| Summary: | <p>Relations among tautological classes on <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M overbar Subscript g comma n">
<mml:semantics>
<mml:msub>
<mml:mover>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi>
</mml:mrow>
<mml:mo accent="false">¯<!-- ¯ --></mml:mo>
</mml:mover>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi>g</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>n</mml:mi>
</mml:mrow>
</mml:msub>
<mml:annotation encoding="application/x-tex">\overline {\mathcal {M}}_{g,n}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> are obtained via the study of Witten’s <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-spin theory for higher <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. In order to calculate the quantum product, a new formula relating the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-spin correlators in genus 0 to the representation theory of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif s sans-serif l Subscript 2 Baseline left-parenthesis double-struck upper C right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="sans-serif">s</mml:mi>
<mml:mi mathvariant="sans-serif">l</mml:mi>
</mml:mrow>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mo stretchy="false">(</mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">C</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{\mathsf {sl}}_2(\mathbb {C})</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is proven. The Givental-Teleman classification of CohFT (cohomological field theory) is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R">
<mml:semantics>
<mml:mi>R</mml:mi>
<mml:annotation encoding="application/x-tex">R</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten’s <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-spin class is obtained (along with tautological relations in higher degrees). As an application, the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r equals 4">
<mml:semantics>
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>4</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">r=4</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> relations are used to bound the Betti numbers of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Superscript asterisk Baseline left-parenthesis script upper M Subscript g Baseline right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:msup>
<mml:mi>R</mml:mi>
<mml:mo>∗<!-- ∗ --></mml:mo>
</mml:msup>
<mml:mo stretchy="false">(</mml:mo>
<mml:msub>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi>
</mml:mrow>
<mml:mi>g</mml:mi>
</mml:msub>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">R^*(\mathcal {M}_g)</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>. At the second semisimple point, the form of the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R">
<mml:semantics>
<mml:mi>R</mml:mi>
<mml:annotation encoding="application/x-tex">R</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-matrix implies a polynomiality property in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of Witten’s <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-spin class.</p>
<p>In Appendix A (with F. Janda), a conjecture relating the <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r equals 0">
<mml:semantics>
<mml:mrow>
<mml:mi>r</mml:mi>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">r=0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> limit of Witten’s <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r">
<mml:semantics>
<mml:mi>r</mml:mi>
<mml:annotation encoding="application/x-tex">r</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-spin class to the class of the moduli space of holomorphic differentials is presented.</p>
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