Modularity, 4d mirror symmetry, and vertex operator algebra modules of 4D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi mathvariant="script">N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow>

The infinite series 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 with central charge relation <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:msub> <d:mi>a</d:mi> <d:mrow> <d:mn>4</d:mn> <d:mi mathvariant="normal">D</d:mi> </d:mrow> </d:msub> <d:mo>=</d:mo> <d:msub> <d:mi>c</d:mi> <d:mrow> <d:mn>4</d:mn> <d:mi mathvariant="normal">D</d:mi> </d:mrow> </d:msub> </d:math> are closely related to the <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:mi mathvariant="script">N</h:mi> <h:mo>=</h:mo> <h:mn>4</h:mn> </h:math> super Yang-Mills theory. In this paper, we study the modular properties of their associated vertex operator algebras <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi mathvariant="double-struck">V</k:mi> <k:mo stretchy="false">[</k:mo> <k:msub> <k:mi mathvariant="script">T</k:mi> <k:mrow> <k:mi>p</k:mi> <k:mo>,</k:mo> <k:mi>N</k:mi> </k:mrow> </k:msub> <k:mo stretchy="false">]</k:mo> </k:math> , where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi mathvariant="script">T</q:mi> <q:mrow> <q:mi>p</q:mi> <q:mo>,</q:mo> <q:mi>N</q:mi> </q:mrow> </q:msub> </q:math> are those <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>a</t:mi> <t:mo>=</t:mo> <t:mi>c</t:mi> </t:math> theories with an <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>S</v:mi> <v:mi>U</v:mi> <v:mo stretchy="false">(</v:mo> <v:mi>N</v:mi> <v:mo stretchy="false">)</v:mo> </v:math> gauge group. We exploit the closed-form formula for the Schur index of the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi mathvariant="script">N</z:mi> <z:mo>=</z:mo> <z:mn>4</z:mn> </z:math> <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>S</cb:mi> <cb:mi>U</cb:mi> <cb:mo stretchy="false">(</cb:mo> <cb:mi>N</cb:mi> <cb:mo stretchy="false">)</cb:mo> </cb:math> theories <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:msub> <gb:mi mathvariant="script">T</gb:mi> <gb:mrow> <gb:mi>S</gb:mi> <gb:mi>U</gb:mi> <gb:mo stretchy="false">(</gb:mo> <gb:mi>N</gb:mi> <gb:mo stretchy="false">)</gb:mo> </gb:mrow> </gb:msub> </gb:math> to derive the space of characters of the vertex operator algebra <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mi mathvariant="double-struck">V</lb:mi> <lb:mo stretchy="false">[</lb:mo> <lb:msub> <lb:mi mathvariant="script">T</lb:mi> <lb:mrow> <lb:mi>p</lb:mi> <lb:mo>,</lb:mo> <lb:mi>N</lb:mi> </lb:mrow> </lb:msub> <lb:mo stretchy="false">]</lb:mo> </lb:math> and the <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:mrow> <rb:mi>S</rb:mi> </rb:mrow> </rb:math> , <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:mrow> <tb:mi>T</tb:mi> </tb:mrow> </tb:math> matrices, and find the (nonmonic) modular linear differential equations that constrain the module characters when possible. We investigate the geometric interpretation of some of these modular data from the viewpoint of 4D mirror symmetry. Using insights from the flavored modular differential equation and defect index, we investigate a map between module characters of <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"> <vb:mi mathvariant="double-struck">V</vb:mi> <vb:mo stretchy="false">[</vb:mo> <vb:msub> <vb:mi mathvariant="script">T</vb:mi> <vb:mrow> <vb:mi>S</vb:mi> <vb:mi>U</vb:mi> <vb:mo stretchy="false">(</vb:mo> <vb:mi>N</vb:mi> <vb:mo stretchy="false">)</vb:mo> </vb:mrow> </vb:msub> <vb:mo stretchy="false">]</vb:mo> </vb:math> and those of <dc:math xmlns:dc="http://www.w3.org/1998/Math/MathML" display="inline"> <dc:mi mathvariant="double-struck">V</dc:mi> <dc:mo stretchy="false">[</dc:mo> <dc:msub> <dc:mi mathvariant="script">T</dc:mi> <dc:mrow> <dc:mi>p</dc:mi> <dc:mo>,</dc:mo> <dc:mi>N</dc:mi> </dc:mrow> </dc:msub> <dc:mo stretchy="false">]</dc:mo> </dc:math> .