In this paper we work out the explicit form of the change of variables that reproduces an arbitrary change of gauge in a higher-order Lagrangian formalism.
In the framework of $Sp(2)$ extended Lagrangian field-antifield BV formalism we study systematically the role of finite field-dependent BRST-BV transformations. We have proved that the Jacobian of a finite BRST-BV transformation is capable of generat
ing arbitrary finite change of the gauge-fixing function in the path integral.
We introduce external sources J_A directly into the quantum master action W of the field-antifield formalism instead of the effective action. The external sources J_A lead to a set of BRST-invariant functions W^A that are in antisymplectic involution
. As a byproduct, we encounter quasi--groups with open gauge algebras.
We introduce classical and quantum antifields in the reparametrization-invariant effective action, and derive a deformed classical master equation.
We provide necessary and sufficient conditions for a bi-Darboux Theorem on triplectic manifolds. Here triplectic manifolds are manifolds equipped with two compatible, jointly non-degenerate Poisson brackets with mutually involutive Casimirs, and with
ranks equal to 2/3 of the manifold dimension. By definition bi-Darboux coordinates are common Darboux coordinates for two Poisson brackets. We discuss both the Grassmann-even and the Grassmann-odd Poisson bracket case. Odd triplectic manifolds are, e.g., relevant for Sp(2)-symmetric field-antifield formulation. We demonstrate a one-to-one correspondence between triplectic manifolds and para-hypercomplex manifolds. Existence of bi-Darboux coordinates on the triplectic side of the correspondence translates into a flat Obata connection on the para-hypercomplex side.
We propose how to incorporate the Leites-Shchepochkina-Konstein-Tyutin deformed antibracket into the quantum field-antifield formalism.
In the field-antifield formalism, we review existence and uniqueness proofs for the proper action in the reducible case. We give two new existence proofs based on two resolution degrees called reduced antifield number and shifted antifield number, re
spectively. In particular, we show that for every choice of gauge generators and their higher stage counterparts, there exists a proper action that implements them at the quadratic order in the auxiliary variables.
We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger-Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth-order term pr
oportional to the Levi-Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann-odd, second-order Delta operator in antisymplectic geometry, which in general has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with the measure density. Finally, we discuss the close relationship with the two-loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space.
Recent works have revealed that the recipe for field-antifield quantization of Lagrangian gauge theories can be considerably relaxed when it comes to choosing a path integral measure rho if a zero-order term u_{rho} is added to the Delta operator. T
he effects of this odd scalar term u_{rho} become relevant at two-loop order. We prove that u_{rho} is essentially the odd scalar curvature of an arbitrary torsion-free connection that is compatible with both the anti-Poisson structure E and the density rho. This extends a previous result for non-degenerate antisymplectic manifolds to degenerate anti-Poisson manifolds that admit a compatible two-form.
