Modified Triplectic Quantization in General Coordinates

Modern covariant quantization methods for general gauge theories are based on the principle of BRST [2, 3], or, more generally, BRST–antiBRST [4, 5, 6, 7] invariance. The consideration of these methods in general coordinates (using appropriate supermanifolds) appears to be very important in order to reveal the geometrical meaning of the basic objects underlying these quantization schemes. The study of the Batalin–Vilkovisky (BV) method [2] in general coordinates was initiated by the work of Witten [8], where the geometrical meaning of the antifields, the antibracket, and the odd second-order operator ∆ was discussed. In [9], it was shown that the geometry of the BV formalism is that of an odd symplectic superspace, endowed with a density function ρ. The quantization schemes based on the BRST–antiBRST symmetry involve additional basic objects. Namely, in the Sp(2)-covariant and triplectic quantization schemes one introduces Sp(2)-doublets of extended antibrackets, as well as doublets of secondand first-order operators ∆a and V a, respectively. In the modified triplectic quantization, an additional Sp(2)-doublet of first-order operators Ua is required. This indicates that the geometrical formulation of these quantization methods in general coordinates, in contrast to the BV quantization, requires more complicated tools. Indeed, in this paper we show that the geometry of the Sp(2)-covariant and triplectic schemes is the geometry of an even symplectic superspace equipped with a density function and a flat symmetric connection (covariant derivative), while the geometry of the modified triplectic quantization also includes a symmetric structure (analogous to a metric tensor). The paper is organized as follows. In Sect. 2, we briefly review the definitions of tensor fields, the covariant derivative, and the curvature tensor on supermanifolds [10]. In Sect. 3, we introduce the notion of a triplectic supermanifold, together with tensor fields and covariant derivatives defined on it. In Sect. 4, an explicit realization of the triplectic algebra of odd differential operators is suggested. In Sect. 5, we find a realization of the modified triplectic algebra and propose a suitable quantization procedure. In Sect. 6, we give a short summary and a few concluding remarks. In Appendix A, we study the connection between even (odd) non-degenerate Poisson structures and even (odd) symplectic structures on supermanifolds, and show their one-to-one correspondence. In Appendix B, the algebra of the generating operators ∆a and V a is presented. We use the condensed DeWitt notation and apply the tensor analysis of Ref. [10]. Derivatives with respect to the variables xi are understood as acting from the left, with the notation ∂iA = ∂A/∂x i. Right-hand derivatives with respect to xi are labelled by the subscript ”r”, and the notation A,i = ∂rA/∂x i is used. Raising the Sp(2)-group indices is performed by the antisymmetric second rank tensor εab (a, b = 1, 2): θa = εθb, ε εcb = δ a b . The Grassmann parity of any quantity A is denoted by ǫ(A).