Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation (1106.2645v1)

Abstract: In this paper the pure spinor formalism is used to obtain a compact expression for the superstring N-point disk amplitude. The color ordered string amplitude is given by a sum over (N-3)! super Yang-Mills subamplitudes multiplied by multiple Gaussian hypergeometric functions. In order to obtain this result, the cohomology structure of the pure spinor superspace is exploited to generalize the Berends-Giele method of computing super Yang-Mills amplitudes. The method was briefly presented in [1], and this paper elaborates on the details and contains higher-rank examples of building blocks and associated cohomology objects. But the main achievement of this work is to identify these field-theory structures in the pure spinor computation of the superstring amplitude. In particular, the associated set of basis worldsheet integrals is constructively obtained here and thoroughly investigated together with the structure and properties of the amplitude in [2].

• The paper derives a concise expression for the complete N-point disk amplitude in superstring theory using the pure spinor formalism to simplify complex calculations.
• It connects superstring amplitudes to super Yang-Mills subamplitudes modulated by Gaussian hypergeometric functions and employs recursive, BRST-invariant building blocks.
• The work enhances calculability for string amplitudes, bridges higher-dimensional theories to 4D, and provides a foundation for exploring loop-level processes and string phenomenology.

Insights into the Complete N–Point Superstring Disk Amplitude

The paper entitled "Complete N–Point Superstring Disk Amplitude I: Pure Spinor Computation" by Carlos R. Mafra, Oliver Schlotterer, and Stephan Stieberger, outlines significant advancements in the computation of superstring disk amplitudes using the pure spinor formalism. The paper's primary objective is to derive a concise expression for the N–point disk amplitude in superstring theory, emphasizing the application of the pure spinor formalism to harness super-Poincaré invariance and simplify calculations that involve substantial algebraic manipulations inherent in previous formalisms.

Key Contributions and Methodology

A central contribution of this work is the formulation of a closed expression for the superstring N–point disk amplitude through the development of systematic approaches that connect superstring amplitudes to super Yang-Mills (SYM) subamplitudes, modulated by Gaussian hypergeometric functions. The authors employ the cohomology structure of the pure spinor superspace to extend the Berends–Giele method—a vital step that facilitates the manipulation and computation of polarizations and momentum dependencies in higher dimensions.

The authors focus on constructing a series of BRST-invariant building blocks, denoted as $T_{123...p}$, which play a pivotal role in expressing the amplitude's components and relate to cubic Feynman diagrams. This recursive formulation serves to simplify the algebraic complexity associated with solving for the N–point tree-level amplitude.

Strong Numerical Results

The paper provides substantial analytical results delineating how the N–point amplitude can be represented in terms of a combination of SYM field-theory amplitudes and pure spinor superspace expressions. In particular, the N–point amplitude is expressed as a sum over (N − 3)! permutations, satisfying the BCJ (Bern, Carrasco, and Johansson) relations, which indicate the degree of structure that can manifest in string theoretic descriptions as proposed and proved in previous work [14-16].

Theoretical Implications and Future Directions

Practically, this research enhances the calculability of string amplitudes necessary for quantum field theory and string theory, characterized by their reliance on fewer supersymmetric constraints. Theoretically, it bridges the gap between higher-dimensional supersymmetric theories and four-dimensional effective theories, presenting avenues for further explorations in string theory when investigating novel compactifications and their ramifications in particle physics.

This work's application of hypergeometric functions in linking field-theory subamplitudes to string theory represents a substantial step in unifying disparate components of theoretical physics into a comprehensive system.

Outlook on Future Developments

Looking ahead, the methods developed herein are potentially applicable in further refining the efficacy of string amplitude computations in more complex settings, such as loop-level processes. As computational techniques advance, extending the pure spinor formalism into multi-loop domains entails significant opportunities for verifying conjectures and establishing foundational frameworks in string phenomenology.

Researchers in the field are expected to further exploit recursive techniques and cohomological methods, as exemplified in this paper, to tackle the challenges present in string topology, non-linear sigma models, and quantum gravity lexicons, where analogous mathematical structures inform the dynamics and solutions of these complex systems. The precise role of pure spinors and their versatile applications warrants continued exploration, spurred by the foundational insights provided by this work.