Supermoduli-space foundations of closed superstring field theory: the single-cover and canonical picture-changing-insertion problem

Erler, Konopka and Sachs (2014) constructed a classical action for the NS-NS sector of type II closed superstring field theory in the small Hilbert space, using recursive multi-string products with explicit picture-changing insertions that satisfy the classical Batalin-Vilkovisky master equation, and posed four next steps: incorporate the Ramond sectors, quantize (higher-genus corrections satisfying the quantum master equation), understand how the vertices and propagators provide a single cover of the supermoduli space of super-Riemann surfaces, and relate to large-Hilbert-space formulations. Later literature resolved the first two and largely the fourth: quantum Batalin-Vilkovisky master actions for heterotic and type II superstring field theory including all Ramond sectors now exist and pass the known consistency checks. The surviving open core is the geometric question, still listed among future directions in the 2024 Sen-Zwiebach review: no closed-superstring field theory has been formulated directly on supermoduli space (only open-string versions exist), it is not understood how the picture-changing-operator-based vertices and propagators of the existing closed theories combine to a single cover or controlled atlas of supermoduli space, and — while chamber decomposition with vertical integration remains the only known systematic way of choosing spurious-pole-free picture-changing-operator locations — a canonical distribution of picture-changing operators on the bulk of Riemann surfaces is not yet available for Sen's heterotic and type II formulations. This problem asks for this supermoduli-geometric foundation of closed superstring field theory.

Background

String perturbation theory expresses scattering amplitudes as integrals over moduli spaces of Riemann surfaces — the spaces parametrizing complex structures and puncture positions of the interacting worldsheets. String field theory reorganizes this expansion into a spacetime gauge theory: one chooses regions of moduli space (string vertices) whose boundaries are exactly compensated by the sewing operations implemented by propagators, which is the geometric content of the Batalin-Vilkovisky (BV) master equation, the condition guaranteeing gauge invariance and, in its quantum version with an extra term for closed-string loops, consistency of the quantized theory.

For superstrings in the Ramond-Neveu-Schwarz (RNS) formalism the world-sheet is a super-Riemann surface: it carries fermionic (odd) coordinates and moduli in addition to the bosonic ones, and the arenas of perturbation theory are supermoduli spaces. A foundational complication, emphasized by Witten (2012), is that supermoduli space is in general not holomorphically projected onto the bosonic moduli space. In the covariant BRST quantization of the RNS string one works in the small Hilbert space (the state space built on the natural vacuum of the superconformal ghosts $\beta,\gamma$), and amplitudes that live naturally on supermoduli space are computed on the bosonic moduli space after inserting picture-changing operators (PCOs): operators built from $\beta,\gamma$ that raise the picture number, a conserved charge bookkeeping the $\beta,\gamma$ vacuum structure whose total value must be fixed (conventionally $-1$ for Neveu-Schwarz-sector fields, which describe spacetime bosons via antiperiodic world-sheet fermions, and $-1/2$ or $-3/2$ for Ramond-sector fields, which describe spacetime fermions via periodic ones) for amplitudes to be nonzero. PCO positions are largely arbitrary, but bad choices produce spurious poles — singularities depending on the PCO placement rather than on physics. The standard remedy divides moduli space into chambers with fixed PCO positions and adds compensating vertical-integration terms on chamber boundaries; Wang and Yin (2022) proved that this prescription reproduces the intrinsic supermoduli-space integrals at the level of world-sheet amplitudes.

Erler, Konopka and Sachs (arXiv:1403.0940, 2014) constructed in this PCO-based framework a classical action for the NS-NS sector of type II closed superstring field theory: recursive multi-string products with explicit $\xi$, $X$ insertions in the small Hilbert space, carrying the same $L_\infty$ (homotopy Lie) algebra structure as bosonic closed string field theory and automatically satisfying the classical BV master equation. They posed as next steps: incorporate the Ramond sectors, quantize, understand how the vertices and propagators provide a single cover of supermoduli space, and relate the construction to large-Hilbert-space formulations (the enlargement of the state space by the $\xi$ zero mode in which Berkovits-type theories live).

Subsequent literature resolved most of this program. Sen constructed one-particle-irreducible (1PI) effective actions including Ramond sectors and then quantum BV master actions for heterotic and type II superstring field theory; Kunitomo and Sugimoto built a complete type II theory with all four sectors (NS-NS, R-NS, NS-R, R-R) on a cyclic $L_\infty$ structure in the small Hilbert space; Pius constructed a quantum closed-superstring theory from hyperbolic string diagrams with a geodesic-based PCO distribution, though this distribution requires a modification near the boundary of the string vertices and yields only an approximate solution of the quantum master equation; and the 2024 review by Sen and Zwiebach records that consistent string field theories now exist for all supersymmetric closed strings and pass all known consistency checks. The open residue is the geometric question that the 2014 paper listed third and that the 2024 review still lists among its future directions: the existing closed theories are all built from bosonic moduli spaces with inserted PCOs. Only the open superstring — not the closed heterotic or type II theories — has been formulated directly on supermoduli space (Ohmori and Okawa in 2017 for the NS sector, Takezaki in 2019 including Ramond, both to quartic order), and for the closed theories chamber decomposition with vertical integration remains the only known systematic way of choosing spurious-pole-free PCO positions, while a canonical distribution of PCOs on the bulk of the moduli spaces is not yet available for Sen's heterotic and type II formulations.

Problem Statement

Give closed superstring field theory a supermoduli-geometric foundation, as posed by Erler-Konopka-Sachs (2014, third listed next step) and still open in the 2024 Sen-Zwiebach review. For the heterotic and/or type II closed superstring, the objective is any one of the following three alternatives: (1) Construct the string field theory — a classical construction satisfying the classical Batalin-Vilkovisky master equation, or the stronger quantum construction satisfying the quantum master equation — whose interaction vertices are defined directly by integration over the supermoduli spaces of super-Riemann surfaces with the appropriate Neveu-Schwarz and Ramond punctures, such that the picture-changing-operator insertions and vertical-integration terms of the existing small-Hilbert-space formulations arise canonically as a choice of section or cover of supermoduli space, with the action satisfying the classical (respectively quantum) Batalin-Vilkovisky master equation and on-shell amplitudes agreeing with first-quantized superstring perturbation theory; or (2) prove that the vertices and propagators of the existing picture-changing-operator-based closed superstring field theories combine to provide a single cover — or an explicit atlas with controlled overlap structure — of the supermoduli spaces, or else identify precisely the obstruction; or (3) exhibit a canonical, spurious-pole-free distribution of picture-changing operators for the closed heterotic/type II theories valid at arbitrary genus — going beyond Pius's hyperbolic geodesic-based distribution, which requires modification near the boundary of the string vertices and yields only an approximate solution of the quantum master equation, and beyond the chamber-by-chamber vertical-integration prescription, which remains the only known systematic way of choosing spurious-pole-free picture-changing-operator locations, a canonical bulk distribution being not yet available for Sen's heterotic and type II formulations per the 2024 review — or give a supermoduli-based construction of a closed theory covering at least one of the heterotic or type II theories at two or more genera with the general structure controlled. The common scientific objective uniting these three alternatives is to replace the chamber-by-chamber picture-changing bookkeeping of closed superstring field theory by an intrinsic supergeometric definition whose relation to the existing constructions is explicit.

References

1. Theodore Erler, Sebastian Konopka, Ivo Sachs, NS-NS Sector of Closed Superstring Field Theory (2014), arXiv:1403.0940, doi:10.48550/arXiv.1403.0940, https://arxiv.org/abs/1403.0940
2. Edward Witten, Notes On Super Riemann Surfaces And Their Moduli (2012), arXiv:1209.2459, doi:10.48550/arXiv.1209.2459, https://arxiv.org/abs/1209.2459
3. Charles Wang, Xi Yin, On the Equivalence between SRS and PCO Formulations of Superstring Perturbation Theory (2022), arXiv:2205.01106, doi:10.48550/arXiv.2205.01106, https://arxiv.org/abs/2205.01106
4. Ashoke Sen, Gauge Invariant 1PI Effective Superstring Field Theory: Inclusion of the Ramond Sector (2015), arXiv:1501.00988, doi:10.48550/arXiv.1501.00988, https://arxiv.org/abs/1501.00988
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