Completing the linear-dilaton regularization of light-cone gauge superstring field theory: heterotic and Type I amplitudes, and a nonperturbative definition

Light-cone gauge superstring field theory, whose action contains only cubic three-string interactions, has long been proposed as a nonperturbative formulation of string theory, but its Feynman amplitudes diverge when worldsheet supercurrent insertions collide at string interaction points. Ishibashi (2016) regularized these contact term divergences by placing the transverse worldsheet theory in a linear dilaton background with imaginary slope, proving all multiloop amplitudes finite for dilaton slope $Q$ with $Q^2>10$ together with the Feynman $i\varepsilon$ prescription; the physical critical-string amplitudes are then to be recovered by analytically continuing $Q$ to zero, in analogy with dimensional regularization. Subsequent work completed this program for Type II superstrings, showing that the continued amplitudes reproduce the standard first-quantized amplitudes for even and odd spin structures and with Ramond-sector external states. Two linked goals remain open: (i) extending the scheme to heterotic and Type I superstrings, where for the heterotic string the light-cone formulation obstructs the holomorphic factorization of the integrand into superstring and bosonic-string parts; and (ii) giving a nonperturbative definition of superstring theory based on this regularized cubic theory, whose Hamiltonian is complex and unbounded below.

Background

In string theory the light-cone gauge fixes the worldsheet reparametrization symmetry completely by using the light-cone coordinates $X^\pm=(X^0\pm X^9)/\sqrt{2}$ and identifying $X^+$ with the worldsheet time. In this gauge Mandelstam's interacting-string picture yields a string field theory whose action contains only cubic terms joining three strings, which was proposed as a candidate nonperturbative definition of closed string theory. In the NSR (Neveu-Schwarz-Ramond) formalism the worldsheet theory includes fermions $\psi^i$ for the transverse directions; their periodicity conditions around the cycles of the Riemann surface define the spin structures (a choice is called even or odd according to the parity of the periodicity pattern), and periodic fermions produce the Ramond sector, which contains the spacetime fermions, while antiperiodic fermions produce the Neveu-Schwarz (NS) sector of spacetime bosons.

The amplitudes of light-cone gauge superstring field theory are, however, ill defined. Greensite and Klinkhamer showed in 1987 that the interaction vertices require insertions of the worldsheet supercurrent at the interaction points, the points on the worldsheet where strings join and split; when such points collide in the moduli space of Riemann surfaces (the parameter space labeling inequivalent worldsheet geometries), the amplitudes diverge. These are the contact term divergences, and they plague even the tree amplitudes. Infrared divergences from infinitely long string propagation are handled by the standard Feynman $i\varepsilon$ prescription with $\varepsilon>0$, but the contact term divergences require a genuine regulator.

Ishibashi's regulator exploits the fact that the light-cone gauge is a complete gauge fixing, so the worldsheet theory of the eight transverse variables can be deformed freely: it is taken to be a free theory in a linear dilaton background $\Phi=-iQX^1$, where the dilaton field varies linearly along a spacelike direction $X^1$ with an imaginary slope $-iQ$. The central charge (the number characterizing the degrees of freedom of the two-dimensional worldsheet theory) of the transverse theory shifts away from its critical value, and for $Q^2>10$ with the Feynman $i\varepsilon$ all multiloop amplitudes are finite. The regulated theory is equivalent to a BRST-invariant (BRST: the Becchi-Rouet-Stora-Tyutin charge encoding gauge symmetry in a gauge-fixed theory) conformal-gauge worldsheet theory with an unusual longitudinal sector, so gauge invariance is not spoiled. As in dimensional regularization of ordinary quantum field theory, the amplitudes are defined as analytic functions of $Q^2$ in the domain $Q^2>10$ and analytically continued to $Q\to0$ to recover the amplitudes of the critical ten-dimensional superstring.

For Type II superstrings this program is complete. Ishibashi and Murakami verified the continuation for multiloop amplitudes with even spin structure and external states in the (NS,NS) sector, then for the odd spin structure contributions, and Ishibashi finally treated external states in the Ramond sectors; in every case the $Q\to0$ limit reproduces the amplitudes of the covariant first-quantized formalism computed with the Sen-Witten prescription, the standard method for multiloop superstring amplitudes that places picture-changing operators (operators converting between the inequivalent fermion-number or picture representations of vertex operators) at constrained positions on the worldsheet. This establishes, for Type II, that the regularized cubic light-cone theory and the covariant formalism define the same string perturbation theory.

For the heterotic string, whose left-moving sector is a superstring and whose right-moving sector is a bosonic string, an additional structural obstacle appears. Closed-string quantities should split into a holomorphic (left-moving) and an antiholomorphic (right-moving) part, and the heterotic amplitude integrand should be the product of a holomorphic superstring factor and an antiholomorphic bosonic-string factor; this is holomorphic factorization. In the light-cone formulation the coordinate $\rho$, which encodes the light-cone time as a function on the Riemann surface, depends on the antiholomorphic moduli parameters, so the split is obstructed. The Type I superstring, the theory of open unoriented superstrings obtained from Type IIB by an orientifold projection (a quotient that reverses the string orientation), is deferred alongside the heterotic string in the conclusions of the Type II paper. Beyond the perturbative completion, the deeper goal of the program is nonperturbative: the regulated Hamiltonian is complex because the dilaton slope is imaginary, the cubic light-cone Hamiltonian is unbounded below, and the region of vanishing longitudinal momentum $p^+$ (the momentum conjugate to $X^-$) is singular in this gauge; these are the obstacles to any nonperturbative use of the construction.

Problem Statement

The overarching objective is to turn the linear-dilaton-regularized light-cone gauge superstring field theory into a complete formulation of superstring theory: perturbatively correct for every superstring theory, and well defined beyond perturbation theory. For Type II superstrings the perturbative part is established: with the transverse worldsheet theory in the linear dilaton background $\Phi=-iQX^1$ and the Feynman $i\varepsilon$ ($\varepsilon>0$), all multiloop amplitudes are finite for $Q^2>10$, and their analytic continuation to $Q\to0$ reproduces the standard first-quantized amplitudes for all spin structures, including Ramond-sector external lines. Two linked goals remain, and both are required to resolve the task: the perturbative construction must be completed for heterotic and Type I superstrings alike, and the nonperturbative definition must be supplied.

(1) Heterotic and Type I amplitudes. Extend the scheme to heterotic and Type I superstrings. This requires formulating the multiloop light-cone gauge amplitudes with the $Q$-regulated transverse worldsheet theory; resolving the holomorphic factorization obstruction, in which the light-cone coordinate $\rho$ depends on the antiholomorphic moduli parameters while the heterotic integrand should factorize into a holomorphic superstring part and an antiholomorphic bosonic-string part; proving finiteness for $Q^2>10$, $\varepsilon>0$ over the full moduli space including all degenerations and collisions of interaction points; and showing that the amplitudes, defined as analytic functions of $Q^2$ and continued to $Q\to0$ (with the physical $\varepsilon\to0$ taken as in the Feynman prescription), reproduce the standard first-quantized heterotic and Type I amplitudes for all spin structures and external sectors, as has been shown for Type II.

(2) Nonperturbative definition. Give a nonperturbative definition of superstring theory along these lines: make the $Q$-regulated theory with only cubic interactions well defined beyond perturbation theory, confronting the obstacles identified in the program: the regulated Hamiltonian is complex because the dilaton slope is imaginary, the cubic light-cone Hamiltonian is unbounded below so the stability of the perturbative vacuum is unclear, and the $p^+=0$ region, where the nonperturbative properties are expected to reside, is singular in this gauge. The nonperturbative definition should have a perturbative expansion that reproduces the regulated amplitudes, and the status of the perturbative vacuum and of the $p^+=0$ region should follow from it.

References

1. Nobuyuki Ishibashi, "Light-cone gauge superstring field theory in linear dilaton background", Prog. Theor. Exp. Phys. (2017), DOI 10.1093/ptep/ptx012, arXiv:1605.04666, https://arxiv.org/abs/1605.04666
2. Nobuyuki Ishibashi and Koichi Murakami, "Multiloop Amplitudes of Light-cone Gauge NSR String Field Theory in Noncritical Dimensions", JHEP 01 (2017) 034, DOI 10.1007/JHEP01(2017)034, arXiv:1611.06340, https://arxiv.org/abs/1611.06340
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