Open & Closed vs. Pure Open String Disk Amplitudes (0907.2211v2)

Abstract: We establish a relation between disk amplitudes involving N_o open and N_c closed strings and disk amplitudes with only N_o+2N_c open strings. This map, which represents a sort of generalized KLT relation on the disk, reveals important structures between open & closed and pure open string disk amplitudes: it relates couplings of brane and bulk string states to pure brane couplings. On the string world-sheet this becomes a non-trivial monodromy problem, which reduces the disk amplitude of N_o open and N_c closed strings to a sum of many color ordered partial subamplitudes of N_o+2N_c open strings. This sum can be further reduced to a sum over (N_o+2N_c-3)! subamplitudes of N=N_o+2N_c open strings only. Hence, the computation of disk amplitudes involving open and closed strings is reduced to computing these subamplitudes in the open string sector. In this sector we find a string theory generalization and proof of the Kleiss-Kuijf and Bern-Carrasco-Johanson relations: All order alpha' identities between open string subamplitudes are derived, which reproduce these field-theory relations in the limit alpha'->0. These identities allow to reduce the number of independent subamplitudes of an open string N-point amplitude to (N-3)!. This number is identical to the dimension of a minimal basis of generalized Gaussian hypergeometric functions describing the full N-point open string amplitude.

• The paper establishes a generalized KLT relation on the disk that maps mixed open and closed string amplitudes to pure open computations.
• It uses a monodromy approach on the world-sheet to reduce complex calculations to a minimal basis via BCJ and KK relations, yielding (N-3)! independent subamplitudes.
• The findings streamline string interaction computations, offering practical insights for gauge theories and supergravity multiplet simulations.

Overview of "Open & Closed vs. Pure Open String Disk Amplitudes"

In the paper "Open & Closed vs. Pure Open String Disk Amplitudes," the author explores the relations between open and closed string disk amplitudes and pure open string disk amplitudes, presenting a generalized Kawai-Lewellen-Tye (KLT) relation on the disk. The paper provides significant insights into the structures of string theory amplitudes involving open and closed strings by reducing these complex interactions to simpler computations in the pure open string sector.

Key Concepts and Methods

The core achievement of the paper is establishing a correspondence that maps disk amplitudes involving both open and closed strings to amplitudes composed solely of open strings. This mapping demonstrates a fundamental relationship between the couplings of brane (open strings) and bulk (closed strings) states to those of pure brane couplings, which are more straightforward to compute.

The author approaches this problem via a monodromy problem on the string world-sheet, which essentially transforms the amplitude calculations into a more manageable sum over color-ordered partial amplitudes of open strings. These can be further reduced to a minimal basis of independent subamplitudes, characterized by the Bern-Carrasco-Johansson (BCJ) and Kleiss-Kuijf (KK) relations. These relations drastically reduce the number of necessary computations by offering an elegant structure to the string theory calculations.

Numerical Results and Implications

A significant numerical outcome from the paper is that the number of independent subamplitudes for an N-point open string amplitude is reduced to $(N-3)!$, aligning with the dimensional basis of generalized Gaussian hypergeometric functions. This reduction is pivotal, as it implies a smaller computational burden in exploring string theory dynamics compared to a naive approach.

The implications of this research are profound, primarily because it provides a clearer computational framework for understanding gauge theories and supergravity multiplets from string theory perspectives. The techniques and results can advance our ability to simulate and predict the behavior of fundamental forces in the universe, leveraging string theory's unique structures.

Theoretical and Practical Impact

Theoretically, the generalized KLT relation on the disk contributes to a deeper understanding of the dualities and symmetries within string theory. Specifically, it highlights how complex closed string interactions, typically harder to compute due to their inherent symmetries and coupling across the world-sheet, can be expressed through the simpler open string interactions which involve fewer entanglements.

Practically, these findings could have ramifications for computing string amplitudes in higher-dimensional theories and provide a groundwork for future explorations into other non-trivial monodromy problems they might encounter. The methodologies developed here could aid in refining Lagrangians in compactified dimensions or for D-brane world-volume dynamics, offering further insight into phenomena such as brane dynamics and the holographic principle.

Future Directions

The advancements presented could be extended to consider amplitudes on more complex surfaces or at higher-loop orders, offering potential new insights into higher-dimensional string theory configurations and their compactifications. Furthermore, similar systematic reductions could be explored for other types of Riemann surfaces or within M-theory contexts.

The paper lays a foundational stone towards refining our computational techniques in string theory, which might bridge some gaps between theoretical predictions and observable phenomena in high-energy physics, potentially offering a path to explore conjectures about the unification of forces.