Families of gauge conditions in BV formalism

Journal of High Energy Physics, Jul 2017

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 group H and Λ is an H-invariant family of Lagrangian submanifold then under certain conditions we construct a form on Λ that descends to a closed form on Λ/H. Integrating the latter form over a cycle in Λ/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 field theories one obtains topological invariants.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP07%282017%29063.pdf

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)


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP07%282017%29063.pdf

Andrei Mikhailov, Albert Schwarz. Families of gauge conditions in BV formalism, Journal of High Energy Physics, 2017, pp. 63, Volume 2017, Issue 7, DOI: 10.1007/JHEP07(2017)063