Witten's construction of the super period matrix of a super Riemann surface with Ramond punctures (2015) left open the geometry of the locus where the matrix develops poles and the construction of well-behaved superstring measures in the Ramond sector. Later work identified the bad locus explicitly — Donagi and Ott (2023) proved it is the intersection of the blow-up divisors of the sign-choice period maps and that every component of it has codimension at most r, with a component of codimension exactly r — and Donagi and Ott (2024) constructed a canonical measure on the moduli space of super Riemann surfaces with 2r Ramond punctures from the super Mumford isomorphism and a choice-free super period map, proving that for r at least 2 the measure extends holomorphically across the bad locus because that locus has codimension at least 2; for r = 0 (no punctures) Deligne proved extension at every genus. The one remaining case, r = 1 (two Ramond punctures — the case arising in Ward-identity applications and of primary physical interest), is explicitly open: the bad locus is then a divisor, the Hartogs-type argument fails, and it is unknown whether the pole of the super period map along the bad divisor, which carries nilpotent residue, is cancelled by the remaining factors of the measure. The problem asks for a proof or disproof of smooth extension for r = 1 at every genus g at least 2, and for a determination of whether the measure extends to a neighborhood of the quasidiagonal, the locus underlying the integration cycle of type-II superstring perturbation theory. Together these would complete a canonically defined, everywhere-regular superstring measure in the Ramond sector, the prerequisite for first-principles multiloop superstring amplitudes with Ramond external states.

Background

In the Ramond-Neveu-Schwarz (RNS) formulation of perturbative superstring theory, an amplitude at loop order g is an integral over the moduli space of super Riemann surfaces of genus g. A super Riemann surface is a complex supermanifold X of dimension 1|1 (one even and one odd coordinate) carrying a subbundle D of the tangent sheaf of rank 0|1 that is maximally non-integrable, meaning D^2 = TX/D; these surfaces are parametrized by moduli superstacks (the algebro-geometric moduli spaces that parametrize families of such surfaces, remembering their automorphisms). The two puncture types of the RNS string enter very differently: a Neveu-Schwarz (NS) puncture is an ordinary marked point, while a Ramond puncture is a divisor along which the superconformal structure degenerates (D becomes integrable there); Ramond punctures occur in pairs, and the moduli space of genus-g surfaces with 2r Ramond punctures and no NS punctures is denoted by the symbol M with subscripts g, 0, 2r. Over a purely even base (the split case) such a surface is equivalent to an ordinary smooth curve C of genus g, an effective divisor D = p_1 + ... + p_{2r} on C, and a twisted spin structure, that is, a line bundle L with an isomorphism of L-squared with the canonical bundle of C twisted by D.

For ordinary Riemann surfaces, the periods of holomorphic differentials over a symplectic basis of 1-cycles form the period matrix, from which Torelli's theorem reconstructs the curve. Witten (2012), building on the two-loop superstring formalism of D'Hoker and Phong, developed the super analog — the super period matrix, a g-by-g symmetric matrix of functions on supermoduli with nilpotent odd entries and positive imaginary part — which for genus at most 3 identifies the underlying bosonic curve and underlies the D'Hoker-Phong computation of the two-loop NS-sector superstring measure. Witten (2015) extended this to Ramond punctures: closed holomorphic 1-forms on the punctured surface, described as sections of an extended Berezinian sheaf (the dualizing sheaf, with zero-residue poles permitted along the Ramond divisor; the Berezinian is the supergeometric replacement of the determinant line), have, besides the usual even periods over 1-cycles, 2r odd (fermionic) periods given by residues along the Ramond divisors; normalizing r of the odd periods produces a matrix of size g|r by g|r that is symmetric in the Z_2-graded sense. The construction is nondegenerate exactly off the bad locus B = {(C, D, L) : h^0(L) > r}, where h^0 denotes the dimension of the space of global sections: there the odd period map acquires a kernel and the period matrix develops poles. For r = 1, Witten showed that B is a divisor and computed the local pole expansion along it, the pole carrying a nilpotent residue; for r > 1 he left the geometry and codimension of B unanalyzed and noted that the choice of fermionic A-periods is then choice-dependent, introducing artificial codimension-1 singularities, suggesting instead that the Lagrangian subspace spanned by all the periods is the more natural object. His genus-2 application — pointwise vanishing, after summing over spin structures, of the two-loop vacuum amplitude in compactifications supersymmetric at tree level, a phenomenon observed in the orbifold computations of D'Hoker and Phong — rests on the pole along B being cancelled by fermion zero modes.

A second ingredient is the super Mumford isomorphism, the analog for supercurves of Mumford's computation of the Picard group of the moduli space of curves: it identifies the Berezinian line of the supermoduli space with the fifth Berezinian power of the Hodge bundle (the vector bundle over moduli space whose fiber over a curve is its space of holomorphic differentials). Donagi and Ott (2024) combine the super Mumford isomorphism with a choice-free super period map — the direct sum of the even period map (Berezin integrals over 1-cycles, the supergeometric integrals that integrate out the odd coordinate along the cycle) and the odd period map (residues at the Ramond punctures), valued in a period local system of rank 2g|2r carrying a supersymplectic pairing, symplectic on even cycles and symmetric of residue type on the odd part — and thereby build, near the diagonal of the product of M with its complex conjugate, a Hermitian pairing on the Berezinian of M, hence a volume form: the superstring measure with Ramond punctures. Regularity of such measures is delicate. In the unpunctured case (r = 0) the super period map itself has a pole along the divisor where the even theta characteristic (the square root of the canonical bundle selected by the even spin sector) acquires a section — the vanishing theta-null divisor — yet Deligne — whose proof is presented in Felder, Kazhdan, and Polishchuk (2024), after the same authors settled genera up to 11 in 2019 — proved that the supermeasure extends holomorphically across it at every genus. Donagi and Ott proved that for r at least 2 the bad locus has codimension at least 2, so their measure extends across it by a Hartogs-type argument (holomorphic functions and volume forms extend automatically across subsets of complex codimension at least 2). For r = 1 the bad locus is a divisor and the question is open.

Two structural facts frame the higher-genus picture. First, for g at least 4 an ordinary period matrix must satisfy the Schottky relations — polynomial relations characterizing period matrices inside the space of symmetric matrices with positive imaginary part — and Felder, Kazhdan, and Polishchuk (2019) showed that the image of the superperiod map is annihilated only by a high power of the Schottky ideal (the g-th, or (g-1)-th, power), so the super period matrix is far from an ordinary one. Second, the supermoduli space admits no holomorphic projection onto the bosonic moduli space in this range: non-projectedness is proven for g at least 5 without punctures (Donagi and Witten, 2013), for the even-spin component in genus 4 (Donagi and Noja, 2026), and with Ramond punctures for g at least 5r+1 (Donagi and Ott, 2023), improved by Wang (2025) to g at least n + 5r + 3 in the presence of n Neveu-Schwarz punctures. Consequently the D'Hoker-Phong genus-2 technique of projecting supermoduli to bosonic moduli via the period matrix cannot extend to genus 4 and beyond, and an intrinsic construction on supermoduli — the Donagi-Ott measure — is the viable route to Ramond-sector measures at all genera. What remains is the regularity of that measure in the missing case r = 1, and its behavior on the quasidiagonal: the locus in the product of M with its conjugate parametrizing pairs whose underlying bosonic curves and Ramond divisors are complex conjugates, with no constraint on the twisted spin structures. The quasidiagonal strictly contains the diagonal, and a thickening of it is the integration cycle of type-II superstring perturbation theory.

Problem Statement

Let M be the moduli space (Deligne-Mumford superstack) of super Riemann surfaces of genus g at least 2 with two Ramond punctures (r = 1), with universal curve and extended Berezinian sheaf (the dualizing sheaf with zero-residue poles permitted along the Ramond divisor), and let the period local system of rank 2g|2 carry its supersymplectic pairing (symplectic intersection pairing on even cycles, symmetric residue pairing on the odd part). The super period map — the direct sum of the even period map (Berezin integrals over a symplectic basis of 1-cycles) and the odd period map (residues at the two Ramond punctures) into the period local system — is injective precisely off the bad locus B = {(C, D, L) : h^0(L) at least 2}, where L-squared is isomorphic to the canonical bundle of C twisted by the Ramond divisor D; for r = 1 and g at least 2, B is a divisor in the reduced (purely bosonic) moduli space. On a neighborhood of the diagonal in the product of M with its complex conjugate, minus the bad diagonal, the Donagi-Ott construction — the Berezinian of the pairing of the super period map with its conjugate against the supersymplectic form, combined with the super Mumford isomorphism — defines a holomorphic volume form: the superstring measure in the Ramond sector.

(1) Core question — prove or disprove: for r = 1 and every genus g at least 2, this measure extends smoothly across the bad-locus divisor B (equivalently, across the bad diagonal), to a holomorphic volume form on a neighborhood of the whole diagonal — the regular extension of the volume form associated with the period pairing, non-degenerate exactly off B. This is the one missing case of the regularity classification of the measure: r = 0 is Deligne's all-genus theorem and r at least 2 is Donagi and Ott's Hartogs-type theorem, while r = 1 is the case arising in Ward-identity applications (a supersymmetry Ward identity is a consistency relation that supersymmetry imposes on amplitudes; Witten's genus-2 argument reduces the two-loop vacuum amplitude, via such an identity, to an integral over the moduli space of genus-2 surfaces with two Ramond punctures). Because the bad locus is a divisor, extension cannot follow from codimension considerations: a proof must exhibit the mechanism by which the pole of the super period map along B — whose residue is nilpotent — is cancelled by the remaining factors of the measure, as happens in Deligne's unpunctured proof and in Witten's genus-2 computation where the pole is cancelled by fermion zero modes; a disproof must exhibit, for some genus g at least 2, a component of B along which the measure has a genuine non-removable singularity, with the singularity made explicit.

(2) Second question, same objective — determine whether the measure extends to an open neighborhood of the quasidiagonal: the locus in the product of M with its conjugate parametrizing pairs whose underlying bosonic curves and Ramond divisors are complex conjugates, with no constraint on the twisted spin structures; the quasidiagonal strictly contains the diagonal, and a thickening of it is the integration cycle of type-II superstring perturbation theory. Prove extension to a neighborhood of the quasidiagonal, or exhibit a point of the quasidiagonal near which the measure fails to extend; off the diagonal the identification of the odd period lattices of the two factors used in the Donagi-Ott construction is unavailable, so new input is required. Both questions serve one objective: a canonically defined, everywhere-regular superstring measure with Ramond punctures, the prerequisite for first-principles multiloop RNS amplitudes with Ramond external states at every genus.

Questions (1) and (2) are not logically independent: since the quasidiagonal strictly contains the diagonal and both concern the extension of one and the same measure, question (2) implies question (1) — a proof of (2) settles (1), and a disproof of (1) settles (2). Each question nonetheless stands on its own as a target.

References

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