Combinatorics and Topology of Kawai-Lewellen-Tye Relations (1706.08527v1)

Abstract: We revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the underlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are the $\alpha'$-corrected bi-adjoint scalar amplitudes recently defined by the author [arXiv:1610.04230]. We show that they naturally arise as intersection numbers of twisted cycles. In this work we focus on the combinatorial and topological description of twisted cycles relevant for string theory amplitudes. In this setting, each twisted cycle is a polytope, known in combinatorics as the associahedron, together with an additional structure encoding monodromy properties of string integrals. In fact, this additional structure is given by higher-dimensional generalizations of the Pochhammer contour. An open string amplitude is then computed as an integral of a logarithmic form over an associahedron. We show that the inverse of the KLT kernel can be calculated from the knowledge of how pairs of associahedra intersect one another in the moduli space. In the field theory limit, contributions from these intersections localize to vertices of the associahedra, giving rise to the bi-adjoint scalar partial amplitudes.

• The paper demonstrates that KLT relations can be derived from twisted period relations within twisted de Rham theory.
• It introduces a novel approach by mapping string amplitudes to associahedra in moduli spaces through combinatorial topology and intersection numbers.
• The findings offer deeper insights for efficiently computing string scattering processes and bridging connections with field-theoretic calculations.

An Overview of Combinatorics and Topology of Kawai–Lewellen–Tye Relations

The paper by Sebastian Mizera offers an in-depth exploration of Kawai–Lewellen–Tye (KLT) relations within the context of string theory amplitudes, elucidating their algebro-topological underpinnings through twisted de Rham theory. To enhance our understanding of these scattering amplitude relations, the work explores the combinatorial and topological frameworks that emerge naturally from string theory, providing a comprehensive and novel approach to resolving the complexities of KLT relations.

Theoretical Framework and Key Results

The primary objective of this paper is to demonstrate that the KLT relations can be derived from twisted period relations within the scope of twisted de Rham theory. This theory accommodates multi-valued functions and their relationships, leveraging tools from algebraic topology. Here, string amplitude computations are translated into pairings between twisted cycles and twisted cocycles, revealing the intrinsic structures of string theory amplitudes beyond conventional interpretations.

A significant result highlighted in the paper is the mapping of string amplitudes to polytope structures, specifically associahedra, within moduli spaces. This correspondence involves expressing both open and closed string amplitudes as integrals over these spaces, leading to the discovery of intersection numbers as the pivotal elements that connect open and closed string amplitudes via the inverse KLT kernel. The intricate polytope interpretations further advance the calculations of string scattering amplitudes by elucidating the geometric and combinatorial properties inherent in their structure.

Implications in Mathematical and Physical Contexts

In the mathematical context, this work enriches the dialogue by integrating combinatorial aspects of geometrical topology with the existing algebraic frameworks used to delineate string theory amplitudes. The sophisticated linkage between string amplitudes and the geometry of moduli spaces analyzed here can pave the way to exploring higher-loop generalizations, enhancing our comprehension of the broader string-theoretic landscape.

From a practical perspective, the findings offer a deeper understanding of string scattering processes, with implications that extend to field theories, rendered by bridging concepts such as twisted homologies into the field of Feynman-like calculations. These insights are expected to facilitate more efficient computations of string scattering processes while also providing clarity to the persistent field-theory limits of string theories, particularly through the lens of the newly evaluated inverse KLT kernel.

Future Direction and Speculations

Looking ahead, this paper points towards potential explorations into higher-genus string amplitudes and their twisted de Rham theory representations. This avenue holds promise for uncovering further structural relationships that could revolutionize the efficiency of string amplitude computations and their applications in theoretical physics. Additionally, investigating the motivic structures of string amplitudes in conjunction with the mathematical formulations presented here could unveil new insights into the deep algebraic properties inherent in string theories.

Moreover, the paper’s approach could ultimately bring forth new tools and methodologies applicable not only in string theory but also in the examination of other physical theories where multi-valued function integrals play a crucial role. The combinatorial framework provided here has the potential to transcend the existing boundaries of amplitude calculations by offering a structured and geometrically enriched perspective that could ripple through various facets of both mathematical physics and theoretical explorations in string theory.

In conclusion, the paper by Mizera illuminates the intrinsic connections between string theoretic phenomena and their algebraic and topological characterizations, steering the community towards novel interpretations and practical implementations within the field of string theory and beyond.