Blowup Equations for Refined Topological Strings

Abstract

Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional $\mathcal{N}=1$ supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on $SU(N)$ geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the $\mathbf{r}$ fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other than the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity $\mathbf{r}$ fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space.

85 pages. v2: Journal version

References in corpus (9)

• The Refined Topological Vertex
• Remodeling the B-model
• Holomorphic Anomaly in Gauge Theories and Matrix Models
• Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs
• BPS states, knots and quivers
• E + E $\to$ H
• Cluster integrable systems, q-Painleve equations and their quantization
• Topological Strings and (Almost) Modular Forms
• On the Elliptic Genus of Three E-strings and Heterotic Strings

Cited by in corpus (24)

• Elliptic Blowup Equations for 6d SCFTs. II: Exceptional Cases
• Blowup Equations for 6d SCFTs. I
• A slow review of the AGT correspondence
• Bootstrapping BPS spectra of 5d/6d field theories
• Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains
• Instantons from Blow-up
• 5d/6d Wilson loops from blowups
• More on topological vertex formalism for 5-brane webs with O5-plane
• Instanton counting and O-vertex
• Elliptic Blowup Equations for 6d SCFTs. IV: Matters
• BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations
• Quantum spectral problems and isomonodromic deformations
• Twisted 6d $(2,0)$ SCFTs on a Circle
• Refined topological vertex with ON-planes
• Topological strings and Wilson loops
• Thermodynamic limit of Nekrasov partition function for 5-brane web with O5-plane
• Seiberg-Witten curves with O7${}^{\pm }$-planes
• Blowup Equations for Little Strings
• Towards Refining the Topological Strings on Compact Calabi-Yau 3-folds
• Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries
• $DE$-type little strings from glued brane webs
• Quantum curves as quantum distributions
• Probing Quantum Curves and Transitions in 5d SQFTs via Defects and Blowup Equations
• More on 5d Wilson Loops in Higher-Rank Theories and Blowup Equations