Towards First Quantisation Formalism for AKSZ Theories

Given an AKSZ theory $\mathbb{T}$ on a manifold $M$, with target a graded vector space $Y$, we formulate a $1$-dimensional theory t on graphs $($the "first quantisation picture for $\mathbb{T}$"$)$, whose partition functions reproduce the Feynman graphs of $\mathbb{T}$. More precisely, the theory $\mathbb{t}$ is itself a $1d$ AKSZ theory with the target built out of $M$, and involving a coupling to $1d$ supergravity. It yields a form on the space of metric graphs $($with length $T$ of an edge and its de Rham differential $\mathrm{d} T$ interpreted as the zero-modes of the graviton and gravitino, respectively$)$; its integral yields the sum of Feynman graphs of $\mathbb{T}$.

We study the theory $\mathbb{t}$ in the BV-BFV formalism; a gauge-fixing of $\mathbb{T}$ corresponds to a gauge-fixing of $\mathbb{t}$. At the classical level, $\mathbb{t}$ assigns to vertices certain Lagrangian submanifolds $L_k$ in Cartesian powers $\Phi^{\times k}$ of the phase space $\Phi$ of $\mathbb{t}$. These submanifolds can be thought of as defining a cyclic $\mathrm{L}_\infty$-algebra in Weinstein's symplectic category $($"dequantising" the cohomological vector field on the target AKSZ dg structure of $\mathbb{T}$$)$. In the path integral construction of $\mathbb{t}$, Lagrangians $L_k$ determine sewing conditions for fields on the incident edges at a $k$-valent vertex.

We give examples of this paradigm, such as when $\mathbb{t}$ on edges is the Witten-Morse supersymmetric quantum mechanics $($which corresponds to a particular type of gauge-fixing for $\mathbb{T}$ and $\mathbb{t}$$)$. In the example where $\mathbb{T}$ is the non-abelian Chern--Simons theory with structure Lie algebra $\mathfrak{su}(2)$, we describe the vertex Lagrangian $L_W$ $($the "Wigner Lagrangian"$)$.

The paper is available to read on the arXiv.