Higher Cluster Categories and QFT Dualities

We present a unified mathematical framework that elegantly describes minimally SUSY gauge theories in even dimension, ranging from $6d$ to $0d$, and their dualities. This approach combines recent developments on graded quiver with potentials, higher Ginzburg algebras and higher cluster categories (also known as $m$-cluster categories). Quiver mutations studied in the context of mathematics precisely correspond to the order $(m+1)$ dualities of the gauge theories. Our work suggests that these equivalences of quiver gauge theories sit inside an infinite family of such generalized dualities, whose physical interpretation is yet to be understood.Comment: 61 pages, 30 figure

T-Path Formula and Atomic Bases for Cluster Algebras of Type D

We extend a T-path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D

Paths to Understanding Birational Rowmotion on Products of Two Chains

Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivariantly on antichains) of a poset $P$, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant) [AST11, BW74, CF95, Pan09, PR13, RuSh12,RuWa15+,SW12, ThWi17, Yil17. In this context, rowmotion appears to be related to Auslander-Reiten translation on certain quivers, and birational rowmotion to $Y$-systems of type $A_m \times A_n$ described in Zamolodchikov periodicity. We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths $r$ and $s$ is $r+s+2$ (first proved by D.~Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.Comment: 31 pages, to appear in Algebraic Combinatoric