Generalized Schur partition functions and RG flows

We revisit a double-scaled limit of the superconformal index of <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 (SCFTs) which generalizes the Schur index. The resulting partition function, <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mover accent="true"> <d:mi mathvariant="script">Z</d:mi> <d:mo stretchy="false">^</d:mo> </d:mover> <d:mo stretchy="false">(</d:mo> <d:mi>q</d:mi> <d:mo>,</d:mo> <d:mi>α</d:mi> <d:mo stretchy="false">)</d:mo> </d:math> , has a standard <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>q</k:mi> </k:math> -expansion with coefficients depending on a continuous parameter <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>α</m:mi> </m:math> . The Schur index is a special case with <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>α</o:mi> <o:mo>=</o:mo> <o:mn>1</o:mn> </o:math> . Through explicit computations we argue that this partition function is an invariant of certain mass deformations and vacuum expectation value (vev) deformations of the SCFT. In particular, two SCFTs residing in different corners of the same Coulomb branch, satisfying certain restrictive conditions, have the same partition function with a nontrivial map of the parameters, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mover accent="true"> <q:mi mathvariant="script">Z</q:mi> <q:mo stretchy="false">^</q:mo> </q:mover> <q:mn>1</q:mn> </q:msub> <q:mo stretchy="false">(</q:mo> <q:mi>q</q:mi> <q:mo>,</q:mo> <q:msub> <q:mi>α</q:mi> <q:mn>1</q:mn> </q:msub> <q:mo stretchy="false">)</q:mo> <q:mo>=</q:mo> <q:msub> <q:mover accent="true"> <q:mi mathvariant="script">Z</q:mi> <q:mo stretchy="false">^</q:mo> </q:mover> <q:mn>2</q:mn> </q:msub> <q:mo stretchy="false">(</q:mo> <q:mi>q</q:mi> <q:mo>,</q:mo> <q:msub> <q:mi>α</q:mi> <q:mn>2</q:mn> </q:msub> <q:mo stretchy="false">(</q:mo> <q:msub> <q:mi>α</q:mi> <q:mn>1</q:mn> </q:msub> <q:mo stretchy="false">)</q:mo> <q:mo stretchy="false">)</q:mo> </q:math> . For example, we show that the Schur index of the SCFTs in the Deligne-Cvitanović series is given by special values of <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>α</eb:mi> </eb:math> of the partition function of the <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>SU</gb:mi> <gb:mo stretchy="false">(</gb:mo> <gb:mn>2</gb:mn> <gb:mo stretchy="false">)</gb:mo> <gb:msub> <gb:mi>N</gb:mi> <gb:mi>f</gb:mi> </gb:msub> <gb:mo>=</gb:mo> <gb:mn>4</gb:mn> </gb:math> <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi mathvariant="script">N</kb:mi> <kb:mo>=</kb:mo> <kb:mn>2</kb:mn> </kb:math> SQCD.