Twistorial chiral algebras in higher dimensions (2501.09627v2)

Abstract: In four spacetime dimensions, the classically integrable self-dual sectors of gauge theory and gravity have associated chiral algebras, which emerge naturally from their description in twistor space. We show that there are similar chiral algebras associated to integrable sectors of gauge theory and gravity whenever the spacetime dimension is an integer multiple of four. In particular, the hyperk\"ahler sector of gravity and the hyperholomorphic sector of gauge theory in $4m$-dimensions have well-known twistor descriptions giving rise to chiral algebras. Using twistor sigma models to describe these sectors, we demonstrate that the chiral algebras in higher-dimensions also arise as soft symmetry algebras under a certain notion of collinear limit. Interestingly, the chiral algebras and collinear limits in higher-dimensions are defined on the 2-sphere, rather than the full celestial sphere.

• The paper extends 4D twistorial chiral algebras to 4m dimensions, identifying integrable sectors in hyperkähler gravity and hyperholomorphic gauge theories.
• It employs twistor sigma models to connect geometric twistor constructions with loop algebras, providing a rigorous framework for symmetry analysis.
• The work shows that collinear limits in 4m-dimensional spacetimes yield infinite-dimensional symmetry algebras, impacting celestial holography and scattering theory.

Analysis of "Twistorial Chiral Algebras in Higher Dimensions"

The paper "Twistorial Chiral Algebras in Higher Dimensions" by Tim Adamo and Iustin Surubaru explores the extension of twistorial chiral algebras, initially observed in four dimensions, to higher spacetime dimensions that are integer multiples of four. These extensions are crucial in understanding integrable sectors of gauge theory and gravity in such dimensions, employing the framework of twistor theory.

Overview and Context

In four-dimensional spacetime, self-dual sectors of gauge theory and gravity admit associated chiral algebras, a phenomenon deeply rooted in the structure of twistor space. Twistor theory, originally developed to provide a geometric interpretation of these self-dual sectors, facilitates the identification of symmetries inherent in these theories, manifesting as infinite-dimensional chiral algebras.

The authors focus on similar constructions in $4m$-dimensional spacetimes, where $m$ is a positive integer, identifying hyperkähler geometry in gravity and hyperholomorphic gauge fields as natural generalizations of the self-dual conditions to higher dimensions. The central thesis is that the symmetries of these higher-dimensional sectors can also be described by infinite-dimensional chiral algebras.

Key Results and Methodology

1. Higher Dimensional Twistor Theory: The authors extend classical twistor constructions to $4m$ dimensions, demonstrating that hyperkähler and hyperholomorphic conditions can be effectively described in twistor space. They exploit the intrinsic twistor structures—complex manifolds with fiber bundle constructs—to extend the chiral algebras known from 4-dimensional spacetime.
2. Chiral Algebras in Twistor Space: By examining twistor spaces $\PT$ of flat $\C^{4m}$, they identify the chiral algebra associated with linearized hyperkähler gravity perturbations as the loop algebra of Hamiltonian vector fields, denoted $\cL\mathfrak{ham}(\C^{2m})$. Similarly, for the hyperholomorphic sector, the chiral algebra is given by $\cL\mathfrak{g}[\C^{2m}]$, the loop algebra of polynomial maps into the Lie algebra of the gauge group.
3. Twistor Sigma Models: The paper leverages twistor sigma models to provide a dynamical perspective on these theories. These models lead to charges that represent modes of the chiral algebras and exhibit operator product expansions (OPEs) consistent with those found in vertex operator algebras.
4. Collinear Limits and Celestial Sphere: In $4m$ dimensions, the usual concept of a celestial sphere is modified, recognizing emergent two-dimensional structures within this higher-dimensional context. The paper outlines how the infinite-dimensional symmetry algebras manifest in collinear limits, akin to holomorphic OPE limits on a celestial $S^2$. This leads to the realization that higher-dimensional chiral algebras emerge as symmetries in this limit.

Implications and Speculations for Future Research

The paper significantly extends our understanding of the symmetries in higher-dimensional gauge theories and gravity, potentially influencing how scattering processes are understood in these dimensions. The results have implications for celestial holography and the AdS/CFT correspondence, offering possible routes to a higher-dimensional generalization of these concepts.

Practically, this work could inform the development of new string theory models, focused on twistor space as a target, further broadening the landscape of theoretical physics frameworks. The results suggest new ways to incorporate twistor theory into non-trivial backgrounds, including those with curvature, supersymmetry, and higher-spin fields.

Conclusion

This paper presents a rigorous extension of four-dimensional twistorial chiral algebras to higher dimensions, leveraging intricate twistor constructions to reveal symmetry algebras in $4m$-dimensional hyperkähler and hyperholomorphic sectors. The findings pave the way for further theoretical developments that could unify disparate areas of theoretical physics through the rich geometric framework provided by twistor theory. Future research is necessary to explore further implications and applications of these results in higher-dimensional quantum field theories and string theory.