nLab local BRST cohomology



Algebraic Quantum Field Theory

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



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Given a Lagrangian field theory (E,L)(E,\mathbf{L}) with field bundle EfbΣE \overset{fb}{\to} \Sigma over some spacetime Σ\Sigma and local Lagrangian density L\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=τ ΣαA = \tau_{\Sigma} \alpha given by functionals on the space of field histories Γ Σ(E) δ EL=0\Gamma_{\Sigma}(E)_{\delta_{EL} = 0} which are transgressions τ Σ\tau_{\Sigma} of variational differential forms αΩ Σ ,(E)\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 ss denotes the BV-BRST differential in a BV-resolution Ω Σ ,(E)| BV\Omega^{\bullet,\bullet}_\Sigma(E)\vert_{\mathcal{E}_{BV}} of the restriction to the shell J Σ (E)\mathcal{E} \hookrightarrow J^\infty_\Sigma(E) of the variational bicomplex Ω Σ ,(E)\Omega^{\bullet,\bullet}_\Sigma(E) with its total spacetime derivative dd (horizontal derivative), then the local BV-BRST cohomology is the cochain cohomology of s+ds + d, hence of the total complex of the double complex given by ss and dd.

Generally, considering variational differential forms up to dd-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.


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

s(ϕ¯ a)=δ ElLδϕ a s(\overline{\phi}_a) = \frac{\delta_{El} L}{\delta \phi^a}

of the Euler-Lagrange form.

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

{J v} d {dJ vι vδ ELL=0} s {v aϕ¯ advol Σ} \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)(s+d)-exact if on-shell the infintiesimal symmetry coincides with an infinitesimal gauge symmetry. To see this, recall:

An infinitesimal gauge symmetry v ϵv_\epsilon of gauge parameter (ϵ α)(\epsilon^\alpha) is a vector field on the jet bundle with components of the form

v ϵϕ aR α aϵ α+R α aμdϵ αdx μ \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

ι v ϵδ ELL =v aδ ELLδϕ advol Σ =ϵ α(R α aδ ELLδϕ addx μ(R α aμδ ELLδϕ a))dvol Σ+d(ϵ αR α aμδ ELLδϕ a)ι μdvol Σ =0+d(...) \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 (ϵ α)(\epsilon^\alpha).

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

s(c¯ α)=R α aϕ¯ addx μ(R α aμϕ¯ a). 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

κ abδ ELLδϕ a \kappa^{a b} \frac{\delta_{EL} L}{\delta \phi^a}

for κ ab=κ ba\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)(s+d)-exact:

v advol Σ =(R α aϵ α+R α aμdϵ αdx μ+κ abδ ELLδϕ a)dvol Σ =s(ϵ αc¯ α12κ abϕ¯ aϕ¯ b)dvol σ+d(ϵ αR α aμ)ι μdvol Σ \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+ds+d-bicomplex like so:

{K} d {dK+ϵ αR α aμδ ELLδϕ a} d {dJ vι vδ ELL=0} s s ϵ αR α aμϕ¯ aι μdvol Σ d {d(ϵ αR α aμϕ¯ a)ι μdvol Σ+(R α aϵ α+R α aμdϵ αdx μ+κ abδ ELLδϕ a)ϕ¯ advol Σ} s (ϵ αc¯ α+12κ abϕ¯ aϕ¯ b)dvol Σ \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 }


The general theory:


Details on the local antibracket:

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

Last revised on January 29, 2024 at 07:37:39. See the history of this page for a list of all contributions to it.