Exact Schur index in closed form for non-Lagrangian theories

The Schur index is a powerful tool to probe the spectrum and dualities of four-dimensional <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, deeply related to two-dimensional vertex operator algebras (VOAs). In this paper, we compute the Schur index in closed form for two series of non-Lagrangian theories. We explore and classify the Argyres-Douglas (AD) theories <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msubsup> <d:mi>D</d:mi> <d:mi>p</d:mi> <d:mi>b</d:mi> </d:msubsup> <d:mo stretchy="false">(</d:mo> <d:msub> <d:mi mathvariant="fraktur">sl</d:mi> <d:mi>N</d:mi> </d:msub> <d:mo>,</d:mo> <d:mo stretchy="false">[</d:mo> <d:mi>Y</d:mi> <d:mo stretchy="false">]</d:mo> <d:mo stretchy="false">)</d:mo> </d:math> realized as the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>S</k:mi> <k:mi>U</k:mi> <k:mo stretchy="false">(</k:mo> <k:mn>2</k:mn> <k:mo stretchy="false">)</k:mo> </k:math> gauging of two AD matter theories, where we identify several infinite families with interesting central charge relations analogous to the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>a</o:mi> <o:mrow> <o:mn>4</o:mn> <o:mi mathvariant="normal">d</o:mi> </o:mrow> </o:msub> <o:mo>=</o:mo> <o:msub> <o:mi>c</o:mi> <o:mrow> <o:mn>4</o:mn> <o:mi mathvariant="normal">d</o:mi> </o:mrow> </o:msub> </o:math> of <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi mathvariant="script">N</s:mi> <s:mo>=</s:mo> <s:mn>4</s:mn> </s:math> theories. We focus on <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:msub> <v:mi>D</v:mi> <v:mrow> <v:mi>N</v:mi> <v:mo>−</v:mo> <v:mn>4</v:mn> </v:mrow> </v:msub> <v:mo stretchy="false">(</v:mo> <v:mi mathvariant="fraktur">sl</v:mi> <v:mo stretchy="false">(</v:mo> <v:mi>N</v:mi> <v:mo stretchy="false">)</v:mo> <v:mo>,</v:mo> <v:mo stretchy="false">[</v:mo> <v:mi>N</v:mi> <v:mo>−</v:mo> <v:mn>4</v:mn> <v:mo>,</v:mo> <v:mn>4</v:mn> <v:mo stretchy="false">]</v:mo> <v:mo stretchy="false">)</v:mo> </v:math> and <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:msub> <eb:mi>D</eb:mi> <eb:mrow> <eb:mi>N</eb:mi> <eb:mo>−</eb:mo> <eb:mn>2</eb:mn> </eb:mrow> </eb:msub> <eb:mo stretchy="false">(</eb:mo> <eb:mi mathvariant="fraktur">sl</eb:mi> <eb:mo stretchy="false">(</eb:mo> <eb:mi>N</eb:mi> <eb:mo stretchy="false">)</eb:mo> <eb:mo>,</eb:mo> <eb:mspace linebreak="goodbreak"/> <eb:mo stretchy="false">[</eb:mo> <eb:mi>N</eb:mi> <eb:mo>−</eb:mo> <eb:mn>3</eb:mn> <eb:mo>,</eb:mo> <eb:mn>3</eb:mn> <eb:mo stretchy="false">]</eb:mo> <eb:mo stretchy="false">)</eb:mo> </eb:math> and compute their flavored and unflavored Schur and Wilson line indices in compact form. We also explore their large- <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"> <ob:mi>N</ob:mi> </ob:math> behavior and show that they arise as special limits of the <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"> <qb:mi>S</qb:mi> <qb:mi>U</qb:mi> <qb:mo stretchy="false">(</qb:mo> <qb:mn>2</qb:mn> <qb:mo stretchy="false">)</qb:mo> </qb:math> SQCD flavored index, also analogous to the relation among the <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"> <ub:msub> <ub:mi>a</ub:mi> <ub:mrow> <ub:mn>4</ub:mn> <ub:mi mathvariant="normal">d</ub:mi> </ub:mrow> </ub:msub> <ub:mo>=</ub:mo> <ub:msub> <ub:mi>c</ub:mi> <ub:mrow> <ub:mn>4</ub:mn> <ub:mi mathvariant="normal">d</ub:mi> </ub:mrow> </ub:msub> </ub:math> theories. We also generalize the elliptic function integration formula in the presence of higher-order poles to compute in closed form the partially flavored indices of the Minahan-Nemeschansky <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"> <yb:msub> <yb:mi>E</yb:mi> <yb:mn>6</yb:mn> </yb:msub> </yb:math> and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"> <ac:msub> <ac:mi>E</ac:mi> <ac:mn>7</ac:mn> </ac:msub> </ac:math> theories. Our results point to a universal structure underlying the residues of elliptic integrands, Wilson loop indices, and nonvacuum modules of the corresponding VOAs.