nLab local BRST cohomology

Idea

Given a Lagrangian field theory $(E,\mathbf{L})$ with field bundle $E \overset{fb}{\to} \Sigma$ over some spacetime $\Sigma$ and local Lagrangian density $\mathbf{L}$, then its local BV-BRST complex (or local BRST complex, for short) is the realization of the BV-BRST complex not on local observables $A = \tau_{\Sigma} \alpha$ given by functionals on the space of field histories $\Gamma_{\Sigma}(E)_{\delta_{EL} = 0}$ which are transgressions $\tau_{\Sigma}$ of variational differential forms $\alpha \in \Omega^{\bullet, \bullet}_\Sigma(E)$ on the jet bundle, but on these variational differential forms themselves (whence “local”, i.e. before transgression).

If $s$ denotes the BV-BRST differential in a BV-resolution $\Omega^{\bullet,\bullet}_\Sigma(E)\vert_{\mathcal{E}_{BV}}$ of the restriction to the shell $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ of the variational bicomplex $\Omega^{\bullet,\bullet}_\Sigma(E)$ with its total spacetime derivative $d$ (horizontal derivative), then the local BV-BRST cohomology is the cochain cohomology of $s + d$, hence of the total complex of the double complex given by $s$ and $d$.

Generally, considering variational differential forms up to $d$-exact terms is the “local” incarnation of what under the integration involved in the transgression is integration by parts and it is in this way that “local BV-BRST cohomology” knows about the actual BV-BRST cohomology on local observables.

Example

Consider local coordinates $(\phi^a)$ on the fibers of the field bundle. The corresponding antifield coordinates are to be denoted $\overline{\phi}_a$ and the BV-BRST differential takes them to the corresponding component

$$s(\overline{\phi}_a) = \frac{\delta_{El} L}{\delta \phi^a}$$

of the Euler-Lagrange form.

In degree $(p+1,0)$ the $s+d$-closed elements in vanishing ghost degree are pairs $(v,J_v)$ consisting of an infinitesimal symmetry of the Lagrangian $v$, regarded as an antifield density $v^a \overline{\phi}_a dvol_\Sigma$, together with a corresponding conserved Noether current $J_v$:

$$\array{ \{J_v\} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \} \\ && \uparrow\mathrlap{-s} \\ && \{ v^a \overline{\phi}_a dvol_\Sigma\} }$$

Such pairs are $(s+d)$-exact if on-shell the infintiesimal symmetry coincides with an infinitesimal gauge symmetry. To see this, recall:

An infinitesimal gauge symmetry $v_\epsilon$ of gauge parameter $(\epsilon^\alpha)$ is a vector field on the jet bundle with components of the form

$$\mathcal{L}_{v_\epsilon} \phi^a \;\coloneqq\; R^a_\alpha \epsilon^\alpha + R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}$$

such that this is an infinitesimal symmetry of the Lagrangian in that

$$\begin{aligned} \iota_{v_\epsilon} \delta_{EL} \mathbf{L} & = v^a \frac{\delta_{EL} L}{\delta \phi^a} dvol_\Sigma \\ & = \epsilon^\alpha \left( R^a_\alpha \frac{\delta_{EL} L}{ \delta \phi^a} - \frac{d}{d x^\mu} \left( R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a} \right) \right) dvol_\Sigma + d\left( \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a} \right) \iota_{\partial_\mu} dvol_\Sigma \\ & = 0 + d(...) \end{aligned}$$

for all $(\epsilon^\alpha)$.

The corresponding antighosts $\overline{c}_\alpha$ are taken by the BV-BRST differential to the antifield-preimage of the term on the left:

$$s\left(\overline{c}_\alpha\right) \;=\; R^a_\alpha \overline{\phi}_a - \frac{d}{d x^\mu} \left( R^{a \mu}_\alpha \overline{\phi}_a \right) \,.$$

Moreover, an on-shell vanishing infinitesimal symmetry of the Lagrangian is a vector field with components of the form

$$\kappa^{a b} \frac{\delta_{EL} L}{\delta \phi^a}$$

for $\kappa^{a b} = - \kappa^{b a}$ a skew-symmetric system of smooth functions on the jet bundle.

The linear combination of such an infinitesimal gauge symmetry and an on-shell vanishing infinitesimal symmetry is $(s+d)$-exact:

$$\begin{aligned} v^a dvol_\Sigma & = \left( R^a_\alpha \epsilon^\alpha + R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu} + \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a } \right) dvol_\Sigma \\ & = s \left( \epsilon^\alpha \overline{c}_\alpha - \tfrac{1}{2}\kappa^{a b} \overline{\phi}_a \overline{\phi}_b \right) dvol_\sigma + d\left( \epsilon^\alpha R^{a \mu}_\alpha \right) \iota_{\partial_\mu} dvol_\Sigma \end{aligned}$$

(Barnich-Brandt-Henneaux 94, p. 20)

It may be useful to organize this expression into the $s+d$-bicomplex like so:

$$\array{ \{K\} &\overset{d}{\longrightarrow}& \{ d K + \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL}\mathbf{L}}{ \delta \phi^a} \} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \} \\ && \mathllap{s}\uparrow && \uparrow\mathrlap{-s} \\ && \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a \iota_{\partial_\mu} dvol_\Sigma &\underset{d}{\longrightarrow}& \left\{ d\left( \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a \right) \iota_{\partial_\mu} dvol_\Sigma + \left( R^a_\alpha \epsilon^\alpha + R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu} + \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a } \right) \overline{\phi}_a \, dvol_\Sigma \right\} \\ && && \uparrow\mathrlap{-s} \\ && && \left( - \epsilon^\alpha \overline{c}_\alpha + \tfrac{1}{2}\kappa^{a b } \overline{\phi}_a \overline{\phi}_b \right) dvol_\Sigma }$$

Related concepts

• variational BV-BRST bicomplex

References

The general theory:

• Glenn Barnich, Friedemann Brandt, Marc Henneaux, Local BRST cohomology in the antifield formalism: I. General theorems, Commun. Math. Phys. 174 (1995) 57-92 [arXiv:hep-th/9405109, doi:10.1007/BF02099464]

Review:

• Glenn Barnich, Friedemann Brandt, Marc Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439-569 [doi:10.1016/S0370-1573(00)00049-1arXiv:hep-th/0002245]

Details on the local antibracket:

• Glenn Barnich, Marc Henneaux, section 2 and appendix B of: Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket, J. Math. Phys. 37 (1996) 5273-5296 [arXiv:hep-th/9601124, doi:10.1063/1.531726]

Application to gravity and/or Yang-Mills theory (Einstein-Yang-Mills theory) is discussed in

• Glenn Barnich, Friedemann Brandt, Marc Henneaux, Local BRST cohomology in Einstein-Yang-Mills theory, Nucl. Phys. B 455 (1995) 357-408 [doi:10.1016/0550-3213(95)00471-4arXiv:hep-th/9505173]