Modularity of the Schur index, modular differential equations, and high-temperature asymptotics

In this paper we analytically explore the modularity of the flavored Schur index of 4d <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> superconformal field theories. We focus on the <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi>A</d:mi><d:mn>1</d:mn></d:msub></d:math> theories of class-<f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi mathvariant="script">S</f:mi></f:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi mathvariant="script">N</i:mi><i:mo>=</i:mo><i:mn>4</i:mn></i:math> theories with <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>S</l:mi><l:mi>U</l:mi><l:mo stretchy="false">(</l:mo><l:mi>N</l:mi><l:mo stretchy="false">)</l:mo></l:math> gauge group. We work out the modular orbit of the flavored index and defect index, compute the dimension of the space spanned by the orbit, and provide complete basis for computing modular transformation matrices. The dimension obtained from the flavored analysis predicts the minimal order of the unflavored modular differential equation satisfied by the unflavored Schur index. With the help of modularity, we also study analytically the high temperature asymptotics of the Schur index. In the high temperature limit <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>τ</p:mi><p:mo stretchy="false">→</p:mo><p:mo>+</p:mo><p:mi>i</p:mi><p:mn>0</p:mn></p:math>, we identified the (defect) Schur index of the genus-zero <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>A</s:mi><s:mn>1</s:mn></s:msub></s:math> theories of class <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi mathvariant="script">S</u:mi></u:math> with the <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"><x:msup><x:mi>S</x:mi><x:mn>3</x:mn></x:msup></x:math> partition function of the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"><z:mi>S</z:mi><z:mi>U</z:mi><z:mo stretchy="false">(</z:mo><z:mn>2</z:mn><z:mo stretchy="false">)</z:mo><z:mo>×</z:mo><z:mi>U</z:mi><z:mo stretchy="false">(</z:mo><z:mn>1</z:mn><z:msup><z:mo stretchy="false">)</z:mo><z:mi>n</z:mi></z:msup></z:math> star-shaped quiver (with Wilson line insertion). In the identification, we observe an interesting relation between the linear-independence of defect indices and the convergence of the Wilson line partition functions.