Combinatorics inspired by Donaldson-Thomas theory

          @misc{ pirsa_PIRSA:09050035,
            doi = {10.48660/09050035},
            url = {https://pirsa.org/09050035},
            author = {Young, Benjamin},
            keywords = {},
            language = {en},
            title = {Combinatorics inspired by Donaldson-Thomas theory},
            publisher = {Perimeter Institute},
            year = {2009},
            month = {may},
            note = {PIRSA:09050035 see, \url{https://pirsa.org}}
          }
          

Abstract

I will describe some combinatorial problems which arise when computing various types of partition functions for the Donaldson-Thomas theory of a space with a torus action. The problems are all variants of the following: give a generating function which enumerates the number of ways to pile n cubical boxes in the corner of a room. Often the resulting generating functions are nice product formulae, as predicted by the recent wall-crossing formulae of Kontsevich-Soibelman. There are now a variety of techniques, both geometric and combinatorial, to compute these formula. My work uses the entirely combinatorial techniques, namely vertex operators and the planar dimer model; these techniques can be applied essentially "bare-handed" and rely very little upon the underlying algebraic geometry.