Superstring Perturbation Theory Revisited

TL;DR

This work develops a covariant perturbative framework for superstring theory by formulating amplitudes on supermoduli space and carefully integrating over both even and odd moduli. It unifies the treatment of vertex insertions, gauge invariance, and the BRST structure across NS and Ramond sectors, while clarifying the roles of picture-changing operators and integration cycles in ensuring finite, gauge-invariant amplitudes. A central achievement is showing how the string propagator, degenerations of worldsheet, and factorization reproduce the expected infrared behavior and unitarity with appropriate iε prescriptions, all within a supergeometric (supermoduli) formalism. The analysis also elaborates on subtleties such as spurious singularities, the DM compactification, and the precise treatment of Ramond punctures, culminating in a coherent account of perturbative superstring amplitudes across open/closed and oriented/unoriented theories. The framework promises a transparent, gauge-covariant understanding of multi-loop string amplitudes and their infrared structure, with implications for consistency checks and future formalisms (e.g., pure spinor) that aim to capture the same physics.

Abstract

Perturbative superstring theory is revisited, with the goal of giving a simpler and more direct demonstration that multi-loop amplitudes are gauge-invariant (apart from known anomalies), satisfy space-time supersymmetry when expected, and have the expected infrared behavior. The main technical tool is to make the whole analysis, including especially those arguments that involve integration by parts, on supermoduli space, rather than after descending to ordinary moduli space.

Figures (41)

• Figure 1: On the locus in moduli space at which naively speaking several punctures in a Riemann surface $\Sigma$ collide, what actually happens in the Deligne-Mumford compactification is that $\Sigma$ splits off a genus zero component that contains the punctures. This is illustrated here for the case of three coinciding punctures, labeled $p_1,p_2,p_3$. The genus zero component in this example has four distinguished points -- the $p_i$ and the "node" at which it meets the other component of $\Sigma$ -- so it has one complex modulus. This modulus is lost if one represents the vertex operator insertions at the $p_i$ (or even 2 of them) in integrated form. Because of this, the use of integrated vertex operators does not treat correctly questions for which this region of the moduli space is important. The difficulties arise when the total momentum flowing through the node is on-shell.
• Figure 2: At infinity in moduli space, a genus 0 surface $\Sigma$ splits into two genus zero components $\Sigma_\ell$ and $\Sigma_r$, joined at a common singularity. The external vertex operators are distributed between the two sides in an arbitrary fashion. In the case of external NS vertex operators, to treat the compactification of moduli space correctly, both $\Sigma_\ell$ and $\Sigma_r$ should have two unintegrated vertex operators -- counting the singularity as one. Regardless of which two of the original vertex operators we take in unintegrated form, this is not always the case, since they might be both on $\Sigma_\ell$ or both on $\Sigma_r$. So the formalism based on integrated vertex operators does not treat correctly the compactification of the moduli space. In the more general formalism of section \ref{['intodd']} based on vertex operators of general picture numbers, one has the same problem. One would like the condition (\ref{['zolot']}) to hold on each side; but no choice of picture numbers ensures this.
• Figure 3: (a) A long strip connecting two parts of an open-string worldsheet. This is meant to be a "flat," purely two-dimensional picture. (b) A long tube connecting two parts of a closed-string worldsheet.
• Figure 4: As explained in the text, a narrow neck in a Riemann surface -- marked here by an arrow -- is conformally equivalent to a long tube. The singularity or "degeneration" that occurs when the neck collapses is said to be separating in (a), and non-separating in (b).
• Figure 5: An annulus, built by gluing together the ends of a long strip, with an operator insertion ${\mathcal{O}}$.