A Note on Gauge Transformations in Batalin-vilkovisky Theory

We give a generally covariant description, in the sense of symplectic geometry, of gauge transformations in Batalin-Vilkovisky quantization. Gauge transformations exist not only at the classical level, but also at the quantum level, where they leave the action-weighted measure dμS ≡ dμe 2S/h̄ invariant. The quantum gauge transformations and their Lie algebra are h̄-deformations of the classical gauge transformation and their Lie algebra. The corresponding Lie brackets [ , ]q, and [ , ]c, are constructed in terms of the symplectic structure and the measure dμS. We discuss closed string field theory as an application. ⋆ E-mail address: [email protected], [email protected] † E-mail address: [email protected], [email protected]. Supported in part by D.O.E. contract DE-AC02-76ER03069. Introduction In the antibracket, or Batalin-Vilkovisky (BV) formalism, the master action has long been known to determine not only the BRST transformations but also the gauge transformations. Indeed, as explained in the original paper of Batalin and Vilkovisky [ 1], and elaborated upon in a recent monograph (Ref.[ 2], §17.4.2), the gauge transformations of a field (or antifield) Φi are δΦ = ( ω∂j∂ r k S ) Λ , (1) where ω is the symplectic form, S is the classical master action, and Λk are field/antifield independent parameters of local gauge transformations with statistics (−)k+1. ‡ This result, however, is not completely general. In addition to leaving only the classical master action invariant, the above formula is not covariant under a change of basis; it requires the use of Darboux coordinates which make the components ωij of the symplectic form constant. More seriously, when Λk is field/antifield dependent, the above transformations do not generally leave the symplectic form invariant, and therefore they do not qualify as true gauge transformations or true invariances (we recall that in BV quantization the physics is determined by the action and the symplectic structure). This is in contrast to ordinary gauge theory where gauge parameters can be chosen to be field dependent, in addition to being spacetime dependent. If the Λk’s are field/antifield independent we find δΦ = ω∂j ( ( ∂ r k S ) Λ ) = { Φ , (∂ r k S) Λ k } (2) showing that the gauge transformations are canonical transformations, and are generated by the hamiltonian K = (∂ r k S) Λ k. This form of the gauge transformations has been widely used since most field theories have been formulated using Darboux coordinates. ‡ ∂j and ∂ r j stand for ∂ ∂Φj and ∂ r ∂Φj respectively, where the supercripts l and r denote left and right derivatives.