3 Corrected spelling, improved formatting.
• the problem of consistent deformation of a theoryand ,
• The BV formalism is specially elegant attractive because it does not require one to make a choice of a gauge fixing and it maintains a manifest spacetime covariance. It can also deals with situation that the traditional BRST formalism can not handle. This is for example the case of gauge theories admitting an open gauge algebra ( a gauge algebra that is closed only modulo the equations of motion). This is for The typical example the case of are supergravity theories.The BV formalism also allows an elegant and powerful mathematical reformulation of certain questions of quantum field theories in the language of homomological algebra.

Mathematically, the BV formalism is just an simply a clever application of homological perturbation theory. In order to understand the relation, I will first explain review the geometry of a physical model given described by a Lagrangian $\mathcal{L}$ depending of fields $\phi^I$ and a finite number of their derivatives and admitting a gauge symmetry $G$. The starting point is the space $\mathcal{M}$ of all possible configurations of fields and their derivatives. This can be formalized using the language of jet-spaces. The Euler-Lagrange equations give the equations of motion of the theory and together with their derivatives, they define a sub-space $\Sigma$ of $\mathcal{M}$ called the stationary space. The on-shell function functions are the function functions relevant for the dynamic of the theory, they are defined on the stationary space $\Sigma$, they can be described a alegebraically as $\mathbb{C}^\infty(\Sigma)=\mathbb{C}^\infty(\mathcal{M})/ \mathcal{N}$ where $\mathcal{N}$ is the ideal of functions that vanish on $\Sigma$.For simple theories like Yang-Mills, we just have $s=\delta+\gamma$ because the gauge algebra closed closes as a group without using the equations of motion. In more complicate situation when the algebra is open there are additional terms in the definition of $s$.

• The group $H^{0,n}(s|d)$ controls the renormalization of the theory by classifying and all the possible counter termsthat can be added to .

• The groups $H^{0,n}(\gamma,d)$ and $H^{1,n}(\gamma, d)$ control the Lagrangianconsistent deformations of the theory.

• 2 Corrected spelling, improved formatting.

The BV formalism provides a (co)homological reformulation of several important questions of quantum field theory. The kind of problems that are usually addressed by the BV formalism are: the determination of gauge invariant operators, the determination of conserved courantscurrents, the problem of consistent deformation of a theory and the determination of possible quantum anomalies (the violation of the gauge invariance due to quantum effects). The BV formalism is specially elegant because it does not require one to make a choice of a gauge fixing and it maintain maintains a manifest spacetime covariance. It can also deals with gauge theories admitting an open gauge algebra ( a gauge algebra that is closed only modulo the equations of motion).

The starting point is the space $\mathcal{M}$ of all possible configurations of fields and their derivatives. This can be formalized using the language of jet-spaces. The Euler-Lagrange equations give the equations of motion of the theory and together with their derivatives, they define a sub-space $\Sigma$ of $\mathcal{M}$ called the stationary space $\Sigma$. . The on-shell function are the function defined on the stationary space $\Sigma$, they can be seen as described a $\mathbb{C}^\infty(\Sigma)=\mathbb{C}^\infty(\mathcal{M})/ \mathcal{N}$ where $\mathcal{N}$ is the ideal of functions that vanish on $\Sigma$.Because of the gauge invariance, the Euler-Lagrange equations are not independent but they satisfy some non-trivial relations called Noether identities. One has to identify different configurations related by a gauge transformation. This step is justified by the fact that Indeed, a gauge symmetry is not a real symmetry of the theory but more a redundancy of the description.

The two steps that we have just described (restriction to the stationary surface and taking the quotient by the gauge transformations) are respectively realized in the BV formalism respectively by the homology of the Koszul-Tate differential $\delta$ and the cohomology of the longitudinal operator $\gamma$. This is done by introducing one antifield $\phi^*_I$ for each field $\phi^I$ of the Lagrangian. The antifields are introduce introduced to ensure that the equations of motion are trivial in the homology of the Koszul-Tate operator. The gauge invariance of the theory is taking care of by the cohomology of the longitidunal longitudinal differential $\gamma$. The In the case of Yang-Mills theories, the cohomology of $\gamma$ is just equivalent to the Lie algebra cohomologyin the case of Yang-Mills theories.

One can generates $s$ using the BV bracket $(\cdot ,\cdot)$ (under which the fields a field and the its associated antifields are dualfields) and a source $S$ such that the BV operator can be expressed as The classifical master equation istherefore
$(S,S)=0$ $(S,S)=0,$$where \Delta is an operator similar to the Laplacian but defined in the space of fields and their antifields. This operator naturally appear appears when one consider considers the invariance of the measure of the path integral under an infinitesimal BRST transformation. Now we can We will now review very quickly how one uses the BV (co)homologial description co)homological interpretation of BV to answer some important questions in quantum field theory: • Non-trivial conserved courantscurrents of the theory are equivalent to the so-called characteristic cohomology H^{n-1}_0(\delta |d) which is the cohomology of the Koszul-Tate operator \delta (in antifield number zero) modulo total derivatives for forms of degree n-1, where n is the dimension of spacetime. • For a short review, I recommend the • For more information on the perturbative homology descriptionof the BV formalism, I would recommend the • The classical reference is the book of • 1 The BV formalism provides a (co)homological reformulation of several important questions of quantum field theory. The kind of problems that are usually addressed by the BV formalism are: the determination of gauge invariant operators, the determination of conserved courants, the problem of consistent deformation of a theory and quantum anomalies (the violation of the gauge invariance due to quantum effects). The BV formalism is specially elegant because it does not require one to make a choice of a gauge fixing and it maintain a manifest spacetime covariance. It can also deals with gauge theories admitting an open gauge algebra ( a gauge algebra that is closed only modulo the equations of motion). This is for example the case of supergravity theories. Mathematically, the BV formalism is just an application of homological perturbation theory. In order to understand the relation, I will first explain the geometry of a physical model given by a Lagrangian \mathcal{L} depending of fields \phi^I and a finite number of their derivatives and admitting a gauge symmetry G. The starting point is the space \mathcal{M} of all possible configurations of fields and their derivatives. This can be formalized using the language of jet-spaces. The Euler-Lagrange equations give the equations of motion of the theory and together with their derivatives, they define a sub-space of \mathcal{M} called the stationary space \Sigma. The on-shell function are the function defined on the stationary space \Sigma, they can be seen as \mathbb{C}^\infty(\Sigma)=\mathbb{C}^\infty(\mathcal{M})/ \mathcal{N} where \mathcal{N} is the ideal of functions that vanish on \Sigma. Because of the gauge invariance, the Euler-Lagrange equations are not independent but they satisfy some non-trivial relations called Noether identities. One has to identify different configurations related by a gauge transformation. This step is justified by the fact that a gauge symmetry is not a real symmetry of the theory but more a redundancy of the description. The two steps that we have just described (restriction to the stationary surface and quotient by the gauge transformations) are realized in the BV formalism respectively by the homology of the Koszul-Tate differential \delta and the cohomology of the longitudinal operator \gamma. The Koszul-Tate operator defines a resolution of the equations of motion in homology. This is done by introducing one antifield \phi^*_I for each field \phi^I of the Lagrangian. The antifields are introduce to ensure that the equations of motion are trivial in the homology of the Koszul-Tate operator. The gauge invariance of the theory is taking care of by the cohomology of the longitidunal differential \gamma. The cohomology of \gamma is just equivalent to the Lie algebra cohomology in the case of Yang-Mills theories. The full BV operator is then given by$$s=\delta + \gamma+\cdots,$$where the dots are for possible additional terms required to ensure that the BV operator s is nilpotent ( s^2=0). The construction of s from \delta and \gamma follows a recursive pattern borrowed from homological perturbation theory. One can trace the need for the antifields and the Koszul-Tate differential to this recursive pattern. For simple theories like Yang-Mills, we just have s=\delta+\gamma because the gauge algebra closed as a group without using the equations of motion. In more complicate situation when the algebra is open there are additional terms in the definition of s. One can generates s using the BV bracket (\cdot ,\cdot) (under which the fields and the antifields are dual fields) and a source S such that$$ s F= (S,F). $$The classifical master equation is therefore (S,S)=0 and it is just equivalent to s^2=0. At the quantum level, the action S is replaced by a quantum action W=S+\sum_ i \hbar^i M_i where the terms M_i are contribution due to the path integral measure. The gauge invariant of quantum expectation values of operators is equivalent to the quantum master equation :$$ \frac{1}{2}(W,W)=i\hbar \Delta W,$$where$\Delta$is an operator similar to the Laplacian but defined in the space of fields and their antifields. This operator naturally appear when one consider the invariance of the measure of the path integral under an infinitesimal BRST transformation. When$\Delta S=0$, we can take$W=S$. Now we can review very quickly how one uses the (co)homologial description of BV to answer important questions in quantum field theory: • The observables of the theory are gauge invariant operators, they are described by the cohomology group$H(s)$in ghost number zero. • Non-trivial conserved courants of the theory are equivalent to the so-called characteristic cohomology$H^{n-1}_0(\delta |d)$which is the cohomology of the Koszul-Tate operator$\delta$(in antifield number zero) modulo total derivatives for forms of degree$n-1$, where$n$is the dimension of spacetime. • The equivalent class of global symmetries is equivalent to$H^n_1(\delta| d)$. • The gauge anomalies are controlled by the group$H^{1,n}(s|d)$(that is$H(s)$in antifield number 1 and in the space of$n$-form modulo total derivative). The conditions that define the cohomology$H^{1,n}(s|d)$are generalization of the famous Wess-Zumino consistency condition. • The group$H^{0,n}(s|d)\$ controls the renormalization of the theory by classifying all the possible counter terms that can be added to the Lagrangian.

## References:

• For a short review I recommend the preprint by Fuster, Henneaux and Maas: hep-th/0506098.

• For more information on the perturbative homology description of the BV formalism, I would recommend the classical book of Marc Henneaux and Claudio Teitelboim (Quantization of Gauge Systems).

• For applications there is also a standard review by Barnich, Brandt and Henneaux: Local BRST cohomology in gauge theories,'' Phys. Rept.338, 439 (2000) [arXiv:hep-th/0002245].