Structures in topological recursion relations

In this paper, we study the basic structures of degree- <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g"> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding="application/x-tex">g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> topological recursion relations on the moduli space of curves <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> : (i) the coefficient of the bouquet class 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> , which gives the answer to a conjecture of T. Kimura and X. Liu [Comm. Math. Phys. 262 (2006), pp. 645–661]; (ii) linear relations among the coefficients of certain rational tails locus of <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> . Three applications of topological recursion relations will be discussed: (i) coefficients of universal equations for Gromov–Witten invariants for any smooth projective variety; (ii) the coefficient of the bouquet class in the double ramification formula of the top Hodge class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Subscript g"> <mml:semantics> <mml:msub> <mml:mi> λ </mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\lambda _g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; (iii) a new recursive formula for computing the intersection numbers on the moduli space of stable curves.