迈向AKSZ理论的第一量子化形式主义
Towards First Quantisation Formalism for AKSZ Theories
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中文总结 AI 辅助
该研究构建了AKSZ理论的第一量子化图像理论$\boldsymbol{\tau}$，其配分函数可重现原理论的费曼图，还给出了Witten-Morse超对称量子力学等实例。
中文摘要 AI 辅助
给定流形$M$上的AKSZ理论$\boldsymbol{\top}$，其靶空间为分次向量空间$Y$，我们在图上构建一个一维理论$\boldsymbol{\tau}$（即$\boldsymbol{\top}$的“第一量子化图像”），其配分函数可重现$\boldsymbol{\top}$的费曼图。更准确地说，理论$\boldsymbol{\tau}$本身是一维AKSZ理论，其靶空间由$M$构建，且涉及与一维超引力的耦合。它在度量图空间上产生一个形式（边的长度$T$及其 de Rham 微分$\boldsymbol{\text{d}}T$分别被解释为引力子和引力微子的零模）；该形式的积分给出$\boldsymbol{\top}$的费曼图之和。我们在BV-BFV形式主义中研究理论$\boldsymbol{\tau}$；$\boldsymbol{\top}$的规范固定对应$\boldsymbol{\tau}$的规范固定。在经典层面，$\boldsymbol{\tau}$将顶点映射到$\boldsymbol{\tau}$的相空间$\boldsymbol{\bar{\boldsymbol{\tau}}}$的笛卡尔幂$\boldsymbol{\bar{\boldsymbol{\tau}}}^{\times k}$中的某些拉格朗日子流形$L_k$。这些子流形可被视为在Weinstein的辛范畴中定义了一个循环$\text{L}_\boldsymbol{\times}$代数（“去量子化”$\boldsymbol{\top}$的靶AKSZ dg结构上的上同调向量场）。在$\boldsymbol{\tau}$的路径积分构建中，拉格朗日量$L_k$确定$k$价顶点处入射边的场的缝合条件。我们给出该范式的例子，例如当$\boldsymbol{\tau}$在边上是Witten-Morse超对称量子力学时（对应$\boldsymbol{\top}$和$\boldsymbol{\tau}$的一种特定类型的规范固定）。在$\boldsymbol{\top}$是具有结构李代数$\boldsymbol{\text{su}(2)}$的非阿贝尔Chern–Simons理论的例子中，我们描述顶点拉格朗日$L_{\text{W}}$（即“Wigner拉格朗日”）。
英文摘要
\noindent Given an AKSZ theory $\mathbb{T}$ on a manifold $M$, with target a graded vector space $Y$, we formulate a 1-dimensional theory $\mathbb{t}$ 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 $Φ^{\times k}$ of the phase space $Φ$ 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 %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 $L_{\mathrm{W}}$ (the ``Wigner Lagrangian'' ).
