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Duality and Axion Wormholes

Edward Witten

TL;DR

This work clarifies how Euclidean axion wormholes mediate a dual description between a massless scalar φ and a two-form gauge field B in four dimensions. The authors show that the classical wormhole exists in the B-description, while the φ-language requires a Poisson-resummed, non-semiclassical treatment in the throat, yet yields the same low-energy effective action. They establish a precise operator mapping, Dirac quantization constraints, and a bilocal effective action that encodes wormhole-induced correlations between distant spacetime regions, with dual descriptions connected by Poisson resummation. The discussion highlights the role of zero-modes, differential-character formalisms, and counterterms needed to equate the scalar and B-field path integrals, advancing understanding of how wormholes influence low-energy constants and quantum vacua in gravity-matter systems.

Abstract

The prototype of a Euclidean wormhole solution of Einstein gravity coupled to matter is the axion wormhole in four spacetime dimensions. In this primarily expository article, we spell out some details about this construction. The axion wormhole has a semiclassical description, found in the original paper [1], in which the matter system is a two-form gauge field B with three-form field strength H=dB. The two-form is dual to a massless scalar, but the wormhole does not have a semiclassical description in terms of the scalar. There is no contradiction here as the duality between the two-form and the scalar is not a simple transformation of classical fields but involves, in Euclidean signature, a Poisson resummation of the sum over fluxes. Because of the need for this Poisson resummation, the scalar field cannot be treated semiclassically in the wormhole throat. Nonetheless, it is straightforward to compute the effective action derived from the wormhole in the scalar (or two-form) language, recovering standard claims.

Duality and Axion Wormholes

TL;DR

This work clarifies how Euclidean axion wormholes mediate a dual description between a massless scalar φ and a two-form gauge field B in four dimensions. The authors show that the classical wormhole exists in the B-description, while the φ-language requires a Poisson-resummed, non-semiclassical treatment in the throat, yet yields the same low-energy effective action. They establish a precise operator mapping, Dirac quantization constraints, and a bilocal effective action that encodes wormhole-induced correlations between distant spacetime regions, with dual descriptions connected by Poisson resummation. The discussion highlights the role of zero-modes, differential-character formalisms, and counterterms needed to equate the scalar and B-field path integrals, advancing understanding of how wormholes influence low-energy constants and quantum vacua in gravity-matter systems.

Abstract

The prototype of a Euclidean wormhole solution of Einstein gravity coupled to matter is the axion wormhole in four spacetime dimensions. In this primarily expository article, we spell out some details about this construction. The axion wormhole has a semiclassical description, found in the original paper [1], in which the matter system is a two-form gauge field B with three-form field strength H=dB. The two-form is dual to a massless scalar, but the wormhole does not have a semiclassical description in terms of the scalar. There is no contradiction here as the duality between the two-form and the scalar is not a simple transformation of classical fields but involves, in Euclidean signature, a Poisson resummation of the sum over fluxes. Because of the need for this Poisson resummation, the scalar field cannot be treated semiclassically in the wormhole throat. Nonetheless, it is straightforward to compute the effective action derived from the wormhole in the scalar (or two-form) language, recovering standard claims.
Paper Structure (12 sections, 95 equations, 2 figures)

This paper contains 12 sections, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Duality
  3. Dirac Quantization and Relation Between Couplings
  4. Derivation of the Duality
  5. Transformation of Local Operators
  6. Dual Description of Vacua
  7. The Wormhole Solution
  8. The Solution
  9. A Practice Exercise
  10. The Effective Action
  11. The Action Revisited
  12. Some More Details About the Path Integral

Figures (2)

  • Figure 1: (a) A wormhole connecting two asymptotically flat worlds. (b) A wormhole that provides a shortcut between distant regions of the same asymptotically flat world.
  • Figure 2: The two-sphere $S$ (shown here as a circle) links the curve $\gamma$.