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relation between BV and BD

Contents

Context

Higher algebra

Algbraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

The concepts of

  1. BV-algebras

  2. BD-algebras

have a superficial similarity and are both related to BV-BRST formalism in physics. Indeed the “BV-operad” was named such because its algebras resemble the structure seen on the BV-complexes in physics. However in the formal deformation quantization picture of BV-formalism used in Costello-Gwilliam, the quantum BV-complexes crucially are BD-algebras and not BV-algebras. On the other hand, if one does not insist on a formal deformation parameter, one may also formalize quantum BV complexes using actual BV-algebras.

The following note means to indicate how all these structures are similar and yet different.

Details

Consider all chain complexes in the following with differential dd of degree +1.

BV

Write E 2E_2 for the little disk operad and E 2 frE_2^{fr} for the framed little disk operad. The chain homology of these is the BV-operad BVBV and the operad P 2P_2 for Poisson 2-algebras, respectively:

BV H (E 2 fr) P 2 H (E 2). \array{ BV &\simeq& H_\bullet(E_2^{fr}) \\ \uparrow && \uparrow \\ P_2 &\simeq& H_\bullet(E_2) } \,.

An algebra over these in chain complexes – a homotopy BV-algebra or Poisson 2-algebra, respectively – is a chain complex (V ,d)(V^\bullet, d) equipped with the following operations of the degree shown:

operation d ()() {,} Δ degree +1 0 1 1, \array{ operation && d && (-)\cdot(-) & & \{-,-\} && \Delta \\ degree && +1 && 0 && -1 && -1 } \,,

where the BV-operator Δ\Delta exists on homotopy BV-algebras but not on Poisson 2-algebras, satisfying the relation

Δ(ab)=(Δa)b+(1) |a|aΔb{a,b}. \Delta (a \cdot b) = (\Delta a) \cdot b + (-1)^{\vert a \vert} a \cdot \Delta b - \{a,b\} \,.

If AA is a commutative smooth Poincaré duality algebra so that its Hochschild cohomology HH (A)HH^\bullet(A) is identified by the HKR theorem with the multi-derivations, then HH (A)HH^\bullet(A) carries a BV-algebra structure and under this identification the above Gerstenhaber bracket {,}\{-,-\} is the Schouten bracket and the BV-operator Δ BV\Delta_{BV} is the image of the de Rham differential dualized. Hence this is then indeed the standard BV-complex as discussed there in the section Multivector fields dual to differential forms.

If we disregard the BV-operator Δ\Delta, then more generally there are Poisson n-algebras for all nn, which are as above but with the bracket of degree 1n1-n

operation {,} degree 1n. \array{ operation && \{-,-\} \\ degree && 1-n } \,.

Notice that this means that P 2P_2 algebras are very similar to P 0P_0-algebras: for the former the backet has degree -1, for the latter it has degree +1.

BD

Now the BD-operad, which is a chain complex of [[]]\mathbb{R}[ [\hbar ] ]-modules (formal power series ring in one variable \hbar) has as chain complex of binary operations

BD(2)( 0 () d operad {,} 0 ) 0 1. BD(2) \;\coloneqq\; \array{ ( \cdots &\to& 0 &\to& \left\langle(-\cdot -)\right\rangle &\stackrel{d_{operad}}{\to}& \left\langle \hbar \{-,-\} \right\rangle &\to& 0 &\to& \cdots ) \\ && && 0 && 1 } \,.

Notice: The differential takes the single generator in degree 0 to the single generator in degree 1. As a differential in chain complexes of \mathbb{R}-modules this would be the null complex, but since the bracket carries a prefactor of \hbar, we get something non-null if we regard this suitably filtered over \hbar… In particular, setting “=0\hbar = 0” by forming the tensor product with [[]]/()\mathbb{R} \simeq \mathbb{R}[ [\hbar] ]/(\hbar) turns it into the chain complex concetrated on a single generator in degree 0. This is the E 0E_0-operad. On the other hand, for finite \hbar, i.e. after tensoring with [[]]/(1)\mathbb{R}[ [\hbar] ]/(\hbar-1) then it becomes the E˜ 0\tilde E_0-operad, which is equivalent to the E 0E_0-operad (whose algebras are simply pointed chain complexes). See also Gwilliam-Haugseng.

E˜ 0 E 0 =1 BD =0 P 0. \array{ && \tilde E_0 & \simeq E_0 \\ & {}^{\mathllap{\hbar = 1}}\nearrow \\ BD \\ & {}_{\mathllap{\hbar = 0}}\searrow \\ && P_0 } \,.

In any case, an algebra over an operad in chain complexes over the BD-operad has operations

operation d {,} degree +1 +1 \array{ operation && d && \hbar \{-,-\} \\ degree && +1 && +1 }

subject to

d(ab)=(da)b+(1) |a|adb{a,b}. d (a \cdot b) = (d a) \cdot b + (-1)^{\vert a \vert} a \cdot d b - \hbar\{a,b\} \,.

This has the same structure as the equation above for BV-algebras if we syntactically replace dd by Δ\Delta and remove the factor of \hbar. But since dd and Δ\Delta are on a different footing, the similarity between these two equations is only superficial.

References

Last revised on January 3, 2018 at 05:04:41. See the history of this page for a list of all contributions to it.