Super Riemann Surfaces Research Articles

Tree-level superstring amplitudes: the Neveu-Schwarz sector

We present a complete computation of superstring scattering amplitudes at tree level, for the case of Neveu-Schwarz insertions. Mathematically, this is to say that we determine explicitly the superstring measure on the moduli space M0,n,0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_{0,n,0} $$\\end{document} of super Riemann surfaces of genus zero with n ≥ 3 Neveu-Schwarz punctures. While, of course, an expression for the measure was previously known, we do this from first principles, using the canonically defined super Mumford isomorphism [1]. We thus determine the scattering amplitudes, explicitly in the global coordinates on M0,n,0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_{0,n,0} $$\\end{document}, without the need for picture changing operators or ghosts, and are also able to determine canonically the value of the coupling constant. Our computation should be viewed as a step towards performing similar analysis on M0,0,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathcal{M}}_{0,0,n} $$\\end{document}, to derive explicit tree-level scattering amplitudes with Ramond insertions.

On the existence of heterotic-string and type-II-superstring field theory vertices

We consider the problem of the existence of heterotic-string and type-II-superstring field theory vertices in the product of spaces of bordered surfaces parameterizing the left- and right-moving sectors of these theories. It turns out that this problem can be solved by proving the existence of a solution to the BV quantum master equation in moduli spaces of bordered spin-Riemann surfaces. We first prove that for arbitrary genus ▪, ▪ Neveu–Schwarz boundary components, and ▪ Ramond boundary components such solutions exist. We also prove that these solutions are unique up to homotopy in the category of BV algebras. Furthermore, we prove that there exists a map in this category under which these solutions are mapped to fundamental classes of Deligne-Mumford stacks of associated punctured spin-Riemann surfaces. These results generalize the work of Costello on the existence of a solution to the BV quantum master equations in moduli spaces of bordered Riemann surfaces which, through the work of Sen and Zwiebach, are related to the existence of bosonic-string vertices, and their relation to fundamental classes of Deligne-Mumford stacks of associated punctured Riemann surfaces. Using the existence of solutions to the BV quantum master equation in moduli spaces of spin-Riemann surfaces, we prove that heterotic-string and type-II-superstring field theory vertices, for arbitrary genus ▪ and an arbitrary number of any type of boundary components, exist. Furthermore, we prove the existence of a solution to the BV quantum master equation in spaces of bordered N=1 super-Riemann surfaces for arbitrary genus ▪, ▪ Neveu–Schwarz boundary components, and ▪ Ramond boundary components.

Super Riemann Surfaces and Fatgraphs

Our goal is to describe superconformal structures on super Riemann surfaces (SRSs) based on data assigned to a fatgraph. We start from the complex structures on punctured (1|1)-supermanifolds, characterizing the corresponding moduli and the deformations using Strebel differentials and certain Čech cocycles for a specific covering, which we reproduce from fatgraph data, consisting of U(1)-graph connection and odd parameters at the vertices. Then, we consider dual (1|1)-supermanifolds and related superconformal structures for N=2 super Riemann surfaces. The superconformal structures, N=1 SRS, are computed as the fixed points of involution on the supermoduli space of N=2 SRS.

On the equivalence between SRS and PCO formulations of superstring perturbation theory

We establish the equivalence between two formulations of superstring perturbation theory, one based on integration over the supermoduli space of super Riemann surfaces (SRS), the other based on integration over the bosonic moduli space with insertions of picture changing operators (PCO) on the worldsheet and the vertical integration prescription, by showing how the latter arises from a specific construction of the supermoduli integration contour.

The bad locus in the moduli of super Riemann surfaces with Ramond punctures

The bad locus in the moduli of super Riemann surfaces with Ramond punctures parametrizes those super Riemann surfaces that have more than the expected number of independent closed holomorphic 1-forms. There is a super period map that depends on certain discrete choices. For each such choice, the period map blows up along a divisor that contains the bad locus. Our main result is that away from the bad locus, at least one of these period maps remains finite. In other words, we identify the bad locus as the intersection of the blowup divisors. The proof abstracts the situation into a question in linear algebra, which we then solve. We also give some bounds on the dimension of the bad locus.

Supermoduli and PCOs at genus two

We illustrate the relation between supermoduli integration and picture changing operators (PCOs) particularly concerning the role of vertical integration, in the context of superstring vacuum amplitudes, by an explicit comparison of different parameterizations of the supermoduli space of genus two super Riemann surfaces.

The supermoduli space of genus zero super Riemann surfaces with Ramond punctures

We give an explicit quotient description of the (nR−3|nR/2−2)-dimensional supermoduli space M0,nR of genus zero super Riemann surfaces with nR≥4 Ramond punctures and prove that it is a Deligne-Mumford superstack. We also give an explicit quotient description of the supermoduli stack of genus zero supercurves with no SUSY structure.

The super Mumford form and Sato Grassmannian

We describe a supersymmetric generalization of the construction of Kontsevich and Arbarello, De Concini, Kac, and Procesi, which utilizes a relation between the moduli space of curves with the infinite-dimensional Sato Grassmannian. Our main result is the existence of a flat holomorphic connection on the line bundle λ3/2⊗λ1/2−5 on the moduli space of triples: a super Riemann surface, a Neveu-Schwarz puncture, and a formal coordinate system. We also prove a superconformal Noether normalization lemma for families of super Riemann surfaces.

Super $$J$$-holomorphic curves: construction of the moduli space

Let $M$ be a super Riemann surface with holomorphic distribution $\mathcal{D}$ and $N$ a symplectic manifold with compatible almost complex structure $J$. We call a map $\Phi\colon M\to N$ a super $J$-holomorphic curve if its differential maps the almost complex structure on $\mathcal{D}$ to $J$. Such a super $J$-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super $J$-holomorphic curves as a smooth subsupermanifold of the space of maps $M\to N$.

An Expansion Formula for Decorated Super-Teichmüller Spaces

Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichm\"uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain formulas for super $\lambda$-lengths associated to arcs in a bordered surface. In the special case of a disk, we are able to give combinatorial expansion formulas for the super $\lambda$-lengths associated to diagonals of a polygon in the spirit of Ralf Schiffler's $T$-path formulas for type $A$ cluster algebras. We further connect our formulas to the super-friezes of Morier-Genoud, Ovsienko, and Tabachnikov, and obtain partial progress towards defining super cluster algebras of type $A_n$. In particular, following Penner-Zeitlin, we are able to get formulas (up to signs) for the $\mu$-invariants associated to triangles in a triangulated polygon, and explain how these provide a step towards understanding odd variables of a super cluster algebra.

Boundary contributions to three loop superstring amplitudes

In type II superstring theory, the vacuum amplitude at a given loop order g can receive contributions from the boundary of the compactified, genus g supermoduli space of curves M̄g. These contributions capture the long distance or infrared behavior of the amplitude. The boundary parameterizes degenerations of genus g super Riemann surfaces. A holomorphic projection of the supermoduli space onto its reduced space would then provide a way to integrate the holomorphic, superstring measure and thereby give the superstring vacuum amplitude at g-loop order. However, such a projection does not generally exist over the bulk of the supermoduli spaces in higher genera. Nevertheless, certain boundary divisors in ∂M̄g may holomorphically map onto a bosonic space upon composition with universal morphisms, thereby enabling an integration of the holomorphic, superstring measure here. Making use of ansatz factorizations of the superstring measure near the boundary, our analysis shows that the boundary contributions to the three loop vacuum amplitude will vanish in closed oriented type II superstring theory with unbroken spacetime supersymmetry.

Volumes And Random Matrices

AbstractThis article is an introduction to newly discovered relations between volumes of moduli spaces of Riemann surfaces or super Riemann surfaces, simple models of gravity or supergravity in two dimensions, and random matrix ensembles. (The article is based on a lecture at the conference on the Mathematics of Gauge Theory and String Theory, University of Auckland, January 2020)

JT supergravity and Brezin-Gross-Witten tau-function

We study thermal correlation functions of Jackiw-Teitelboim (JT) supergravity. We focus on the case of JT supergravity on orientable surfaces without time-reversal symmetry. As shown by Stanford and Witten recently, the path integral amounts to the computation of the volume of the moduli space of super Riemann surfaces, which is characterized by the Brezin-Gross-Witten (BGW) tau-function of the KdV hierarchy. We find that the matrix model of JT supergravity is a special case of the BGW model with infinite number of couplings turned on in a specific way, by analogy with the relation between bosonic JT gravity and the Kontsevich-Witten (KW) model. We compute the genus expansion of the one-point function of JT supergravity and study its low-temperature behavior. In particular, we propose a non-perturbative completion of the one-point function in the Bessel case where all couplings in the BGW model are set to zero. We also investigate the free energy and correlators when the Ramond-Ramond flux is large. We find that by defining a suitable basis higher genus free energies are written exactly in the same form as those of the KW model, up to the constant terms coming from the volume of the unitary group. This implies that the constitutive relation of the KW model is universal to the tau-function of the KdV hierarchy.

JT gravity and the ensembles of random matrix theory

We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer torsion. Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.

Superstring field theory, superforms and supergeometry

Inspired by superstring field theory, we study differential, integral, and inverse forms and their mutual relations on a supermanifold from a sheaf-theoretical point of view. In particular, the formal distributional properties of integral forms are recovered in this scenario in a geometrical way. Further, we show how inverse forms “extend” the ordinary de Rham complex on a supermanifold, thus providing a mathematical foundation of the Large Hilbert Space used in superstrings. Last, we briefly discuss how the Hodge diamond of a supermanifold looks like, and we explicitly compute it for super Riemann surfaces.

String geometry and nonperturbative formulation of string theory

We define string geometry: spaces of superstrings including the interactions, their topologies, charts, and metrics. Trajectories in asymptotic processes on a space of strings reproduce the right moduli space of the super-Riemann surfaces in a target manifold. Based on the string geometry, we define Einstein–Hilbert action coupled with gauge fields, and formulate superstring theory nonperturbatively by summing over metrics and the gauge fields on the spaces of strings. This theory does not depend on backgrounds. The theory has a supersymmetry as a part of the diffeomorphisms symmetry on the superstring manifolds. We derive the all-order perturbative scattering amplitudes that possess the super moduli in type IIA, type IIB and SO(32) type I superstring theories from the single theory, by considering fluctuations around fixed backgrounds representing type IIA, type IIB and SO(32) type I perturbative vacua, respectively. The theory predicts that we can see a string if we microscopically observe not only a particle but also a point in the space–time. That is, this theory unifies particles and the space–time. This paper is a summary version of Ref. 1.

On the super Mumford form in the presence of Ramond and Neveu–Schwarz punctures

We generalize the result of Voronov (1988) to give an expression for the super Mumford form μ on the moduli spaces of super Riemann surfaces with Ramond and Neveu–Schwarz punctures. In the Ramond case we take the number of punctures to be large compared to the genus. We consider for the case of Neveu–Schwarz punctures the super Mumford form over the component of the moduli space corresponding to an odd spin structure. The super Mumford form μ can be used to create a measure whose integral computes scattering amplitudes of superstring theory. We express μ in terms of local bases of H0(X,ωj) for ω the Berezinian line bundle of a family of super Riemann surfaces.

On Ramond Decorations

We impose constraints on the odd coordinates of super Teichm\"uller space in the uniformization picture for the monodromies around Ramond punctures, thus reducing the overall odd dimension to be compatible with that of the moduli spaces of super Riemann surfaces. Namely, the monodromy of a puncture must be a true parabolic element of the canonical subgroup $PSL(2,\mathbb{R})$.

On the problem of splitting deformations of super Riemann surfaces

An odd deformation of a super Riemann surface $\mathcal S$ is a deformation of $\mathcal S$ by variables of odd parity. In this article we study the obstruction theory of these odd deformations $\mathcal X$ of $\mathcal S$. We view $\mathcal X$ here as a complex supermanifold in its own right. Our objective in this article is to show, when $\mathcal X$ is a deformation of second order of $\mathcal S$ with genus $g>1$: if the primary obstruction class to splitting $\mathcal X$ vanishes, then $\mathcal X$ is in fact split. This result leads naturally to a conjectural characterisation of odd deformations of $\mathcal S$ of any order.

Projective superspaces in practice

We study the supergeometry of complex projective superspaces $\mathbb{P}^{n|m}$. First, we provide formulas for the cohomology of invertible sheaves of the form $\mathcal{O}_{\mathbb{P}^{n|m}} (\ell)$, that are pull-back of ordinary invertible sheaves on the reduced variety $\mathbb{P}^n$. Next, by studying the even Picard group $\mbox{Pic}_0 (\mathbb{P}^{n|m})$, classifying invertible sheaves of rank $1|0$, we show that the sheaves $\mathcal{O}_{\mathbb {P}^{n|m}} (\ell)$ are not the only invertible sheaves on $\mathbb{P}^{n|m}$, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the $\Pi$-Picard group $\mbox{Pic}_\Pi (\mathbb{P}^{n|m})$, classifying $\Pi$-invertible sheaves of rank $1|1$, proving that there are also non-split $\Pi$-invertible sheaves on supercurves $\mathbb{P}^{1|m}$. Further, we investigate infinitesimal automorphisms and first order deformations of $\mathbb{P}^{n|m}$, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special special attention is paid to the meaningful case of supercurves $\mathbb{P}^{1|m}$ and of Calabi-Yau's $\mathbb{P}^{n|n+1}$. Last, with an eye to applications to physics, we show in full detail how to endow $\mathbb{P}^{1|2}$ with the structure of $\mathcal{N}=2$ super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra $\mathfrak{osp} (2|2)$. A particular effort has been devoted to keep the exposition as concrete and explicit as possible.