermanifold", \group" instead of \supergroup", etc. We understand an element of
super Lie algebra as a linear combination P ATA where TA are even or odd generators of
Z2 -graded Lie algebra and A are even or odd elements of some Grassmann algebra; hence
in our understanding an element of super Lie algebra is always an even object (see [
2, 3
]
for the de nitions of supermanifold, super Lie algebra, etc. that we are using).
We work in BV-formalism assuming that the BV action functionals are de ned on
odd symplectic manifold M equipped with volume element (SP-manifold in terminology
of [4, 5]). In this situation the odd Laplacian
is de ned on the space of functions on M .
It was noticed in [6] that in the absence of the volume element the odd Laplacian is de ned
on semidensities; this allows the reformulation of BV-formalism for any odd symplectic
manifold. In appendix C we show how to prove our main results in this more general
setting. Some basic formulas of BV-formalism are listed in appendix A.
The space of (smooth) functions on a supermanifold M is denoted Fun(M ).This space
is Z2-graded: Fun(M ) = Fun0(M ) + Fun1(M ). Functions on
T M (on the space of
tangent bundle with reversed parity of bers) are called pseudodi erential forms (PDF) on
M . (Di erential forms can be considered as polynomial functions on
T M .) Di stands
for the group of di eomorphisms, Vect for its Lie algebra (the algebra of vector elds),
Weyl for the group of Weyl transformations. As we have noticed an element of any super
Lie algebra (and hence a vector eld) is considered an even object.
We use the term \canonical transformation" for a transformation of (odd) symplectic
manifold preserving the symplectic form (another word for this notion is
\symplectomorphism"). On a simply connected manifold in nitesimal canonical transformations can be
characterized as Hamilt (...truncated)