The perturbation of the differential in the passage from a classical BV-complex to a quantum BV-complex is called a *BV-Laplacian*. As discussed at *BV-complex* the BV Laplacian is a homological incarnation of a measure.

In the archetypical example of the BV-complex of a smooth manifold of finite dimension $n$ and equipped with a volume form $vol$, the BV-Laplacian is the operation on multivector fields which is the image of the de Rham differential under the isomorphism between multivector fields and differential forms induced by contraction with the volume form.

Specifically over a Cartesian space $\mathbb{R}^n$ with its canonical volume form and with $\{x^i, \xi_i\}$ the canonical basis elements of the algebra of multivectorfields, the BV-Laplacian has the expression

$\Delta = \sum_{i = 1}^n \frac{\partial}{ \partial x^i} \frac{\partial}{\partial \xi_i}$

(it differentiates the function and “removes a vector field” for each derivative, being the dual operation to “wedging with $d x^i$”).

action functional | kinetic action | interaction | path integral measure |
---|---|---|---|

$\exp(-S(\phi)) \cdot \mu =$ | $\exp(-(\phi, Q \phi)) \cdot$ | $\exp(I(\phi)) \cdot$ | $\mu$ |

BV differential | elliptic complex + | antibracket with interaction + | BV-Laplacian |

$d_q =$ | $Q$ + | $\{I,-\}$ + | $\hbar \Delta$ |

See at *BV-BRST formalism* for general references.

Discussion of the BV-operation in renormalization and effective field theory is in section 10.1 of

- Kevin Costello,
*Renormalisation and the Batalin-Vilkovisky formalism*(arXiv:0706.1533)

Last revised on January 13, 2021 at 14:56:27. See the history of this page for a list of all contributions to it.