Refined Invariants and Quantum Curves from Supersymmetric Localization

Abstract

We study an Aganagic-Vafa brane supported on a special Lagrangian submanifold $\mathcal{L}$ in a non-compact toric Calabi-Yau threefold $\mathcal{X}$. From the perspective of geometric engineering, the Aganagic-Vafa branes give rise to a special class of half-BPS codimension-two defects in 5d $\mathcal{N}=1$ supersymmetric field theories in the presence of $Ω$-background. We propose that the defect partition functions give generating functions of refined, non-negative, integral open BPS invariants of the pair $(\mathcal{X},\mathcal{L})$, across different Kähler moduli chambers they are expanded in. In the Nekrasov-Shatashvili limit, the partition function provides a partially resummed solution to a $q$-difference equation that quantizes the mirror curve of $\mathcal{X}$ in an unambiguous fashion, in a polarization determined by the discrete labels of the Aganagic-Vafa brane. We demonstrate our method at examples of $\mathbb{C}^3$, resolved conifold, resolved $A_1$-singularity, local $F_0$, and local $F_1$.

Figures (16)

• Figure 1: ADHM quiver with gauge node $K=\mathbb{C}^k$ and framing node $N=\mathbb{C}^n$.
• Figure 2: Transformation of a classical curve (corresponding to the Calabi-Yau $\mathbb C^3$), represented by a toric diagram in the real $xz$-plane, under $S$ and $T$-transformations according to \ref{['STf']}. The colors are only to help identify the same edge across transformations.
• Figure 3: A $(p,q)$-web (black) represents a state for a quantized probe (red) constrained to the corresponding curve inside $(\mathbb C^\times)^2$. The three subfigures correspond to the states $\ket{\psi_0}$, $\ket{\psi_f}$, and $\ket{\psi_f^S}$ related by $\hat{T}$ and $\hat{S}$-transformations according to \ref{['transformedState']}. A probe attached to a vertical (resp. horizontal) edge represents the associated wave function in the $X$ (resp. $Z$)-polarization.
• Figure 4: The abstract curves, probes, and wave functions of Fig. \ref{['fig:wavefunctions']} realized as concrete $(p,q)$-web, D3 branes, and expectation values of codimension-2 defects in type IIB string theory. The $(p,q)$-branes engineer 5d $\mathcal{N}=1$ theories, the D3 branes (red) create codimesion-2 defects therein, and their expectation values realize the wave functions quantizing the Seiber-Witten curves. Expectation values of certain defects before and after S-duality coincide.
• Figure 5: (a) Toric diagram of $\mathbb{C}^3$ at framing $f=0$ and its graph-dual; (b) $(p,q)$-web of fivebranes dual to $\mathbb{C}^3$ and D3-branes dual to Aganagic-Vafa branes. The cases (I) and (II) give rise to a $Q$-observable, while the case (III) engineers an $H$-observable.

Theorems & Definitions (3)

• Remark 2.1
• Remark 2.2
• Remark 3.1