[Paper Review] Kac-Moody instantons in space-time foam as an alternative solution to the black hole information paradox

Andrea Addazi, Pisin Chen+2 more|arXiv (Cornell University)|Jul 2, 2017

Computability, Logic, AI Algorithms2 references3 citations

TL;DR

This paper proposes that Kac-Moody instantons in quantum space-time foam—specifically in $S^2 \times S^2$ gravitational bubbles—provide a new mechanism for resolving the black hole information paradox. By showing that self-dual Yang-Mills and gravitational fields in foam backgrounds generate an infinite-dimensional Kac-Moody algebra without central charge, the authors identify instantonic moduli as quantum hairs that encode information, subtly compatible with the no-hair theorem and distributed throughout the foam rather than at the horizon.

ABSTRACT

Hawking, Perry and Strominger recently invoked BMS symmetry charges in an attempt to resolve the black hole information paradox. Here we propose an alternative scenario that is based on the Kac-Moody charges. We show that the role of BMS charges can be played by an infinite set of symmetries that emerge from the space-time foam predicted by quantum gravity. Specifically, we focus on Yang-Mills fields embedded in the gravity described by the Holst formulation, and argue that the Yang-Mills and gravitational self-duality conditions in space-time bubbles are related to a new infinite dimensional global symmetry, hidden in the Lagrangian. Such a symmetry is manifested by the Kac-Moody algebra, with zero central charges. This implies the existence, in the space-time foam, of an infinite number of different instantons that are interconnected by the Kac-Moody symmetry. These modes puncture the horizons of the building block of the space-time bubbles. On the other hand, the same Kac-Moody symmetry is retried in non-perturbative regime at the level of the gravitational quantum loops. The new result carries consequences on the no-hair theorem and on the study of quantum black holes. In particular, instantonic moduli of the Kac-Moody charges are quantum hairs encoding the missing black hole information, subtly compatible with the no hair theorem.

Motivation & Objective

• To address the black hole information paradox by proposing an alternative to BMS symmetry-based solutions.
• To investigate the emergence of infinite-dimensional symmetries in quantum space-time foam at mesoscopic scales near the Planck scale.
• To establish a connection between self-dual Yang-Mills theories on complexified spacetime and Kac-Moody algebras in the context of gravitational bubbles.
• To demonstrate that instantonic moduli in space-time foam can act as quantum hairs, preserving information without violating the no-hair theorem.
• To explore the implications of Kac-Moody symmetry for the quantum structure of black holes and the finiteness of Hilbert space entropy.

Proposed method

• Analyze self-dual Yang-Mills and Holst gravity actions in the background of $S^2 \times S^2$ space-time bubbles, representing virtual black hole–white hole pairs.
• Identify the emergence of an infinite-dimensional Kac-Moody algebra from the self-duality condition in complexified Euclidean spacetime, with zero central charge.
• Relate the Kac-Moody symmetry to an infinite family of instantons with the same standard moduli, interconnected via Kac-Moody transformations.
• Study the quantum behavior of zero modes $a$ perturbing instantonic backgrounds, showing they acquire a mass gap due to non-perturbative interactions.
• Connect the Kac-Moody level $M$ to energy levels $E_M \sim M\Lambda$, where $\Lambda$ is the Yang-Mills confinement scale.
• Relate the Hilbert space dimension and entropy to the Chern-Simons theory on the horizon boundary, with area gap proportional to $l_P^2$.

Experimental results

Research questions

• RQ1 Can Kac-Moody symmetries in space-time foam provide a viable alternative to BMS symmetries in resolving the black hole information paradox?
• RQ2 How do self-dual Yang-Mills and gravitational fields in $S^2 \times S^2$ bubbles lead to an infinite-dimensional Kac-Moody algebra without central charge?
• RQ3 What is the role of instantonic moduli in encoding quantum information in a way compatible with the no-hair theorem?
• RQ4 How do quantum fluctuations in the foam background lead to a mass gap for zero modes in the instantonic sector?
• RQ5 Can the Hilbert space of quantum black holes be finite and regulated by the Kac-Moody level and horizon area gap?

Key findings

• The self-duality condition in Yang-Mills and Holst gravity on $S^2 \times S^2$ bubbles leads to an infinite-dimensional Kac-Moody algebra with zero central charge.
• An infinite number of distinct instantons with the same standard moduli are interconnected by Kac-Moody symmetry, forming a new class of quantum hair.
• Zero modes $a$ perturbing the instantonic background acquire a mass gap $\sim \Lambda$, the confinement scale of the $SU(N)$ Yang-Mills theory.
• Energy levels of Kac-Moody excitations scale as $E_M \sim M\Lambda$, with $M$ the level, indicating a discrete spectrum due to space-time discretization.
• The Hilbert space dimension and associated entropy are finite, regulated by the Chern-Simons theory on the horizon boundary, with area gap $\propto l_P^2$.
• Quantum information is stored not at the horizon but throughout the foam, in virtual instantonic modes, offering a non-local resolution of the information paradox.