BF theory as an AKSZ theory

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\newcommand{\gf}{\mathcal{g}}
\newcommand{\Ci}{C^\infty}
\newcommand{\FF}{\mathcal{F}}
\newcommand{\AA}{\mathcal{A}}
\newcommand{\TT}{\mathcal{T}}
\newcommand{\BB}{\mathcal{B}}
\newcommand{\ev}{\mathrm{ev}}
\newcommand{\Map}{\mathrm{Map}}
\newcommand{\lra}{\longrightarrow}
\newcommand{\pair}[2]{\left\langle #1, #2 \right\rangle}
$
While the previous introduction to AKSZ-type field theories completely lacked any practical examples, I already mentioned some popular TQFTs that can be formulated as AKSZ theories. In this article I will focus on $\textbf{BF theories}$, a certain type of TQFT that can be formulated in any dimension, a rare feat for a TQFT. Again the exposition will mainly follow the lecture notes by Prof. Pavel Mnev while giving some additional insights and comments.

Let $G$ be a Lie-group and $\gf$ its Lie-algebra. For the sake of simplicity we will focus on classical BF theory defined over a trivial $G$-bundle. Thus let $M$ be any compact oriented $n$-manifold without boundary and denote by $\mathrm{Conn}(G)$ the connections on the trivial $G$-bundle over $M$. Further define the $\textbf{curvature}$ of a connection $A \in \mathrm{Conn}(G)$ by
$$ F_A := dA + \frac{1}{2} [A,A] \in \Omega^2(M; \gf), $$
where $d$ denotes the de Rham differential, $\Omega^\bullet(M; \gf)$ the differential complex of $\gf$-valued differential forms and $[\cdot , \cdot]$ the Lie-bracket of $\gf$. Now if we denote by $\langle \cdot, \cdot \rangle$ the canonical non-degenerate pairing of $\gf$ and its dual $\gf^*$ [in particular this defines an invariant polynomial], we can define the following action functional:
$$\begin{aligned}
S_{cl} \colon \mathrm{Conn}(G) \times \Omega^{n-2}(M; \gf) &\longrightarrow \mathbb{R} \\
(A, B) & \longmapsto \int_M \pair{B}{F_A} .
\end{aligned}$$
While we won't need the explicit form of the action functional for the construction of the BV action in the AKSZ formalism, it will be an interesting exercise to consider its form once we have constructed the latter. Now the $\textbf{source data}$ is already clear: It is the compact oriented $n$-manifold without boundary $M$ together with the de Rham differential $d_M$. For the target, consider the following data:

• Define $N := \gf[1] \oplus \gf[n-2]$ as the target manifold with de Rham differential $\delta$, denote by $\{t_a\}$ a basis of $\gf$ with $\{t^a\}$ the dual basis of $\gf^*$ and by $\phi^a$ and $\xi_a$ the coordinates in degree $1$ and $(n-2)$ respectively. This lets us denote the generating functions for degree $1$ and $n-2$ respectively as
  $$ \phi := \phi^a t_a, \quad \quad \quad \quad \xi := \xi_a t^a. $$
  Lastly denote by $f^a_{bc}$ the structure constants of $\gf$ such that $[t_b, t_c] = f^a_{bc} t_a$.
• Define a symplectic form of internal grading degree $(n-1)$ via
  $$ \omega_N := \pair{ \delta \xi}{\delta \phi}. $$
  This lets us immediately read of not only that $\omega_N$ is exact, i.e. $\omega_N = \delta \alpha_N$, but that
  $$ \alpha_N = \pair{\xi}{\delta \phi}. $$
• Observing the isomorphism $\Ci(N) = \Ci(\gf[1] \oplus \gf[n-2]) \cong C^\bullet_{CE}( \gf, \mathrm{Sym}^\bullet(\gf^*[n-2]))$, where $C^\bullet_{CE}$ denotes the Chevalley-Eilenberg complex, we can simply take the Chevalley-Eilenberg differential to be the cohomological vector field $Q_N$:
  $$ Q_N := d_{CE} := \frac{1}{2} \pair{ [\phi, \phi]}{\frac{\partial}{\partial \phi}} +\pair{ \mathrm{ad}^*_\phi(\xi)}{\frac{\partial}{\partial \xi}  }.$$
• Since $Q_N$ is to be the Hamiltonian vector field of the Hamiltonian $\Theta_N$ on the target, i.e. $\imath_{Q_N} \omega_N = \delta \Theta_N$, we can infer
  $$ \Theta_N =\frac{1}{2} \pair{ \xi  }{[\phi, \phi]  }. $$
  Since $Q_N$ is cohomological, $\{\Theta_N, \Theta_N\}_{\omega_N}$ holds.

All in all the tuple $(N, Q_N, \omega_N, \Theta_N)$ poses the necessary target data properties to start the AKSZ construction! For the space of fields we can utilise the following helpful property of the mapping space: Since $\FF := \mathrm{Map}(T[1]M, N)$ and $N = \gf[1] \oplus \gf[n-2]$ is a graded vector space, it admits a global coordinate chart. We can thus identify the mapping space as
$$ \FF = \Omega^\bullet(M) \otimes N \cong \Omega^\bullet(M;\gf)[1] \oplus \Omega^\bullet(M;\gf^*)[n-2], $$
i.e. the space of $N$-valued differential forms on $M$. Next we pull back the generating functions $\phi$ and $\xi$, identified with $x^1$ and $x^2$ from the AKSZ construction, via $\ev \colon T[1]M \times \FF \lra N$ obtaining
$$\begin{aligned}
\AA := X^1 &= \ev^* \phi = c + A + B^\dagger + \sum_{i = 1}^{n-2} \tau_i^\dagger, \\
\BB := X^2 &= \ev^*\xi = c^\dagger + A^\dagger + B + \sum_{i = 1}^{n-2} \tau_i.
\end{aligned}$$
At this point we should carefully analyse the above "superfields" $\AA$ and $\BB$ in terms of their components:

• $A$ and $B$ denote the classical fields of the theory. They live in the internal grading degree $0$ and are of de Rham degree $1$ and $(n-2)$ respectively, just as in the classical BF theory. To each there is a corresponding anti-field: $A^\dagger$ is of de Rham degree $(n-1)$ and of internal grading degree $-1$ while $B^\dagger$ has de Rham degree $2$ and internal grading degree $-1$.
• The first ghost field $c$ sits in de Rham degree $0$ and internal degree $1$ with its antighost $c^\dagger$ in de Rham degree $n$ and internal degree $(n-2)$.
• The higher order ghosts $\tau_i$ which live in de Rham degree $(n-2-i)$ and internal degree $i$. To each there is an associated anti-ghost $\tau_i^\dagger$ in de Rham degree $(i+2)$ and internal degree $(-i-1)$.

A discussion of the physical meaning of anti-fields, ghost fields, anti-ghosts and the ghost number (i.e. internal degree) demands its own article. For now note that the components in negative internal degree correspond to the Koszul-Tate generators of the BV complex. A special role is assigned to $\tau_1$ and $c$ which are the generators of the gauge symmetries of BF theory. Meanwhile the higher order ghosts generate a particular reducibility the previously mentioned gauge symmetries exhibit. In particular this implies that in higher dimensions, the gauge symmetries are higher-order reducible, more on that in another article on BF theory. Note that the anti-fields and anti-ghosts arise in the usual construction of the BV complex to make its cohomology acyclic except for degree $0$. This is a rather strict demand and can lead to obstructions when trying to quantise the system.

Using the AKSZ construction we march forward to define the $(-1)$-symplectic form on $\FF$ as
$$\begin{aligned}
\omega :&= \TT(\omega_N) = \int_M \pair{ \delta \BB }{ \delta \AA }\\
&= \int_M \pair{\delta A}{\delta A^\dagger} + \pair{\delta B}{\delta B^\dagger} + \pair{\delta c}{\delta c^\dagger} + \sum_{i=1}^{n-2} \pair{\delta \tau_i}{\delta \tau_i^\dagger}
\end{aligned}$$
In particular we get
$$ \alpha := \TT(\alpha_N) = \int_M \pair{  \BB }{ \delta \AA }. $$
All that's left is to construct the BV action:
$$ S^{BF}_{BV} := \imath_{d^\dagger_M} \circ \TT(\alpha_N) + \TT(\Theta_N) = \int_M \pair{ \BB }{ \delta \AA + \frac{1}{2} [\AA, \AA] }. $$
Since we now see that the thus constructed tuple $(\FF, \omega, Q, S^{BF}_{BV})$ where $Q$ is the Hamiltonian vector field associated to $S^{BF}_{BV}$ is the data of a classical BV theory, we have successfully solved the classical gauge problem for the described BF theory!

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