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Refined BPS state counting from Nekrasov's formula and Macdonald functions

Hidetoshi Awata, Hiroaki Kanno

TL;DR

Awata and Kanno develop a refined BPS counting framework by introducing a Macdonald-function–based refined topological vertex that serves as a building block for Nekrasov’s partition function with two equivariant parameters. They formulate the 5D (K-theoretic) lift of Nekrasov’s function with a Chern–Simons coupling, derive multiple equivalent representations, and prove a Spin$(4)$ character symmetry, connecting refined BPS multiplicities to GV-type invariants. The paper provides diagrammatic gluing rules on web diagrams, explicit four-point and one-loop constructions, and concrete examples for $U(1)$ and $SU(N_c)$ theories, including cases with fundamental matter; it also links these amplitudes to geometric engineering, Macdonald theory, and potential homological invariants of links. The results yield a robust computational framework for refined topological strings and gauge theories, with implications for refined GV invariants and connections to Hilbert schemes and Hopf-link homologies. Overall, the work unifies refined BPS counting, Macdonald functions, and toric geometry into a coherent, calculable formalism with broad applications in nonperturbative string/gauge dynamics.

Abstract

It has been argued that the Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi-Yau spaces. We show that a refined version of the topological vertex we proposed before (hep-th/0502061) is a building block of the Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal-Kozcaz-Vafa (hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on C^2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang-Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

Refined BPS state counting from Nekrasov's formula and Macdonald functions

TL;DR

Awata and Kanno develop a refined BPS counting framework by introducing a Macdonald-function–based refined topological vertex that serves as a building block for Nekrasov’s partition function with two equivariant parameters. They formulate the 5D (K-theoretic) lift of Nekrasov’s function with a Chern–Simons coupling, derive multiple equivalent representations, and prove a Spin character symmetry, connecting refined BPS multiplicities to GV-type invariants. The paper provides diagrammatic gluing rules on web diagrams, explicit four-point and one-loop constructions, and concrete examples for and theories, including cases with fundamental matter; it also links these amplitudes to geometric engineering, Macdonald theory, and potential homological invariants of links. The results yield a robust computational framework for refined topological strings and gauge theories, with implications for refined GV invariants and connections to Hilbert schemes and Hopf-link homologies. Overall, the work unifies refined BPS counting, Macdonald functions, and toric geometry into a coherent, calculable formalism with broad applications in nonperturbative string/gauge dynamics.

Abstract

It has been argued that the Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi-Yau spaces. We show that a refined version of the topological vertex we proposed before (hep-th/0502061) is a building block of the Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal-Kozcaz-Vafa (hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on C^2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang-Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

Paper Structure

This paper contains 31 sections, 210 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Structure of Nekrasov's Partition Function
  3. Partition function with Chern-Simons coupling
  4. Another form of the partition function
  5. Symmetry as a character of $Spin(4)$
  6. Geometric Engineering and Toric Geometry
  7. Refined Topological Vertex
  8. Refined topological vertex
  9. Gluing rules
  10. Four-Point Functions
  11. Building blocks
  12. OPE formula
  13. Computations of four-point functions
  14. Flop operation
  15. Finite $N$ Macdonald polynomial and homological invariants
  16. ...and 16 more sections

Figures (13)

  • Figure 1: Ladder diagram for $SU(N)$ gauge theory. There are $N+1$ possible toric diagrams ($m= 0, \cdots N$).
  • Figure 2: Subdiagram of the $k$-th divisor of $SU(N)_m$ geometry ($1 \leq k \leq N-1$)
  • Figure 3: Refined topological vertex: the represantation for the preferred direction, i.e. the middle index $\lambda$, is indicated by the white arrow.
  • Figure 4: Gluing rules
  • Figure 5: Four-point function: the framing indices of the internal lines are $1$, $0$, $0$ and $-1$ from the left.
  • ...and 8 more figures