相关论文: Exact WKB analysis and cluster algebras

Let $V$ be a vertex operator superalgebra and $g=\left(1\ 2\ \cdots k\right)$ be a $k$-cycle which is viewed as an automorphism of the tensor product vertex operator superalgebra $V^{\otimes k}$. In this paper, we construct an explicit…

In this paper we study the Gauss and Kummer hypergeometric equations in depth. In particular, we focus on the confluence of two regular singularities of the Gauss hypergeometric equation to produce the Kummer hypergeometric equation with an…

We use exact WKB analysis to derive some concrete formulae in singular quantum perturbation theory, for Schr\"odinger eigenvalue problems on the real line with polynomial potentials of the form $(q^M + g q^N)$, where $N>M>0$ even, and…

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups…

We use Bonahon-Wong's trace map to study character varieties of the once-punctured torus and of the 4-punctured sphere. We clarify a relationship with cluster algebra associated with ideal triangulations of surfaces, and we show that the…

Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative…

We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.

Recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool. In this article we show how to generalise the QR decomposition and SVD…

Hamiltonians whose symbols are not simply real valued, but matrix or, more generally, endomorphism valued functions appear in many places in physics, examples being the Dirac equation, multicomponent wave equations like electrodynamics in…

We consider an $n\times n$ system of ODEs on $\mathbb{P}^1$ with a simple pole $A$ at $z=0$ and a double pole $u={\rm diag}(u_1, \dots, u_n)$ at $z=\infty$. This is the simplest situation in which the monodromy data of the system are…

We study the topology associated with physical vector and scalar fields. A mathematical object, e.g., a ball, can be continuously deformed, without tearing or gluing, to make other topologically equivalent objects, e.g., a cube or a solid…

In the present paper, we examine in detail the method of "graph compactifications" of topological groups. The graph and Ellis methods of constructing proper compactifications of topological groups are applied for the investigation of…

This paper studies clustering algorithms for dynamically evolving graphs $\{G_t\}$, in which new edges (and potential new vertices) are added into a graph, and the underlying cluster structure of the graph can gradually change. The paper…

We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…

This paper contributes to the theory of singularities of meromorphic linear ODEs in traceless $2\times2$ cases, focusing on their deformations and confluences. It is divided into two parts: The first part addresses individual singularities…

Stokes' theorem turns Abelian Berry phases into curvature fluxes, whereas path ordering precludes such a simple formula for non-Abelian holonomies. We show that a quantitative form of this intuition survives: arbitrary Wilczek--Zee…

Snake graphs appear naturally in the theory of cluster algebras. For cluster algebras from surfaces, each cluster variable is given by a formula which is parametrized by the perfect matchings of a snake graph. In this paper, we identify…

Deformations of quantum field theories which preserve Poincar\'e covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an…

Given a maximal rigid object $T$ of the cluster tube, we determine the objects finitely presented by $T$. We then use the method of Keller and Reiten to show that the endomorphism algebra of $T$ is Gorenstein and of finite representation…

This paper studies the validity of Stokes' theorem for differential subcomplexes naturally adapted to the noncommutative geometry of positively graded Lie groups, with particular emphasis on Carnot groups. We introduce geometric conditions…