群胚、Frobenius代数与Poisson西格玛模型

有限量子群胚的双代数观

The double algebraic view of finite quantum groupoids

这是一个双代数的介绍，双代数是由霍普夫代数、弱霍普夫代数和霍普夫代数体中的卷积积的性质所模拟的结构。证明了具有Frobenius积分的Hopf代数胚可以刻画为两个乘法满足分配性的双代数。我们讨论的问题的对偶性，对极，Maschke定理和例子。

Frobenius objects in the category of spans

We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids. Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

A Functorial Construction of Quantum Subtheories

We apply the geometric quantization procedure via symplectic groupoids proposed by E. Hawkins to the setting of epistemically restricted toy theories formalized by Spekkens. In the continuous degrees of freedom, this produces the algebraic structure of quadrature quantum subtheories. In the odd-prime finite degrees of freedom, we obtain a functor from the Frobenius algebra in \textbf{Rel} of the toy theories to the Frobenius algebra of stabilizer quantum mechanics.

Generating functions for local symplectic groupoids and non-perturbative semiclassical quantization

This paper contains three results about generating functions for Lie-theoretic integration of Poisson brackets and their relation to quantization. In the first, we show how to construct a generating function associated to the germ of any local symplectic groupoid and we provide an explicit (smooth, non-formal) universal formula $S_π$ for integrating any Poisson structure $π$ on a coordinate space. The second result involves the relation to semiclassical quantization. We show that the formal Taylor expansion of $S_{tπ}$ around $t=0$ yields an extract of Kontsevich's star product formula based o…

李代数体泊松西格玛模型

The Lie algebroid Poisson sigma model

我们最近提出的Poisson-Weil西格玛模型是从Poisson西格玛模型的目标空间的Hamilton李群对称性出发的。在BV主作用量的规范固定后，它产生了有趣的拓扑场论，如二维Donaldson-Witten拓扑规范理论和规范A拓扑西格玛模型。在本文中，我们推广了上述构造，构造了李代数体泊松西格玛模型。这是通过测量泊松西格玛模型目标空间的哈密顿李群胚对称性得到的。我们在AKSZ几何版本中使用BV量化方法以确保一致的量化和目标空间协方差。该模型具有非常丰富的几何和复杂的BV上同调，这是详细的研究。

AKSZ西格玛模型的高阶Chern-Weil推导

A higher Chern-Weil derivation of AKSZ sigma-models

Chern-Weil理论为李代数g上的每个不变多项式提供了一个从g-联络到微分上圈的映射，微分上圈的体积完整性是相应的Chern-Simons理论作用泛函。Kotov和Strobl已经观察到，这自然地从李代数推广到dg-流形和dg-丛，并且以这种方式与$n$ -辛流形相关联的Chern-Simons作用泛函是AKSZ $σ$ -模型的作用泛函，其目标空间是给定的$n$ -辛流形（这方面的例子是泊松σ模型或柯朗σ模型，包括普通的陈-西蒙斯理论，或高维阿贝尔陈-西蒙斯理论）。在这里，我们展示了如何在光滑无穷广群胚中的高阶Chern-Weil理论的框架内，这个结果可以自然地恢复并增强到高阶栈的态射，就像普通的Chern-Simons理论被增强到从带连接的主G-丛的栈到带连接的线3-丛的3-栈的态射一样。

Stokes manifolds and cluster algebras

Stokes' manifolds, also known as wild character varieties, carry a natural symplectic structure. Our goal is to provide explicit log-canonical coordinates for these natural Poisson structures on the Stokes' manifolds of polynomial connections of rank $2$, thus including the second Painlevé\ hierarchy. This construction provides the explicit linearization of the Poisson structure first discovered by Flaschka and Newell and then rediscovered and generalized by Boalch. We show that, for a connection of degree $K$, the Stokes' manifold is a cluster manifold of type $A_{2K}$. The main idea is then …