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Carroll symmetries in field theory and gravity

Florian Ecker

Abstract

This thesis explores several facets of Carroll symmetries through their applications to field theories and gravity. The geometric description of curved Carroll manifolds is developed from a Cartan-geometric viewpoint, reviewed at the outset. On these backgrounds, we study various field theories, including scalar and vector Carroll swiftons. Imposing causality and locality, we derive a universal sector of the commutators between Carroll stress-energy tensor components valid for any Carroll quantum field theory. In two dimensions, we confirm the connection to holography by showing that a Carroll boost anomaly gives rise to additional Schwinger-like terms in these brackets, sourcing the familiar central extensions of the asymptotic symmetries of three-dimensional asymptotically flat Einstein gravity. Afterwards, we come to theories of Carroll gravity which, as we argue, provide a valuable playground for understanding quantum gravity in a specific scaling limit which we refer to as the tantum gravity limit. At first, we review Carroll gravity in general dimensions and subsequently restrict to two spacetime dimensions where we introduce Carroll dilaton gravity. We define Carroll black holes as massive vacuum solutions to these theories that admit well-defined thermodynamic properties but have a Carroll extremal surface instead of an event horizon. After investigating several models and their solutions we finally add quantum matter to these backgrounds and study how the thermodynamic properties of a Carroll black hole reflect in its vacuum states. For the Carroll-Schwarzschild black hole we find a non-vanishing asymptotic energy density. We refer to this phenomenon as the Carroll-Hawking effect.

Carroll symmetries in field theory and gravity

Abstract

This thesis explores several facets of Carroll symmetries through their applications to field theories and gravity. The geometric description of curved Carroll manifolds is developed from a Cartan-geometric viewpoint, reviewed at the outset. On these backgrounds, we study various field theories, including scalar and vector Carroll swiftons. Imposing causality and locality, we derive a universal sector of the commutators between Carroll stress-energy tensor components valid for any Carroll quantum field theory. In two dimensions, we confirm the connection to holography by showing that a Carroll boost anomaly gives rise to additional Schwinger-like terms in these brackets, sourcing the familiar central extensions of the asymptotic symmetries of three-dimensional asymptotically flat Einstein gravity. Afterwards, we come to theories of Carroll gravity which, as we argue, provide a valuable playground for understanding quantum gravity in a specific scaling limit which we refer to as the tantum gravity limit. At first, we review Carroll gravity in general dimensions and subsequently restrict to two spacetime dimensions where we introduce Carroll dilaton gravity. We define Carroll black holes as massive vacuum solutions to these theories that admit well-defined thermodynamic properties but have a Carroll extremal surface instead of an event horizon. After investigating several models and their solutions we finally add quantum matter to these backgrounds and study how the thermodynamic properties of a Carroll black hole reflect in its vacuum states. For the Carroll-Schwarzschild black hole we find a non-vanishing asymptotic energy density. We refer to this phenomenon as the Carroll-Hawking effect.
Paper Structure (105 sections, 2 theorems, 604 equations, 10 figures, 3 tables)

This paper contains 105 sections, 2 theorems, 604 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Black holes and a first glimpse of Carroll symmetry
  3. The many faces of Carroll symmetry
  4. Holography
  5. Carroll field theories
  6. Carroll gravity
  7. Carroll hydrodynamics
  8. Condensed matter systems
  9. Outline of this thesis
  10. Carroll geometry and symmetries
  11. Carroll algebra as contraction of the Poincaré algebra
  12. Adding a cosmological constant
  13. Minkowski spacetime in the limit ctozero
  14. Gauging kinematical Lie algebras
  15. Klein geometry
  16. ...and 90 more sections

Key Result

Lemma 2.3.1

Given a principal $G$-bundle and an associated vector bundle $(P\times _\rho V)$ there is a one-to-one correspondence between sections of $(P\times _\rho V)$ and $G$-equivariant, $V$-valued functions on $P$, where the property of $G$-equivariance is defined by $\phi (u\cdot g)=\rho (g^{-1})\phi (u)$.

Figures (10)

  • Figure 1: The lightcones of Lorentzian spacetime close in the Carroll limit $c\to 0$.
  • Figure 2: The choice of a principal bundle connection allows to decompose each vector $X$ at some point $u\in \pi ^{-1}(p)$ into its horizontal and vertical components. We also drew an example for the right action of $G$ on a fiber $\pi ^{-1}(p')$.
  • Figure 3: Bronstein cube with tantum gravity limit highlighted as TG and its antipodal as TG$^\ast$.
  • Figure 4: Left (orange): Lorentzian theories. Right (yellow): Carroll theories.
  • Figure 5: Kruskal diagram of Lorentzian eternal black hole
  • ...and 5 more figures

Theorems & Definitions (13)

  • Example 2.3.1
  • Definition 2.3.1: Kinematical Lie algebra
  • Definition 2.3.2: Homogeneous kinematical spacetime
  • Example 2.3.2
  • Example 2.3.3
  • Definition 2.3.3: Vector bundle
  • Definition 2.3.4: Principal bundle
  • Definition 2.3.5: Connection on a principal bundle
  • Lemma 2.3.1
  • Definition 2.3.6: Solder form
  • ...and 3 more