M2-branes and $\mathfrak{q}$-Painlevé equations

Abstract

In this paper we investigate a novel connection between the effective theory of M2-branes on $({\mathbb{C}}^2/{\mathbb{Z}}_2\times {\mathbb{C}}^2/{\mathbb{Z}}_2)/{\mathbb{Z}}_k$ and the $\mathfrak{q}$-deformed Painlevé equations, by proposing that the grand canonical partition function of the corresponding four-nodes circular quiver $\mathcal{N}=4$ Chern-Simons matter theory solves the $\mathfrak{q}$-Painlevé VI equation. We analyse how this describes the moduli space of the topological string on local ${\text{dP}}_5$ and, via geometric engineering, five dimensional $N_f=4$ $\text{SU}(2)$ $\mathcal{N}=1$ gauge theory on a circle. The results we find extend the known relation between ABJM theory, $\mathfrak{q}$-Painlevé ${\text{III}}_3$, and topological strings on local ${\mathbb{P}}^1\times {\mathbb{P}}^1$. From the mathematical viewpoint the quiver Chern-Simons theory provides a conjectural Fredholm determinant realisation of the $\mathfrak{q}$-Painlevé VI $\tau$-function. We provide evidence for this proposal by analytic and numerical checks and discuss in detail the successive decoupling limits down to $N_f=0$, corresponding to $\mathfrak{q}$-PainlevéIII${}_3$.

arXiv comment63 pages, 7 figures

References in corpus (51)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter
• Geometric Engineering of Quantum Field Theories
• Five Dimensional SUSY Field Theories, Non-trivial Fixed Points and String Dynamics
• Fractional M2-branes
• Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams
• N=5,6 Superconformal Chern-Simons Theories and M2-branes on Orbifolds
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Non-perturbative effects and the refined topological string
• Matrix Models, Geometric Engineering and Elliptic Genera
• On the moduli space of elliptic Maxwell-Chern-Simons theories
• Instanton Effects in ABJM Theory from Fermi Gas Approach
• A $q$-anaolg of the sixth Painlevé equation
• A Massive Study of M2-brane Proposals
• Quantum Algebraic Approach to Refined Topological Vertex
• Isomonodromic tau-functions from Liouville conformal blocks
• M5-branes, toric diagrams and gauge theory duality
• Geometric Aspects of Painlevé Equations
• Matrix models from operators and topological strings, 2
• On Painlevé/gauge theory correspondence
• Exact results on ABJ theory and the refined topological string
• Matrix models from operators and topological strings
• Direct derivation of "mirror" ABJ partition function
• Exact ABJM Partition Function from TBA
• ABJ Fractional Brane from ABJM Wilson Loop
• The exact Schur index of $\mathcal{N}=4$ SYM
• Instanton effects in ABJM theory with general R-charge assignments
• The Partition Function of ABJ Theory
• Partition Functions of Superconformal Chern-Simons Theories from Fermi Gas Approach
• Bilinear equations on Painleve tau functions from CFT
• Cluster integrable systems, q-Painleve equations and their quantization
• Exact Instanton Expansion of Superconformal Chern-Simons Theories from Topological Strings
• Q-deformed Painleve tau function and q-deformed conformal blocks
• Operators and higher genus mirror curves
• The $\mathcal{N}=2$ Schur index from free fermions
• Riemann-Hilbert correspondence and blown up surface defects
• Large N behavior of mass deformed ABJM theory
• Instanton Effects in Rank Deformed Superconformal Chern-Simons Theories from Topological Strings
• Cluster Toda chains and Nekrasov functions
• Superconformal Chern-Simons Theories from del Pezzo Geometries
• Painleve equations from Nakajima-Yoshioka blowup relations
• On a q-analogue of the multiple gamma functions
• Symmetry Breaking in Quantum Curves and Super Chern-Simons Matrix Models
• Mass Deformed ABJM Theory on Three Sphere in Large N limit
• BPS quivers of five-dimensional SCFTs, Topological Strings and q-Painlevé equations
• Hanany-Witten Transition in Quantum Curves
• Instantons to the people: the power of one-form symmetries
• Supersymmetry Breaking in a Large N Gauge Theory with Gravity Dual
• Brane Transitions from Exceptional Groups