Relations on $\overline{\mathcal{M}}_{g,n}$ and the negative $r$-spin Witten conjecture

Abstract We construct and study various properties of a negative spin version of the Witten $r $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> -spin class. By taking the top Chern class of a certain vector bundle on the moduli space of spin curves that parametrises $r $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> -th roots of the anticanonical bundle, we construct a non-semisimple cohomological field theory (CohFT) that we call the Theta class $\Theta ^{r} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Θ</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:math> . This CohFT does not have a flat unit and its associated Dubrovin–Frobenius manifold is nowhere semisimple. Despite this, we construct a semisimple deformation of the Theta class, and using the Teleman reconstruction theorem, we obtain tautological relations on $\overline{\mathcal{M}}_{g,n} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>‾</mml:mo> </mml:mover> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:math> . Furthermore, we prove that the descendant potential of the Theta class is the unique solution to a set of $\mathcal{W} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>W</mml:mi> </mml:math> -algebra constraints, which implies a recursive formula for all the descendant integrals. Using this result for $r = 2 $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> , we prove Norbury’s conjecture which states that the descendant potential of $\Theta ^{2} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Θ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> coincides with the Brézin–Gross–Witten tau function of the KdV hierarchy. Furthermore, we conjecture that the descendant potential of $\Theta ^{r} $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>Θ</mml:mi> <mml:mi>r</mml:mi> </mml:msup> </mml:math> is the $r $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> -BGW tau function of the $r $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> </mml:math> -KdV hierarchy and prove the conjecture for $r = 3 $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>r</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:math> .