The spindle index from localization
Matteo Inglese, Dario Martelli, Antonio Pittelli
Abstract
Abstract We present a new supersymmetric index for three-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">N</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> gauge theories defined on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="double-struck">Σ</mml:mi> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="double-struck">Σ</mml:mi> </mml:mrow> </mml:math> is a spindle, with twist or anti-twist for the R -symmetry background gauge field. We start examining general supersymmetric backgrounds of Euclidean new minimal supergravity admitting two Killing spinors of opposite R -charges. We then focus on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="double-struck">Σ</mml:mi> </mml:mrow> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> and demostrate how to realise twist and anti-twist. We compute the supersymmetric partition functions on such backgrounds via localization and show that these are captured by a general formula, depending on the type of twist, which unifies and generalises the superconformal and topologically twisted indices.