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Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory

Amer Iqbal, Can Kozcaz, Khurram Shabbir

TL;DR

The paper builds a bridge between refined topological vertex calculus, 5D $U(1)$ gauge theory with an adjoint hypermultiplet, and the combinatorics of cylindric/periodic partitions. By employing a transfer-matrix/crystal framework and two natural slicings of toric geometries, it shows the refined partition function equals a generating function for cylindric partitions, or equivalently a trace over vertex-operator constructions; the cylinder size is set by the adjoint mass and Omega-background parameters. It also derives nontrivial $(q,t)$ identities from slicing independence, connects to the periodic Schur process (K=1) and to Hilbert-scheme vertex operators in the 4D limit, and demonstrates that the combinatorics of these gauge-theory partitions encode fiber-base dualities of the associated toric geometries. These results unify topological strings, gauge theory, and algebraic combinatorics, and provide new $(q,t)$ identities with potential applications in refined enumerative geometry and representation theory.

Abstract

We study the partition function of the compactified 5D U(1) gauge theory (in the Omega-background) with a single adjoint hypermultiplet, calculated using the refined topological vertex. We show that this partition function is an example a periodic Schur process and is a refinement of the generating function of cylindric plane partitions. The size of the cylinder is given by the mass of adjoint hypermultiplet and the parameters of the Omega-background. We also show that this partition function can be written as a trace of operators which are generalizations of vertex operators studied by Carlsson and Okounkov. In the last part of the paper we describe a way to obtain (q,t) identities using the refined topological vertex.

Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory

TL;DR

The paper builds a bridge between refined topological vertex calculus, 5D gauge theory with an adjoint hypermultiplet, and the combinatorics of cylindric/periodic partitions. By employing a transfer-matrix/crystal framework and two natural slicings of toric geometries, it shows the refined partition function equals a generating function for cylindric partitions, or equivalently a trace over vertex-operator constructions; the cylinder size is set by the adjoint mass and Omega-background parameters. It also derives nontrivial identities from slicing independence, connects to the periodic Schur process (K=1) and to Hilbert-scheme vertex operators in the 4D limit, and demonstrates that the combinatorics of these gauge-theory partitions encode fiber-base dualities of the associated toric geometries. These results unify topological strings, gauge theory, and algebraic combinatorics, and provide new identities with potential applications in refined enumerative geometry and representation theory.

Abstract

We study the partition function of the compactified 5D U(1) gauge theory (in the Omega-background) with a single adjoint hypermultiplet, calculated using the refined topological vertex. We show that this partition function is an example a periodic Schur process and is a refinement of the generating function of cylindric plane partitions. The size of the cylinder is given by the mass of adjoint hypermultiplet and the parameters of the Omega-background. We also show that this partition function can be written as a trace of operators which are generalizations of vertex operators studied by Carlsson and Okounkov. In the last part of the paper we describe a way to obtain (q,t) identities using the refined topological vertex.

Paper Structure

This paper contains 21 sections, 141 equations, 17 figures.

Table of Contents

  1. Introduction
  2. 3D Partitions, Refined Vertex and Crystals
  3. Transfer matrix approach
  4. Refined topological vertex
  5. Crystal models and $(t,q)$ parameters
  6. Double-$\mathbb{P}^{1}$ and closed refined topological vertex
  7. Adjoint Theory and Periodic Schur Process
  8. Geometric Engineering of $U(1)$ Theory with Adjoint Hypermultiplet
  9. Refined partition function
  10. Field theory limit
  11. Periodic Schur Process
  12. Hilbert schemes and Vertex Operators
  13. Refined Topological Vertex and $(q,t)$ Identities
  14. Identities
  15. Example 1: $X_{0}:={\cal O}(-1)\oplus {\cal O}(-1)\mapsto \mathbb{P}^{1}$
  16. ...and 6 more sections

Figures (17)

  • Figure 1: The hook length of a box in $\nu^{c}$.
  • Figure 2: The diagonal slicing of the plane partition.
  • Figure 3: Three inner corners $\{v_{1},v_{2},v_{3}\}=\{-4,-2,3\}$, two outer corners $\{u_{1},u_{2}\}=\{-3,0\}$.
  • Figure 4: The parameters $q$ and $t$ are assigned based on the partition $\nu=(5,4,4,3,1,1)$.
  • Figure 5: 3D partition and choice of slicing.
  • ...and 12 more figures