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$S^3$ partition functions and Equivariant CY$_4 $/ CY$_3$ correspondence from Quantum curves

Kiril Hristov, Naotaka Kubo, Yi Pang

Abstract

We study the perturbative large-$N$ expansion of the round three-sphere partition function in a class of M2-brane theories, including flavored SYM and ABJM theories as well as more general 3d theories admitting dual $(p,q)$ 5-brane web descriptions. Using the Fermi gas formalism and quantum curve techniques, we derive the Airy-function representation of the partition function and find exact agreement with predictions based on equivariant constant maps in topological string theory proposed in [1]. In particular, we provide affirmative tests of this proposal for the toric geometries $\mathbb{C} \times \mathcal{C}$ (the conifold), the cone over the Sasakian space $Q^{1,1,1}$, and $\mathbb{C} \times \mathrm{SPP}$ (the suspended pinch point). Motivated by a recent conjecture in [2], we further propose a novel equivariant correspondence between distinct toric Calabi-Yau manifolds of the form $\mathrm{CY}_4 \leftrightarrow \mathbb{C} \times\mathrm{CY}_3$, arising from relations between the corresponding quantum curves under specific constraints. This correspondence suggests an equivariant extension and points toward a geometric origin of the topological string/spectral theory (TS/ST) correspondence, while offering new insight into the structure of the holography duality.

$S^3$ partition functions and Equivariant CY$_4 $/ CY$_3$ correspondence from Quantum curves

Abstract

We study the perturbative large- expansion of the round three-sphere partition function in a class of M2-brane theories, including flavored SYM and ABJM theories as well as more general 3d theories admitting dual 5-brane web descriptions. Using the Fermi gas formalism and quantum curve techniques, we derive the Airy-function representation of the partition function and find exact agreement with predictions based on equivariant constant maps in topological string theory proposed in [1]. In particular, we provide affirmative tests of this proposal for the toric geometries (the conifold), the cone over the Sasakian space , and (the suspended pinch point). Motivated by a recent conjecture in [2], we further propose a novel equivariant correspondence between distinct toric Calabi-Yau manifolds of the form , arising from relations between the corresponding quantum curves under specific constraints. This correspondence suggests an equivariant extension and points toward a geometric origin of the topological string/spectral theory (TS/ST) correspondence, while offering new insight into the structure of the holography duality.
Paper Structure (32 sections, 236 equations, 10 figures, 1 table)

This paper contains 32 sections, 236 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of results and propositions
  3. From brane webs to quantum curves and equivariant geometry
  4. Brane setup and quiver theory
  5. The quantum curve at large N
  6. Computing the number of states
  7. Toric CY geometry and equivariant intersection numbers
  8. Equivariant volume
  9. Geometric prediction for the S3 partition function
  10. Holographic duality
  11. R-charge identification
  12. S3 partition functions
  13. C4 geometry
  14. C-Conifold geometry
  15. Q111 geometry
  16. ...and 17 more sections

Figures (10)

  • Figure 1: An example of the CY$_{4}$/CY$_{3}$ correspondence. The Minkowski sum of the two Newton polygons in the curve connects two Calabi-Yau manifolds given by their respective toric diagrams (left and right).
  • Figure 2: Schematic diagram of the duality web we explore. Apart from the red arrow, which we discuss here for the first time, we establish the holographic relations within the dashed red circle for several new non-trivial examples of dual pairs.
  • Figure 3: A $\left(p,q\right)$ web which leads to a Lagrangian theory.
  • Figure 4: The $\mathcal{N}=2$ quiver diagrams. Left: the flavored SYM theory, which is the worldvolume theory of \ref{['eq:fSYM-BC']}. Center: the flavored $\widetilde{L^{020}}$ theory, which is the worldvolume theory of \ref{['eq:fABJM-BC']}. Right: the flavored ABJM theory, which is the worldvolume theory of \ref{['eq:fABJM-BC2']}.
  • Figure 5: The toric diagram of the conifold, together with its two possible resolutions that correspond to two different triangulations.
  • ...and 5 more figures