Abstract
We study linear Batalin–Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of infty -categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of mathrm {E}_{n}-algebras to produce a “higher BV quantization” functor, and when restricted to formal moduli problems, it behaves like a determinant. Along the way we also use our machinery to give an algebraic construction of mathrm {E}_{n}-enveloping algebras for shifted Lie algebras.
Highlights
A well-known adage in mathematical physics is that “quantization is not a functor,” but with suitable restrictions, there are situations where quantization is functorial
Our construction produces the simplest possible examples of Batalin–Vilkovisky quantization. This homological approach to quantization of field theories was introduced by Batalin and Vilkovisky [4,5,6] as a generalization of the BRST formalism, in an effort to deal with complicated field theories such as supergravity
We expect that it is closely related to recent work [7,9,49] on vanishing cycles on stacks. (In a sense, the BV formalism is an obfuscated version of the twisted de Rham complex, as explained in Sect. 1.4, and closely related to vanishing cycles.)
Summary
A well-known adage in mathematical physics is that “quantization is not a functor,” but with suitable restrictions, there are situations where quantization is functorial. This assignment can formulated as a functor, known as Weyl quantization, from symplectic vector spaces (or more generally, vector spaces with a skew-symmetric pairing) to associative algebras For us this is the model case of functorial quantization. The universal enveloping algebra U Heis(V, ω) can be viewed as a deformation quantization of the Poisson algebra Sym(V ) This procedure is at the core of all approaches to “free theories,” and the base case for the more challenging and more interesting interacting theories. Our construction produces the simplest possible examples of Batalin–Vilkovisky quantization This homological approach to quantization of field theories was introduced by Batalin and Vilkovisky [4,5,6] as a generalization of the BRST formalism, in an effort to deal with complicated field theories such as supergravity.
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