Towards First Quantisation Formalism for AKSZ Theories

Abstract

\noindent 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 $t$ 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 $t$ in the BV-BFV formalism; a gauge-fixing of $\mathbb{T}$ corresponds to a gauge-fixing of $t$. At the classical level, $t$ assigns to vertices certain Lagrangian submanifolds $L_k$ in Cartesian powers $\hat{I}{¦}^{\times k}$ of the phase space $\hat{I}¦$ of $t$. 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 %target AKSZ dg structure of $\mathbb{T}$). In the path integral construction of $t$, 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 $t$ on edges is the Witten-Morse supersymmetric quantum mechanics (which corresponds to a particular type of gauge-fixing for $\mathbb{T}$ and $t$). 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 $L_{\mathrm{W}}$ (the ``Wigner Lagrangian'' ).

51 pages, 7 figures

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