nLab BV-BRST variational bicomplex

Idea

In variational calculus, the variational bicomplex of a fiber bundle serves to give a local description of the variational derivative $\delta$. In applications one is typically interested in 1) specifying a differential equation which carves out its solution locus (diffiety) inside the jet bundle and 2) forming the quotient by infinitesimal symmetries of the PDE. Moreover, it is usually useful to perform both these operations in homotopy theory/higher differential geometry. Using tools from homological algebra, under suitable conditions the resulting combined homotopy intersection in and homotopy quotient is modeled by a combination of the variational bicomplex with a local BV-BRST complex.

Definition

Definition

(variational BV-bicomplex)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (this def.) equipped with a gauge parameter bundle $\mathcal{G}$ ([this def.](geometry+of+physics++A+first+idea+of+quantum+field+theory#GaugeParametrizedInfinitesimalGaugeTransformation)) which is closed (this def..). Consider the from this example, whose is the of the theory.]

Then its Weil algebra $W(E/(\mathcal{G} \times_\Sigma T \Sigma))$ has as differential the variational derivative (this def.) plus the BRST differential

$$\begin{aligned} d_{W} & = \mathbf{d} - (d - s_{BRST}) \\ & = \delta + s_{BRST} \end{aligned} \,.$$

Therefore we speak of the variational BRST-bicomplex and write

$$\Omega^\bullet_\Sigma( E/(\mathcal{G} \times_\Sigma T \Sigma) ) \,.$$

Similarly, the Weil algebra of the derived prolonged shell $\mathcal{E}^\infty_{BV}$ (this def.) has differential

$$\begin{aligned} d_W & = \mathbf{d} - (d - s) \\ & = \delta + s \end{aligned} \,.$$

Since $s$ is the BV-BRST differential this defines the “BV-BRST variational bicomplex”.