[Paper Review] Complex Geometry and Supergeometry

Eric D’Hoker, D. H. Phong|ArXiv.org|Dec 15, 2005

Black Holes and Theoretical Physics40 references4 citations

TL;DR

This paper presents a breakthrough in superstring perturbation theory by resolving the long-standing supermoduli problem at genus 2, achieving gauge-slice independent amplitudes through a novel interplay of holomorphic and superholomorphic forms. The key result is the explicit computation of the 4-point function for massless NS bosons, confirming non-renormalization theorems and yielding a modular covariant measure $Ω_6[\delta]$, which provides a foundation for higher-genus amplitudes and dualities in string theory.

ABSTRACT

Complex geometry and supergeometry are closely entertwined in superstring perturbation theory, since perturbative superstring amplitudes are formulated in terms of supergeometry, and yet should reduce to integrals of holomorphic forms on the moduli space of punctured Riemann surfaces. The presence of supermoduli has been a major obstacle for a long time in carrying out this program. Recently, this obstacle has been overcome at genus 2, which is the first loop order where it appears in all amplitudes. An important ingredient is a better understanding of the relation between geometry and supergeometry, and between holomorphicity and superholomorphicity. This talk provides a survey of these developments and a brief discussion of the directions for further investigation.

Motivation & Objective

• To overcome the supermoduli problem in superstring theory at genus 2, where odd supermoduli obstruct standard gauge-fixing procedures.
• To establish gauge-slice independence of scattering amplitudes, a critical requirement previously unmet in prior formulations.
• To derive a modular covariant measure $\Xi_6[\delta]$ from $\vartheta$-constants, enabling explicit computation of amplitudes.
• To verify non-renormalization theorems for massless NS states at two-loop order using first-principles computations.
• To lay the groundwork for higher-genus amplitudes and effective action computations via factorization and BRST formalism.

Proposed method

• Utilizes a refined gauge-fixing procedure based on super period matrices to eliminate odd supermoduli and reduce integrals to the moduli space $\mathcal{M}_2$.
• Employs Dolbeault and de Rham cohomology to show that gauge-slice changes yield exact forms, ensuring gauge-slice independence.
• Constructs the measure $d\mu_2[\delta]$ as a product of $\vartheta$-constants and a modular form $\Xi_6[\delta]$, which transforms covariantly under modular transformations.
• Extracts holomorphic forms $\mathcal{H}$ from superholomorphic structures by isolating Dolbeault-exact terms in one insertion point.
• Applies factorization limits to constrain candidate forms for the 3-loop superstring measure, using known 2-loop results as a benchmark.
• Uses chiral splitting of matter fields and superholomorphic forms to extend the formalism to odd spin structures, though with added complexity.

Experimental results

Research questions

• RQ1 How can the supermoduli problem at genus 2 be resolved to yield gauge-slice independent amplitudes in superstring theory?
• RQ2 What is the explicit form of the modular covariant measure $\Xi_6[\delta]$ on the moduli space $\mathcal{M}_2$?
• RQ3 Can the 4-point function for massless NS bosons be computed explicitly and shown to be gauge-slice independent?
• RQ4 Do the computed amplitudes confirm non-renormalization theorems for 0-, 1-, 2-, and 3-point functions at two-loop order?
• RQ5 How can the 3-loop superstring measure be constructed using factorization constraints and modular invariance?

Key findings

• The 4-point function for massless NS bosons is computed explicitly and found to be gauge-slice independent, with a surprisingly simple form.
• The 0-, 1-, 2-, and 3-point functions for massless NS states vanish identically at two-loop order, confirming non-renormalization theorems from first principles.
• The measure $d\mu_2[\delta]$ is constructed as $\Xi_6[\delta](\Omega)$, a modular covariant form of weight 6, expressible in terms of $\vartheta$-constants.
• Gauge-slice independence is proven by showing that changes in gauge slice produce de Rham-exact forms at all insertion points, ensuring invariance of the amplitude.
• The 3-loop superstring measure is constrained by factorization: in the limit $t \to 0$, $\Xi_6[\Delta](\Omega^{(3)})$ must reduce to $\eta(\Omega^{(1)})^{12} \Xi_6[\delta](\Omega^{(2)})$, guiding candidate constructions.
• The formalism is extended to odd spin structures, where a new superholomorphic form $\hat{\omega}_0$ emerges, complicating the structure of the super period matrix.