What is the Batalin-Vilkovisky formalism, and what are its uses in mathematics?

Question

I checked Wikipedia, I know it is a powerful quantization in physics, but I am wondering what is its relation in mathematics (like mirror symmetry as in wikipedia). A related thing is quantum master equation, what's its use in mathematics? Any reference or background? Thanks!

Answer 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 currents,
• the problem of consistent deformation of a theory,
• the determination of possible quantum anomalies (the violation of the gauge invariance due to quantum effects).

The BV formalism is specially 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). The typical example 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 simply a clever application of homological perturbation theory. In order to understand the relation, I will first review the geometry of a physical model 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 functions are the functions relevant for the dynamic of the theory, they are defined on the stationary space $\Sigma$, they can be described alegebraically as $\mathbb{C}^\infty(\Sigma)=\mathbb{C}^\infty(\mathcal{M})/ \mathcal{N}$ where $\mathcal{N}$ is the ideal of functions that vanish on $\Sigma$. 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. Indeed, a gauge symmetry is not a real symmetry of the theory but 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 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 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 longitudinal differential $\gamma$. In the case of Yang-Mills theories, the cohomology of $\gamma$ is equivalent to the Lie algebra cohomology.

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 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 $s$. One can generates $s$ using the BV bracket $(\cdot ,\cdot)$ (under which a field and its associated antifields are dual) and a source $S$ such that the BV operator can be expressed as $$ s F= (S,F). $$ The classifical master equation is
$$(S,S)=0,$$ 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 appears when one considers the invariance of the measure of the path integral under an infinitesimal BRST transformation. When $\Delta S=0$, we can take $W=S$.

We will now review the BV (co)homological interpretation of some 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 currents 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 and all the possible counter terms.

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

References:

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

• The classical reference is the 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].