Constructs solutions to higher-rank oper Riemann-Hilbert problems via a single non-linear integral equation, proving that the oper generating function equals the Toda Yang-Yang function and thereby establishing the Nekrasov-Rosly-Shatashvili conjecture.
Introduction to 1-summability and resurgence
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abstract
This text is about the mathematical use of certain divergent power series. The first part is an introduction to 1-summability. The definitions rely on the formal Borel transform and the Laplace transform along an arbitrary direction of the complex plane. Given an arc of directions, if a power series is 1-summable in that arc, then one can attach to it a Borel-Laplace sum, i.e. a holomorphic function defined in a large enough sector and asymptotic to that power series in Gevrey sense. The second part is an introduction to Ecalle's resurgence theory. A power series is said to be resurgent when its Borel transform is convergent and has good analytic continuation properties: there may be singularities but they must be isolated. The analysis of these singularities, through the so-called alien calculus, allows one to compare the various Borel-Laplace sums attached to the same resurgent 1-summable series.In the context of analytic difference-or-differential equations, this sheds light on the Stokes phenomenon. A few elementary or classical examples are given a thorough treatment (the Euler series, the Stirling series, a less known example by Poincar\'e). Special attention is devoted to non-linear operations: 1-summable series as well as resurgent series are shown to form algebras which are stable by composition. As an application, the resurgent approach to the classification of tangent-to-identity germs of holomorphic diffeomorphisms in the simplest case is included. An example of a class of non-linear differential equations giving rise to resurgent solutions is also presented. The exposition is as self-contained as can be, requiring only some familiarity with holomorphic functions of one complex variable.
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Derives weakest quantization conditions in terms of monodromy data for higher-order DEs tied to quantum Toda chain and proves duality predictions for deformed Schrödinger operators.
At large chemical potential the Gross-Neveu model enters a crystalline phase in which a-particle bound states condense, producing a periodically oscillating chiral condensate governed by two new scales Λ_n and Λ_c that replace the usual Λ.
Lecture notes review exact WKB analysis for ODEs and its combination with topological recursion and isomonodromy to compute monodromy and resurgent structures for Painlevé equations.
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Higher-Rank Mathieu Opers, Toda Chain, and Analytic Langlands Correspondence
Constructs solutions to higher-rank oper Riemann-Hilbert problems via a single non-linear integral equation, proving that the oper generating function equals the Toda Yang-Yang function and thereby establishing the Nekrasov-Rosly-Shatashvili conjecture.
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Higher-Rank Connections and Deformed Schr\"odinger Operators
Derives weakest quantization conditions in terms of monodromy data for higher-order DEs tied to quantum Toda chain and proves duality predictions for deformed Schrödinger operators.
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Perturbative, Nonperturbative and Exact Aspects of Crystalline Phases in the Gross-Neveu Model
At large chemical potential the Gross-Neveu model enters a crystalline phase in which a-particle bound states condense, producing a periodically oscillating chiral condensate governed by two new scales Λ_n and Λ_c that replace the usual Λ.
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Les Houches Lectures on Exact WKB Analysis and Painlev\'e Equations
Lecture notes review exact WKB analysis for ODEs and its combination with topological recursion and isomonodromy to compute monodromy and resurgent structures for Painlevé equations.