Les Houches lectures on non-perturbative Seiberg-Witten geometry

Abstract

In these lectures we detail the interplay between the low-energy dynamics of quantum field theories with four supercharges and the exact WKB analysis. This exposition may be the first comprehensive account of this connection and includes new arguments and results. The lectures start with the introduction of massive two-dimensional $\mathcal{N}=(2,2)$ theories and their spectra of BPS solitons. We place these theories in a two-dimensional cigar background with supersymmetric boundary conditions labelled by a phase $\zeta =e^{i\vartheta }$, while turning on the two-dimensional $\Omega$-background with parameter~$\epsilon$. We show that the resulting partition function ${\mathcal{Z}}_{2d}^{\vartheta }(\epsilon )$ can be characterized as the Borel-summed solution, in the direction $\vartheta$, to an associated Schrödinger equation. The partition function ${\mathcal{Z}}_{2d}^{\vartheta }(\epsilon )$ is locally constant in the phase $\vartheta$ and jumps across phases ${\vartheta }_{\text{BPS}}$ associated with the BPS solitons. Since these jumps are non-perturbative in the parameter~$\epsilon$, we refer to $Z_{2d}^{\vartheta }(\epsilon )$ as the non-perturbative partition function for the original two-dimensional $\mathcal{N}=(2,2)$ theory. We completely determine this partition function ${\mathcal{Z}}_{2d}^{\vartheta }(\epsilon )$ in two classes of examples, Landau-Ginzburg models and gauged linear sigma models, and show that ${\mathcal{Z}}_{2d}^{\vartheta }(\epsilon )$ encodes the well-known vortex partition function at a special phase ${\vartheta }_{\text{FN}}$ associated with the presence of self-solitons. This analysis generalizes to four-dimensional $\mathcal{N}=2$ theories in the $\frac{1}{2}\Omega$-background.

114pp, 37 figures, any feedback welcome! v2: added overview of original results

References in corpus (65)

• Generalized Global Symmetries
• Solutions Of Four-Dimensional Field Theories Via M Theory
• Mirror Symmetry is T-Duality
• On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)
• D-Branes And Mirror Symmetry
• Self-Dual Strings and N=2 Supersymmetric Field Theory
• Partition functions of N=(2,2) gauge theories on S^2 and vortices
• Localization techniques in quantum field theories
• Exact Results in D=2 Supersymmetric Gauge Theories
• Vortex Counting and Lagrangian 3-manifolds
• Quantization of Integrable Systems and Four Dimensional Gauge Theories
• The Omega Deformation, Branes, Integrability, and Liouville Theory
• Spectral networks
• D-Branes in Landau-Ginzburg Models and Algebraic Geometry
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters
• TASI Lectures on Solitons
• Wall-Crossing in Coupled 2d-4d Systems
• Equivariant Gromov - Witten Invariants
• Exact Kahler Potential from Gauge Theory and Mirror Symmetry
• Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary
• Conformal field theory of Painlevé VI
• BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters
• Exact Results in Supersymmetric Field Theories on Manifolds with Boundaries
• Riemann-Hilbert problems from Donaldson-Thomas theory
• A mini-course on topological strings
• tt* Geometry in 3 and 4 Dimensions
• On F-term contribution to effective action
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions
• Exact WKB and abelianization for the $T_3$ equation
• Exponential Networks and Representations of Quivers
• Quantization condition from exact WKB for difference equations
• Exact WKB methods in $SU(2)$ $N_f=1$
• Exploring 5d BPS Spectra with Exponential Networks
• Mathematical structures of non-perturbative topological string theory: from GW to DT invariants
• Seiberg-Witten Prepotential From Instanton Counting
• Supersymmetric localization in two dimensions
• Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence
• Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I
• A geometric recipe for twisted superpotentials
• 2d Partition Function in Omega-background and Vortex/Instanton Correspondence
• Exact results for boundaries and domain walls in 2d supersymmetric theories
• Intersection Theory, Integrable Hierarchies and Topological Field Theory
• ADE Spectral Networks and Decoupling Limits of Surface Defects
• Exact WKB analysis and cluster algebras
• Seiberg-Witten Geometry of Four-Dimensional $\mathcal{N}=2$ Quiver Gauge Theories
• Bispectral duality and separation of variables from surface defect transition
• Givental J-functions, Quantum integrable systems, AGT relation with surface operator
• Mirror Symmetry
• Jena lectures on generalized global symmetries: principles and applications
• Holomorphic Floer Theory and the Fueter Equation
• Les Houches lectures on non-perturbative topological strings
• Advanced topics in gauge theory: mathematics and physics of Higgs bundles
• Categorical Wall-Crossing in Landau-Ginzburg Models
• Lectures on invertible field theories
• Family Floer theory, non-abelianization, and Spectral Networks
• Holomorphic Floer theory and Donaldson-Thomas invariants
• Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions
• Higher length-twist coordinates, generalized Heun's opers, and twisted superpotentials
• 3d spectral networks and classical Chern-Simons theory
• Notes on Factorization Algebras and TQFTs
• Opers and nonabelian Hodge: numerical studies
• Monodromies of Second Order $q$-difference Equations from the WKB Approximation
• On the Algebra of the Infrared with Twisted Masses
• Geometry and Resurgence of WKB Solutions of Schrödinger Equations