AKSZ Theories
While reading about the BV-formalism for gauge theories, I recently found this great script by Prof. Pavel Mnev posing a great starting point for a variety of related topics. The lecture notes, written down for a lecture at the University of Notre Dame in the Fall of 2016, give a widespread yet comprehensible introduction to a multitude of topics like TQFTs, path integrals, the BV-formalism and a particular type of sigma models or $\sigma$-models, so-called AKSZ theories, which I want to focus on in this article. This article aspires to be a comprehensive compactification of the exposition of AKSZ in Mnev's lecture notes while some nods and comments to embed the main ideas in a physical and categorial context.
In their original paper, Alexandrov, Kontsevich, Schwarz and Zaboronsky use their motivating example, $2$-dimensional Chern-Simons theory, to foreshadow a general construction of BV data for field theories that can be sensibly formulated as $\sigma$-models. The tl;dr is that by considering fitting field theories as a certain split between source and target data with fields defined as the mapping space between them [this is characteristic for $\sigma$-models], one can literally and immediately pull-back the target data to define the BV data on the space of fields. But why should a physicist or mathematician getting into gauge theories care about such a construction?
First and foremost the AKSZ construction has the virtue that constructing the BV action from a fitting $\sigma$-model QFT is immediate; no complicated explicit construction including a possibly infinite number of Ghost and Anti-Ghost fields is required.
Now one can rightfully ask, if the class of "fitting $\sigma$-model QFTs" has any relevant members. The answer is yes! Among its member are popular physical TQFTs like Chern-Simons theory, topological Yang-Mills theory and BF theory which, in $3$ dimensions can be used to formulate Palatini Gravity.
From a mathematicians perspective the AKSZ construction, defined on $\sigma$-models with target space a symplectic Lie $n$-algebroid, provides a naturally arising graded geometry view on many TQFTs as well as a beautiful description in terms of category theory.
The following data constitutes the base of the AKSZ construction:
$\textbf{Source data:}$ The source is comprised of a compact, oriented $n$-manifold $M$ without boundary. It harbours the de Rham differential $d_M$.
- $\textbf{Target data:}$ The target is a hamiltonian differential graded $(n-1)$-manifold, namely a tuple $(N, Q_N, \omega_N, \Theta_N)$ where
- $(N, Q_N)$ is a differential graded manifold with de Rham differential $\delta$,
- $\omega_N$ is a graded symplectic form with internal (grading) degree $(n-1)$, namely $\omega_N \in \Omega^2(N)_{n-1}$,
- $\omega_N$ is exact, thus $\omega_N = \delta \alpha_N$ for a fixed $a \in \Omega^1(N)_{n-1}$,
- $\Theta_N \in C^\infty(N)_n$ is the Hamiltonian associated to the cohomological vector field $Q_N$, namely $Q_N := \{\Theta_N, \cdot\}_{\omega_N}$ and $\{\Theta_N, \Theta_N\}_{\omega_N} = 0$.
Some comments: In mathematical literature one often refers to compact manifolds without boundaries as "closed" manifolds. While this term has become somewhat standard, it can conflict with the notion of closedness in the topological sense, hence the spelled-out version. Also note that the data of the target can be summarised under the notion of a symplectic Lie n-algebroid. Lastly I want to point out that the target in the presented definition is strongly reminiscient of the data of ordinary classical mechanics in the Hamiltonian formulation.
Since we want to construct a field theory, the last basic ingredient we need is the $\textit{space of fields}$, namely a suitable space of mappings between the shifted tangent space of the source, $T[1]M$, and the target $N$. Since both are graded manifolds, a tentative definition would be
$$ \mathcal{F} := \mathrm{Mor}(T[1]M, N), $$
i.e. degree preserving mappings, the morphisms of the category of graded manifolds. However for three arbitrary graded vector spaces $U,V,W$, this space fails the following desireable adjunction property:
$$ \mathrm{Mor}(U, \mathrm{Mor}(V,W)) \neq \mathrm{Mor}(U \times V, W). $$
Thus it also fails for graded manifolds. If however we fall back to the full space of continous mappings $\mathrm{Map}(V,W) := W^V$, the adjunction property holds, even for graded manifolds. We thus define the space of fields as
$$ \mathcal{F} := \mathrm{Map}(T[1]M, N). $$
At this point it seems mandatory to adress the general spirit of the AKSZ construction. In their $1995$ paper they motivate the approach using a hamiltonian dg-manifold instead of data containing a classical action functional. Their main point is that one and the same action, say the trivial action $S = 0$, can yield physically different BV theories, if the group describing the trivial and gauge symmetries is chosen differently. For classical gauge theories, the action functional, with all gauge fixing terms added, is required to satisfy the $\textbf{Classical Master Equation}$ or $\textbf{CME}$
$$ \{S, S\} = 0, $$
after the process of gauge fixing has been applied. Thus the CME is used as a defining equation for the solution of the gauge problem. It was however common to put the classical, ungauged action $S_{cl}$ as the defining quantity for a physical theory, even in the BV-formalism. AKSZ argued that due to the obvious degeneracies following from this focus, one should rather put a solution to the CME at the center of attention. While this might seem like a given rule nowadays, it sheds some light onto the thought process behind the AKSZ construction.
The goal is now to construct sensible data on the space of fields $\mathcal{F}$ to mimic that of the BV-formalism. Since $d_M$ defines a cohomological vector field on $T[1]M$ and $Q_N$ one on $N$, we can lift these two to the space of fields $\mathcal{F} = \mathrm{Map}(T[1]M, N)$ by considering their left and right action on fields respectively. We thus obtain the lifted $d_M^\dagger, Q_N^\dagger \in \Gamma(T\mathcal{F})_1$. To no surprise this defines a cohomological vector field on $\mathcal{F}$ via
$$ Q := d_M^\dagger + Q_N^\dagger. $$
To see that $Q$ is cohomological, note that $d_M$ and $Q_N$ anti-commute, so $Q$ squares to $0$ due to its individual parts squaring to $0$. We also see that the source part, namely $d_M^\dagger$ and the target part $Q_N^\dagger$ are split in this cohomological vector field. Hence it is to be expected that there is a way to make the resulting cohomology split into a bi-complex hosting two form-degrees, one associated to the source and one to the target. To this end, consider the following diagram:
$\require{AMScd}$
\begin{CD}
T[1]M \times \mathcal{F} @>ev>> N\\
@V \pi V V \\
\mathcal{F}
\end{CD}
Here, $\mathrm{ev} \colon T[1]M \times \mathcal{F} \longrightarrow N$ denotes the obvious evaluation mapping and $\pi \colon T[1]M \times \mathcal{F} \longrightarrow \mathcal{F}$ the natural porjection onto the second factor. Since our goal is to move structure from $N$ to $\mathcal{F}$, we consider the pullback by $\mathrm{ev}$ and the pushforward $\pi_*$ which can be canonically defined.
Using $\mathrm{ev}^*$ we can take a form $\beta \in \Omega^k(N)_i$ and pull it back to $T[1]M \times \mathcal{F}$, choosing the $(0,k)$-form part. Since on $T[1]M$ there exists a canonical Berezinian $\mu$, we can integrate $\mathrm{ev}^*\beta$ over $T[1]M$ yielding a form in $\Omega^k(\mathcal{F})_{i-n}$. We denote this integration process by $\pi_*$. Note that the integration sends $n$-forms on $M$ to numbers, effectively dropping the degree of the final form by $n$. All in all we obtain a map
$$ \mathcal{T} := \pi_* \mathrm{ev}^* \colon \Omega^k(N)_i \longrightarrow \Omega^k(\mathcal{F})_{i-n}. $$
This map has some peculiar properties which we will briefly discuss. First let $\{x^a\}$ be the set of local coordinates on the target space $N$. We define local coordinates on $T[1]M \times \mathcal{F}$ by $X^a = \mathrm{ev}^* x^a$. Written in these local coordinates, we can express the action of $\mathcal{T}$ on a differential form $\beta \in \Omega^k(N)_i$ as
$$ \mathcal{T}(\beta) = \mathcal{T} \left( \sum_{b_1, \ldots, b_k} \beta_{b_1 \ldots b_k}(x) \delta x^{b_1} \wedge \ldots \wedge \delta x^{b_k} \right) = \int_M \left( \sum_{b_1, \ldots, b_k} \beta_{b_1 \ldots b_k}(X) \delta X^{b_1} \wedge \ldots \wedge \delta X^{b_k} \right). $$
This has one immediate consequence: The map $\mathcal{T}$ intertwines the de Rham differential $\delta$ on $N$ and $\delta_\mathcal{F}$ on $\mathcal{F}$ by rule of Stoke's theorem. Namely $\mathcal{T}$ defines a homomorphism between the chain complexes on the target and on the space of fields. Using this we can sensibly define the following objects:
- $\omega = \mathcal{T} (\omega_N) \in \Omega^2(\mathcal{F})_{-1}$, a $(-1)$-symplectic closed form. We can directly infer weak non-degenerate from the non-degeneracy of $\omega_N$. Furthermore we use that $\mathcal{T}$ is a chain map to note
$$ \delta_\mathcal{F} \omega = \delta_\mathcal{F} \circ \mathcal{T}(\omega_N) = \mathcal{T}(\delta \omega_N) = 0.$$ - $S = \imath_{d^\dagger_M} \mathcal{T}(\alpha_N) + \mathcal{T} (\Theta_N) \in C^\infty(\mathcal{F})_0$, an action satisfying the CME and inducing the Hamiltonian vector field $Q$. The latter can be seen from
$$ \imath_Q \omega = (\imath_{d^\dagger_M} \circ \mathcal{T} + \imath_{Q^\dagger_N} \circ \mathcal{T}) \omega_N = \imath_{d^\dagger_M} \circ \delta_\mathcal{F} \circ \mathcal{T}(\alpha_N) + \mathcal{T}(\imath_{Q_N} \omega_N) = \delta_\mathcal{F} \circ \imath_{d^\dagger_M} (\alpha_N) + \delta_\mathcal{F} \circ \mathcal{T}(\Theta_N) = \delta_\mathcal{F} S.$$
Since we already showed that $Q^2 = 0$, this immediately shows that $\{S,S\}_\omega = 0$.
Collecting all of the above constructions, we see that the tuple $(\mathcal{F}, \omega, Q, S)$ constitutes a classical BV theory!
The last point I want to make is the connection to the classical field theory contained within the AKSZ construction. Classical fields correspond to the morphisms of graded manifolds contained within the mapping space. Thus we set
$$ \mathcal{F}_{cl} := \mathrm{Mor}(T[1]M, N), $$
to be the space of classical fields. From the BV action $S$, we obtain the classical action via the restriction to classical fields
$$ S_{cl} = \left. S \right|_{\mathcal{F}_{cl}}. $$
With this short introduction to the AKSZ construction done, the next related article will treat its application to BF theories.
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