Donaldson-Thomas invariants, torus knots, and lattice paths

In this paper we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory -- we find explicit formulae for classical generating functions and Donaldson-Thomas invariants of an arbitrary symmetric quiver. We then focus on quivers corresponding to $(r,s)$ torus knots and show that their classical generating functions, in the extremal limit and framing $rs$, are generating functions of lattice paths under the line of the slope $r/s$. Generating functions of such paths satisfy extremal A-polynomial equations, which immediately follows after representing them in terms of the Duchon grammar. Moreover, these extremal A-polynomial equations encode Donaldson-Thomas invariants, which provides an interesting example of algebraicity of generating functions of these invariants. We also find a quantum generalization of these statements, i.e. a relation between motivic quiver generating functions, quantum extremal knot invariants, and $q$-weighted path counting. Finally, in the case of the unknot, we generalize this correspondence to the full HOMFLY-PT invariants and counting of Schr\"oder paths.Comment: 45 pages. Corrected typos in new versio

Extended Fermion Representation of Multi-Charge 1/2-BPS Operators in AdS/CFT -- Towards Field Theory of D-Branes --

We extend the fermion representation of single-charge 1/2-BPS operators in the four-dimensional N=4 super Yang-Mills theory to general (multi-charge) 1/2-BPS operators such that all six directions of scalar fields play roles on an equal footing. This enables us to construct a field-theorectic representation for a second-quantized system of spherical D3-branes in the 1/2-BPS sector. The Fock space of D3-branes is characterized by a novel exclusion principle (called `Dexclusion' principle), and also by a nonlocality which is consistent with the spacetime uncertainty relation. The Dexclusion principle is realized by composites of two operators, obeying the usual canonical anticommutation relation and the Cuntz algebra, respectively. The nonlocality appears as a consequence of a superselction rule associated with a symmetry which is related to the scale invariance of the super Yang-Mills theory. The entropy of the so-called superstars, with multiple charges, which have been proposed to be geometries corresponding to the condensation of giant gravitons is discussed from our viewpoint and is argued to be consistent with the Dexclusion principle. Our construction may be regarded as a first step towards a possible new framework of general D-brane field theory.Comment: 43 pages, 4 figures; version 2, corrected typos and added reference

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete