Les Houches lectures on non-perturbative Seiberg-Witten geometry

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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>𝒩</mml:mi> </mml:mstyle> <mml:mo>=</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>ζ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mi>i</mml:mi> <mml:mi>ϑ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , while turning on the two-dimensional \Omega <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>Ω</mml:mi> </mml:math> -background with complex parameter \epsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ϵ</mml:mi> </mml:math> . We show that the resulting partition function \mathcal{Z}_{\mathrm{2d}}^\vartheta(\epsilon) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mstyle mathvariant="script"> <mml:mi>𝒵</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mstyle> <mml:mi>ϑ</mml:mi> </mml:msubsup> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> can be characterized as the Borel-summed solution, in the direction \vartheta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ϑ</mml:mi> </mml:math> , to an associated Schrödinger equation. The partition function \mathcal{Z}_{\mathrm{2d}}^\vartheta(\epsilon) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mstyle mathvariant="script"> <mml:mi>𝒵</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mstyle> <mml:mi>ϑ</mml:mi> </mml:msubsup> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> is locally constant in the phase \vartheta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ϑ</mml:mi> </mml:math> and jumps across phases \vartheta_\textrm{BPS} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ϑ</mml:mi> <mml:mtext mathvariant="normal">BPS</mml:mtext> </mml:msub> </mml:math> associated with the BPS solitons. Since these jumps are non-perturbative in the parameter \epsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ϵ</mml:mi> </mml:math> , we refer to Z^\vartheta_\mathrm{2d}(\epsilon) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> </mml:mstyle> <mml:mi>ϑ</mml:mi> </mml:msubsup> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> as the non-perturbative partition function for the original two-dimensional \mathcal{N}=(2,2) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>𝒩</mml:mi> </mml:mstyle> <mml:mo>=</mml:mo> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow>