Intrinsic proof of perturbative unitarity (cutting rules) in covariant string perturbation theory via the Feynman iε integration cycle

Witten's 'The Feynman iε in String Theory' (JHEP 04 (2015) 055) defined perturbative string amplitudes in physical Lorentzian kinematics as integrals of a holomorphic form over a noncompact cycle $\widetilde{\Gamma}$ in complexified moduli space: near each worldsheet degeneration that puts a string on shell, the Schwinger proper time is continued to Lorentzian values, mirroring the Feynman $i\varepsilon$ of field theory. This makes unitarity plausible — the singularities match those of field-theory propagators — but the paper states that a direct proof is missing: no covariant string analog of cutting rules or the greatest-time equations is known, and the special degenerations, where the momentum through a degenerating cycle is automatically on shell, are not tamed by the $i\varepsilon$ alone. Partial results are known: partial-wave unitarity of tree-level superstring amplitudes has been proven in $D\le 6$, genus-one unitarity cuts have been computed exactly at finite string tension, and the cycle prescription has been shown equivalent at all orders to unitary superstring field theory; the direct, intrinsic, all-genus cutting-rules proof — including a uniform treatment of special degenerations — remains open.

Background

Setup: string amplitudes as moduli-space integrals. In perturbative string theory, a scattering amplitude with $n$ external strings at genus $g$ (the loop order) is computed as an integral over the moduli space $\mathcal{M}_{g,n}$ — the space of complex structures on a genus-$g$ Riemann surface with $n$ marked punctures where vertex operators (worldsheet fields that create the external particles) are inserted. The integrand is a holomorphic differential form built from worldsheet correlators; for open or unoriented strings one works on the fixed-point set of an involution (a symmetry of the doubled closed surface whose square is the identity) inside its moduli space. The field-theory content of the amplitude emerges at the boundary of moduli space: at a degeneration, the surface develops a long strip (open string) or a long tube (closed string) and locally factorizes. Each such degeneration carries a sewing or gluing parameter $q$, with $q=e^{-t+i\varphi}$ for a closed-string tube, where $t$ is a Schwinger proper time (the Euclidean propagation time of the exchanged string, as in the field-theory representation of a propagator) and $\varphi$ is a twist angle; the region $t\to\infty$ corresponds to the pole of a propagator $-i/(p^2+m^2)$ for every string state of mass $m$ flowing through the tube.

Why an $i\varepsilon$ is needed. In quantum field theory, amplitudes in physical Lorentzian kinematics are defined by the Feynman $i\varepsilon$: each propagator becomes $1/(p^2+m^2-i\varepsilon)$. This prescription both makes the loop integrals well defined and underlies perturbative unitarity — the statement that the S-matrix obeys $SS^{\dagger}=1$, or, writing $S=1+iT$, that $\mathrm{Im}\,T=\tfrac12 T T^{\dagger}$ holds order by order in the coupling. Field-theory proofs use cutting rules (Cutkosky's discontinuity formulae: the discontinuity of an amplitude across an on-shell threshold is obtained by replacing the cut propagators with on-shell phase-space integrals), or the compact 'equation of greatest time' argument for Feynman diagrams (a bookkeeping argument showing that the imaginary part of any diagram is obtained by cutting exactly the propagators attached to the vertex of greatest time), which is the field-theory proof cited in the source paper. Naive string amplitudes — integrals of the holomorphic form over the real Euclidean slice of moduli space — converge only for unphysical (spacelike, Euclidean) kinematics; in the physical region the $t$-integrals diverge, and a manifestly real, divergent integral could never reproduce the imaginary parts that unitarity demands.

Witten's prescription. The source paper (Witten, JHEP 04 (2015) 055) constructed the string-theoretic analog: a string only goes on shell near a degeneration, so the worldsheet is given Lorentzian signature precisely there while remaining Euclidean elsewhere. Concretely, for each nearly degenerating tube or strip one integrates the proper time over the real axis up to a large cutoff $t_0$ and then continues $t=t_0+i\tau$ with $0\le\tau<\infty$, keeping the twist angles real, and inserts a convergence factor $e^{-\varepsilon\Phi}$ with $\Phi$ growing linearly in the total proper time. Equivalently, the amplitude is the integral of a holomorphic form over a noncompact cycle $\widetilde{\Gamma}$ in a cover of the complexified moduli space (a cover is needed because the analytically continued form is multivalued; the original Euclidean slice is the fixed locus of an anti-holomorphic involution). At tree level this parallels what classical Pochhammer contours — closed contours winding around the branch points of Euler-type integrals, used since the 19th century to analytically continue beta-function integrals such as the Veneziano amplitude (the original tree-level string scattering amplitude) — achieve, but with the Feynman $i\varepsilon$ built in; crucially, the construction generalizes to all genera. Its initial limitation is that it is defined only when the kinematic invariants (the dot products of external momenta) are real.

Generic and special degenerations; the missing proof. A generic degeneration carries momentum that is generically off shell; on $\widetilde{\Gamma}$ the tube region contributes a damped oscillatory integral and there is no boundary term. A special degeneration carries momentum that is automatically on shell (for example, when all punctures lie on one side of a separating tube, momentum conservation forces the tube momentum on shell); the propagator degenerates to $\sim 1/\varepsilon$, the $i\varepsilon$ alone does not help, and the region is tamed only by cancellations relying on worldsheet supersymmetry and the GSO projection (the Gliozzi–Scherk–Olive projection: the sum over worldsheet spin structures that removes tachyons and implements spacetime supersymmetry). Special degenerations make bosonic string perturbation theory break down already at one loop and are the main subtlety of superstring perturbation theory, where the supermoduli space (the moduli space of super Riemann surfaces, i.e. worldsheets with anticommuting fermionic coordinates) is, by a theorem of Donagi and Witten, not holomorphically projected — its fermionic moduli cannot be eliminated in favor of an ordinary complex manifold. Before this program, unitarity of string perturbation theory rested on light-cone gauge quantization (a gauge in which only transverse string oscillators appear, making unitarity manifest at the cost of manifest Lorentz covariance), on factorization analyses such as those of Aoki, D'Hoker and Phong for closed superstrings at low genus, and on Berera's construction of unitary string amplitudes. With the $i\varepsilon$ cycle in place, unitarity becomes 'more or less obvious' by matching field-theory singularities — but, as the paper stresses, something is still missing: a proof of unitarity directly in the covariant moduli-space formulation, with no detour through field-theory or string-field-theory Feynman diagrams. The genus-one work of Eberhardt and Mizera makes the target concrete: pinching the integration contour around the non-separating divisor (the boundary locus where a handle degenerates without splitting the surface) of the complexified moduli space reproduces exactly the expected cuts, and the natural formulation of unitarity for a purely holomorphic integral is a holomorphic cutting rule, in which the conjugation in $T^{\dagger}$ is eliminated by solving iteratively for $T^{\dagger}$ in terms of $T$, at the price of summing over more and more simultaneous cuts at higher genus.

Problem Statement

Prove perturbative unitarity of string scattering amplitudes directly and intrinsically in the covariant (Polyakov/moduli-space) formulation, for amplitudes defined by Witten's $i\varepsilon$ prescription — integrals of the holomorphic string integrand over the noncompact Lorentzian cycle $\widetilde{\Gamma}$ in the complexified moduli space — at arbitrary genus $g$ and arbitrary external multiplicity.

Concretely, establish a string-theoretic analog of field-theory cutting rules: show, order by order in the string coupling, that the imaginary part (equivalently, the discontinuity across on-shell thresholds) of a genus-$g$ amplitude equals the sum over physically allowed on-shell intermediate string states — with their exact level degeneracies (the number of distinct string states at each mass level) — of phase-space integrals of products of lower-genus amplitudes, i.e. $\mathrm{Im}\,T=\tfrac12\,T T^{\dagger}$ (or the equivalent purely holomorphic, conjugation-free version in which $T^{\dagger}$ is solved for iteratively at the price of multiple simultaneous cuts). The equality must be derived from the geometry of $\widetilde{\Gamma}$: from contour deformation and pinching around the divisors of the complexified moduli space at which degenerating tubes or strips go on shell.

The proof must be intrinsic to the moduli-space formulation — not routed through light-cone gauge, through the no-ghost theorem (the statement that the covariant string Fock space contains no negative-norm states) plus factorization, or through the Feynman diagrams of string field theory (whose all-order equivalence with the cycle prescription is already known) — and it must control both generic degenerations and the special degenerations in which the exchanged momentum is automatically on shell, the propagator degenerates to $1/\varepsilon$, and the required cancellations rest on the GSO projection and spacetime supersymmetry. A uniform cycle-based treatment of both degeneration types is the source's separately posed refinement; the proof must control both generic degenerations and the special degenerations, whether by a uniform argument for both types or by two separate arguments.

Scope notes: the related questions posed alongside this one in the source — the analytic continuation of $\widetilde{\Gamma}$ in the external momenta (crossing symmetry and sheet structure of loop amplitudes) and the identification of dominant complex saddles at high energy — are tied to the same integration cycle but are logically separate questions with independent targets and are not part of the target of this problem. Likewise, the target requires an all-genus, all-multiplicity structural argument, as opposed to an argument restricted to tree level, to genus $\le 1$, to fixed low spacetime dimension, or a genus-by-genus computational algorithm without an all-genus structural argument.

References

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