ENUMERATIVE GEOMETRY VIA THE MODULI SPACE OF SUPER RIEMANN SURFACES PAUL NORBURY Abstract. In this paper we relate volumes of moduli spaces of super Riemann surfaces to integrals over the moduli space of stable Riemann surfaces Mg;n. This allows us to use a recursion between the super volumes recently proven by Stanford and Witten to deduce recursion relations of a natural collection of ∗ cohomology classes Θg;n 2 H (Mg;n). We give a new proof that a generating function for the intersection numbers of Θg;n with tautological classes on Mg;n is a KdV tau function. This is an analogue of the Kontsevich-Witten theorem ∗ where Θg;n is replaced by the unit class 1 2 H (Mg;n). The proof is analogous to Mirzakhani's proof of the Kontsevich-Witten theorem replacing volumes of moduli spaces of hyperbolic surfaces with volumes of moduli spaces of super hyperbolic surfaces. Contents 1. Introduction 2 ∗ 2. The classes Θg;n 2 H (Mg;n). 7 2.1. Cohomological field theories 9 3. Hyperbolic geometry and spin structures 10 3.1. Fuchsian representations 10 3.2. Flat bundles 16 3.3. Higgs bundles 23 3.4. Euler class of Ebg;n 31 4. Mirzakhani's volume recursion 32 4.1. Hyperbolic geometry 32 4.2. Volume recursion 35 5. Moduli space of super hyperbolic surfaces 38 5.1. Supermanifolds 38 5.2. Super hyperbolic surfaces 40 5.3. Recursion for super volumes 43 5.4. Calculations 45 6. KdV tau functions 46 6.1. Intersection numbers 48 6.2. Hyperbolic cone angles 52 7. Topological recursion 54 References 63 Date: May 18, 2020. 2010 Mathematics Subject Classification. 32G15; 14H81; 58A50. 1 2 PAUL NORBURY 1. Introduction Mumford initiated a systematic approach to calculating intersection numbers of tautological classes on the moduli space of stable Riemann surfaces Mg;n in [45]. Witten conjectured a recursive structure on a collection of these intersec- tion numbers [62] and Kontsevich proved the conjecture in [33], now known as the Kontsevich-Witten theorem. Other proofs followed [32, 43, 49]. The proof by Mirzakhani [43] deduced the Kontsevich-Witten theorem by proving recursion relations between Weil-Petersson volumes of moduli spaces of smooth hyperbolic surfaces. Wolpert had proven earlier in [63, 64] that the Weil-Petersson symplectic form !WP extends to the moduli space of stable curves, and related it to a tauto- logical class, κ1, which was studied by Mumford in [45]. This enabled Mirzakhani to relate integrals over Mg;n and Mg;n. Stanford and Witten [57] proved recursive relations between volumes of moduli spaces of smooth super hyperbolic surfaces using methods analogous to those of Mirzakhani. In this paper we prove an analogue of Wolpert's results which relates super volumes to integrals over the moduli space of stable curves. This uses classes ∗ Θg;n 2 H (Mg;n) previously studied by the author [48]. Super Riemann surfaces have been studied over the last thirty years [8, 22, 34, 53, 57, 61]. Underlying any super Riemann surface is a Riemann surface equipped with a spin structure. In this paper we consider Riemann surfaces of finite type Σ = Σ−D where Σ is a compact curve with boundary divisor D = fp1; :::; png ⊂ Σ, and equivalently study pairs (Σ;D). The moduli space of super Riemann surfaces can be defined algebraically, complex analytically and using hyperbolic geometry, building on the same approaches to the moduli space of Riemann surfaces. The last of these approaches regards a super Riemann surface as a super hyperbolic surface, which is a quotient of super hyperbolic space Hb defined in (45). A Riemann surface equipped with a spin structure, or spin surface, has a well- 1 1 1 2 2 2 ∼ defined square root bundle, TΣ , of the tangent bundle, so TΣ ⊗ TΣ = TΣ, which is 1 2 ∼ also a real subbundle of the rank two bundle of spinors TΣ ⊗R C = SΣ. It is a flat SL(2; R)-bundle, and the flat structure defines the sheaf of locally constant sections 1 1 2 1 2 of TΣ with sheaf cohomology HdR(Σ;TΣ ). The sheaf cohomology can be calculated via the cohomology of the twisted de Rham complex defined by the flat connection 1 2 associated to the flat bundle TΣ , justifying the subscript. The deformation theory of a super Riemann surface with underlying spin surface Σ defines a bundle spin Eg;n !Mg;n 1 1 2 with fibres Eg;nj(Σ,θ) = HdR(Σ;TΣ ) over the moduli space of smooth genus g spin Riemann surfaces Σ = Σ − D with jDj = n. The total space of Eg;n gives the underlying smooth moduli space of super Riemann surfaces.1 One new contribution of this paper to the study of the moduli space of super Riemann surfaces is the extension of the bundle Eg;n to the moduli space of stable spin spin curves Mg;n . A stable spin curve is a stable orbifold curve with n labeled 1Donagi and Witten proved in [14] that the moduli space of super Riemann surfaces can- not be represented as the total space of a holomorphic bundle over the moduli space of super Riemann surfaces. Here we consider the moduli space of super Riemann surfaces as a smooth supermanifold/orbifold which can always be represented as the total space of a smooth bundle. ENUMERATIVE GEOMETRY VIA SUPER RIEMANN SURFACES 3 2 log points (C;D), equipped with a spin structure θ satisfying θ = !C = !C(D). The points of D = fp1; :::; png, and nodal points of C are orbifold points with isotropy group Z2|see Section 2. There is a map from C to its underlying coarse curve which forgets the orbifold structure. 1 _ 2 When C is smooth, C − D = Σ is a Riemann surface and θ jΣ = TΣ . Using a theorem of Simpson [55, 56] we prove in Section 3.3 a canonical isomorphism for C smooth: 1 1 2 ∼ 1 _ _ (1) HdR(Σ;TΣ ) = H (C; θ ) : The cohomology groups H1(C; θ_) are well-defined on any stable spin curve (C; θ) 1 _ spin spin and dim H (C; θ ) is locally constant on Mg;n , hence the bundle Eg;n !Mg;n spin 1 _ _ is the restriction of a bundle Ebg;n ! Mg;n with fibres H (C; θ ) . The total space of Ebg;n gives the compactification of the moduli space of super Riemann surfaces studied by Witten in [61, Section 6]. The extension of the bundle Eg;n to a compactification is a crucial ingredient for enumerative methods such as the relation of intersection numbers to the KdV hierarchy in Corollary 2 below. 1 _ ∼ 2 The isomorphism (1) is non-trivial even in the case D = ? where θ = TΣ . The left hand side of (1) uses the sheaf of locally constant sections while the right hand side uses the sheaf of locally holomorphic sections, and we take the sheaf cohomology in both cases. The difference between the sheaf structures on each side of (1) is 1 _ ∼ 2 demonstrated clearly in the following case. As a bundle θ jΣ = TΣ , however _ the sheaf of locally holomorphic sections of θ jΣ is trivial when n > 0, whereas 1 1 2 1 2 the sheaf of locally constant sections of TΣ is non-trivial, since HdR(Σ;TΣ ) 6= 0. One particularly satisfying aspect of applying Simpson's parabolic Higgs bundles techniques to the pair (Σ;D) is that it naturally gives rise to the orbifold curve (C;D) ! (Σ;D). Parabolic bundles over the coarse curve Σ correspond to the push-forward of bundles over C. In the Neveu-Schwarz case, given by Definition 3.1, 1 the push-forward of θ_ to Σ is T 2 (−D) which embeds in a parabolic bundle, as Σ described in 3.3.5. In particular, we can express (1) in terms of the coarse curve 1 (C;D) ! (Σ;D) via H1(C; θ_) =∼ H1(Σ;T 2 (−D)). Σ spin Under the forgetful map p : Mg;n ! Mg;n, define the push-forward classes g−1+n 4g−4+2n Θg;n := 2 p∗c2g−2+n(Ebg;n) 2 H (Mg;n) for g ≥ 0, n ≥ 0 and 2g − 2 + n > 0. These classes are shown in [48] to pull back naturally under the gluing maps φirr φh;I Mg−1;n+2 −! Mg;n; Mh;jIj+1 × Mg−h;jJj+1 −! Mg;n;I t J = f1; :::; ng π and the forgetful map Mg;n+1 −! Mg;n as follows. ∗ ∗ (2) φirrΘg;n = Θg−1;n+2; φh;I Θg;n = Θh;jIj+1 ⊗ Θg−h;jJj+1; ∗ (3) Θg;n+1 = n+1 · π Θg;n 2 where n+1 2 H (Mg;n+1; Q) is a natural class, defined in (9) in Section 2. Prop- R 1 erties (2), (3) and a single calculation Θ1;1 = are enough to uniquely M1;1 8 4 PAUL NORBURY determine the intersection numbers Z n N Y mi Y `j Θg;n i κj Mg;n i=1 j=1 via a reduction argument|see Section 2 for details. In particular, we restrict to the case of only κ1 classes below. The volume of the moduli space of super hyperbolic surfaces is shown in [57] from general considerations to coincide with the integral Z WP WP (4) Vbg;n = e(Eg;n) exp ! spin Mg;n where e(Eg;n) is a differential form representing the the Euler class of the bundle WP WP Eg;n.