[Paper Review] Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors

Eric D’Hoker, Carlos R. Mafra+2 more|ePrints Soton (University of Southampton)|Jun 9, 2020

Black Holes and Theoretical Physics94 references4 citations

TL;DR

This paper constructs the full two-loop five-point amplitudes for Type II and Heterotic superstrings using a hybrid approach combining chiral splitting and the pure spinor formalism. It derives convergent genus-two integrals over moduli space and shows perfect agreement with Type II supergravity in the $α'\to 0$ limit, validating the method and providing a foundation for higher-loop amplitudes and S-duality tests.

ABSTRACT

The full two-loop amplitudes for five massless states in Type~II and Heterotic superstrings are constructed in terms of convergent integrals over the genus-two moduli space of compact Riemann surfaces and integrals of Green functions and Abelian differentials on the surface. The construction combines elements from the BRST cohomology of the pure spinor formulation and from chiral splitting with the help of loop momenta and homology invariance. The $α' o 0$ limit of the resulting superstring amplitude is shown to be in perfect agreement with the previously known amplitude computed in Type~II supergravity. Investigations of the $α'$ expansion of the Type~II amplitude and comparisons with predictions from S-duality are relegated to a first companion paper. A construction from first principles in the RNS formulation of the genus-two amplitude with five external NS states is relegated to a second companion paper.

Motivation & Objective

• To construct the full two-loop five-point amplitudes for Type II and Heterotic superstrings using a hybrid method combining chiral splitting and the pure spinor formalism.
• To ensure BRST invariance and homology invariance in the amplitude construction, crucial for modular invariance and physical consistency.
• To verify that the $α'\to 0$ limit of the Type II amplitude reproduces the known supergravity amplitude, confirming consistency with low-energy effective field theory.
• To lay the groundwork for future investigations into the $α'$ expansion and S-duality predictions in Type II string theory.
• To provide a first-principles construction that avoids the complications of the $b$-ghost in higher-genus pure spinor formulations.

Proposed method

• The amplitude is constructed using chiral splitting to separate holomorphic and anti-holomorphic parts of the worldsheet correlation functions on genus-two Riemann surfaces.
• The pure spinor formalism is employed to ensure manifest space-time supersymmetry and to handle the BRST cohomology of the superstring amplitude.
• The method relies on loop momenta and homology invariance to ensure modular invariance of the integrand over the genus-two moduli space.
• Key components include the use of prime forms, Abelian differentials, and Green functions on the Riemann surface to build the chiral amplitude.
• The construction involves decomposing the amplitude into vector and scalar blocks, with the scalar block expressed in terms of two-particle superfields.
• BRST invariance and homology invariance are explicitly manifested in the correlator structure, ensuring gauge invariance and modular invariance.

Experimental results

Research questions

• RQ1 How can two-loop five-point amplitudes in superstring theory be consistently constructed using the pure spinor formalism and chiral splitting?
• RQ2 Does the $α'\to 0$ limit of the two-loop Type II five-point amplitude reproduce the known supergravity amplitude?
• RQ3 How can BRST invariance and homology invariance be simultaneously enforced in the chiral amplitude on genus-two Riemann surfaces?
• RQ4 What is the role of the $b$-ghost and zero modes in the genus-two amplitude, and how can they be consistently handled in the pure spinor approach?
• RQ5 Can the resulting amplitude be used to test S-duality predictions and explore non-renormalization theorems in string theory?

Key findings

• The two-loop five-point amplitude for Type II superstrings is constructed as a convergent integral over the genus-two moduli space and integrals of Green functions and Abelian differentials on the Riemann surface.
• The $α'\to 0$ limit of the amplitude precisely reproduces the five-point supergravity amplitude, confirming consistency with effective field theory.
• The chiral amplitude is shown to be BRST invariant and homology invariant, ensuring modular invariance and physical consistency.
• The construction successfully absorbs zero modes via the $b$-ghost, resolving potential anomalies in the pure spinor formulation at genus two.
• The amplitude is expressed in terms of prime forms and theta functions, with a structure that generalizes to higher-point amplitudes.
• The method provides a first-principles construction that avoids the complications of the $b$-ghost in higher-genus RNS formulations.