Infinite Distances in Field Space and Massless Towers of States (1802.08264v2)

Abstract: It has been conjectured that in theories consistent with quantum gravity infinite distances in field space coincide with an infinite tower of states becoming massless exponentially fast in the proper field distance. The complex-structure moduli space of Calabi-Yau manifolds is a good testing ground for this conjecture since it is known to encode quantum gravity physics. We study infinite distances in this setting and present new evidence for the above conjecture. Points in moduli space which are at infinite proper distance along any path are characterised by an infinite order monodromy matrix. We utilise the nilpotent orbit theorem to show that for a large class of such points the monodromy matrix generates an infinite orbit within the spectrum of BPS states. We identify an infinite tower of states with this orbit. Further, the theorem gives the local metric on the moduli space which can be used to show that the mass of the states decreases exponentially fast upon approaching the point. We also propose a reason for why infinite distances are related to infinite towers of states. Specifically, we present evidence that the infinite distance itself is an emergent quantum phenomenon induced by integrating out at one-loop the states that become massless. Concretely, we show that the behaviour of the field space metric upon approaching infinite distance can be recovered from integrating out the BPS states. Similarly, at infinite distance the gauge couplings of closed-string Abelian gauge symmetries vanish in a way which can be matched onto integrating out the infinite tower of charged BPS states. This presents evidence towards the idea that also the gauge theory weak-coupling limit can be thought of as emergent.

• The paper rigorously applies advanced geometric and algebraic techniques to show that infinite distances in moduli spaces correspond to an infinite tower of exponentially light BPS states.
• It uses the Nilpotent Orbit and Sl₂-Orbit theorems to analyze infinite monodromy matrices and the behavior of period vectors near infinite loci.
• The findings reinforce the Swampland Distance Conjecture, offering new insights into gauge symmetry emergence and implications for quantum gravity.

Essay on "Infinite Distances in Field Space and Massless Towers of States"

The paper presented in the paper "Infinite Distances in Field Space and Massless Towers of States" explores crucial aspects of quantum gravity by examining the moduli spaces of Calabi-Yau manifolds, formulated within the framework of the Swampland Distance Conjecture (SDC). The central thesis is a geometric and algebraic examination of infinite distances in field spaces—a pivotal notion in string theory facilitating potential insights into constraints imposed by quantum gravity. This conjecture asserts that such infinite distances correspond to an infinite tower of states that become massless, which is explored through mathematical rigor and illustrative case studies.

Key Contributions and Findings

The authors methodically investigate infinite distance loci in moduli spaces and the occurrence of massless BPS states in Type IIB string theory on Calabi-Yau threefolds. They apply advanced mathematical formulations such as the Nilpotent Orbit and $Sl_2$-Orbit theorems of Schmid, providing a solid basis for examining infinite order monodromies. These mathematical structures drive the analysis by delineating the behavior of period vectors and field metrics near infinite loci. In particular, crucial algebraic properties of infinite monodromy matrices come to light, which have symposia with the presence of massless towers.

A pivotal achievement of the paper is the rigorous demonstration that infinite distance manifests geometrically as a result of integrating out an infinite tower of states. Structurally, the analysis is centered upon discrete monodromy transformations and their action on state charges. The BPS mass and an electric-magnetic duality aspect play influential roles, primarily because the BPS states behave according to well-established central charge relations in $\mathcal{N}=2$ supersymmetric settings. The authors show that the mass of these BPS states falls exponentially upon approaching infinite distance loci, consistent with the SDC.

Theoretical and Practical Implications

The paper provides not only a refined understanding of infinite distances but also aligns these profound quantum phenomena with practical implications for gauge theories and field-theoretic models. The work reiterates the fundamental conjecture that any emergent global symmetry must coincide with the compactification of gravity to infinite distance loci, while the distance-dependent behavior of mass spectra ties these local symmetries back to classical string theory groundings.

Furthermore, the analysis advises significant insights into the Weak Gravity Conjecture (WGC), revealing that any decline in gauge coupling, especially in the vanishing limit, is matched by a characteristic light tower of states, hinting at deeper symmetries and unifying structures over the landscape of string theory. This supports a vision of symmetry and unification for various conjectures, such as emergent gauge symmetries and magnetically charged states within effective quantum field theories.

Future Directions in AI and Quantum Gravity

The research establishes a theoretical scaffold for experimental tests and practical implementations through collaborative efforts across quantum gravity and machine learning. AI-driven data exploration could supplement these theoretical results by identifying pattern regularities across larger vacua datasets. The integration of AI can also refine predictions of stability and identify new symmetries in computational geometry, opening new chapters in field space topology and its correspondence to physical observables.

In conclusion, the paper represents a significant contribution to our understanding of quantum gravity. By revealing the intricate relationship between infinite distances in field space and the consequent hierarchy of massless states, it broadens the traditional dimensions of string theoretical research, setting a new foundation for many future studies.