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The Sum Over Topological Sectors and $θ$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $σ$-Model

Daniel S. Freed, Zohar Komargodski, Nathan Seiberg

TL;DR

This work analyzes the sum over topological sectors in the 2+1 dimensional CP^1 sigma-model and shows that locality and unitarity constrain the theta parameter to θ=0 or θ=π, yielding bosonic/skyrmion statistics. By embedding CP^N in a linear UV theory with U(1) gauge fields and CS couplings, the authors derive a Wess-Zumino term in the IR (for N>1) and a mod 2 invariant (for N=1), linking theta to the spin-statistics of Skyrmions via the Hopf invariant. They then demonstrate how coupling to a TQFT or modifying the UV gauge structure can produce anyon-like behavior, but argue that purely local, well-defined theories demand θ∈{0,π} unless extended by invertible field theories, with a rigorous classification using spin and spin^c bordism via Freed–Hopkins results. The combination of geometric and topological methods establishes precise consistency conditions for theta-terms in the CP^1 model on general 3-manifolds and clarifies when fractional statistics can emerge only through coupling to topological sectors. The findings have implications for condensed-mmatter constructions and for understanding global consistency constraints in quantum field theories with nontrivial topological sectors.

Abstract

We discuss the three spacetime dimensional $\mathbb{C}\mathbb{P}^N$ model and specialize to the $\mathbb{C}\mathbb{P}^1$ model. Because of the Hopf map $π_3(\mathbb{C}\mathbb{P}^1)=\mathbb{Z}$ one might try to couple the model to a periodic $θ$ parameter. However, we argue that only the values $θ=0$ and $θ=π$ are consistent. For these values the Skyrmions in the model are bosons and fermions respectively, rather than being anyons. We also extend the model by coupling it to a topological quantum field theory, such that the Skyrmions are anyons. We use techniques from geometry and topology to construct the $θ=π$ theory on arbitrary 3-manifolds, and use recent results about invertible field theories to prove that no other values of $θ$ satisfy the necessary locality.

The Sum Over Topological Sectors and $θ$ in the 2+1-Dimensional $\mathbb{C}\mathbb{P}^1$ $σ$-Model

TL;DR

This work analyzes the sum over topological sectors in the 2+1 dimensional CP^1 sigma-model and shows that locality and unitarity constrain the theta parameter to θ=0 or θ=π, yielding bosonic/skyrmion statistics. By embedding CP^N in a linear UV theory with U(1) gauge fields and CS couplings, the authors derive a Wess-Zumino term in the IR (for N>1) and a mod 2 invariant (for N=1), linking theta to the spin-statistics of Skyrmions via the Hopf invariant. They then demonstrate how coupling to a TQFT or modifying the UV gauge structure can produce anyon-like behavior, but argue that purely local, well-defined theories demand θ∈{0,π} unless extended by invertible field theories, with a rigorous classification using spin and spin^c bordism via Freed–Hopkins results. The combination of geometric and topological methods establishes precise consistency conditions for theta-terms in the CP^1 model on general 3-manifolds and clarifies when fractional statistics can emerge only through coupling to topological sectors. The findings have implications for condensed-mmatter constructions and for understanding global consistency constraints in quantum field theories with nontrivial topological sectors.

Abstract

We discuss the three spacetime dimensional model and specialize to the model. Because of the Hopf map one might try to couple the model to a periodic parameter. However, we argue that only the values and are consistent. For these values the Skyrmions in the model are bosons and fermions respectively, rather than being anyons. We also extend the model by coupling it to a topological quantum field theory, such that the Skyrmions are anyons. We use techniques from geometry and topology to construct the theory on arbitrary 3-manifolds, and use recent results about invertible field theories to prove that no other values of satisfy the necessary locality.

Paper Structure

This paper contains 11 sections, 4 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. The $2+1$ dimensional ${\mathbb C\mathbb P}^N$ model from a linear model
  3. $N>1$
  4. $N=1$
  5. Skyrmion operators and conditions on $\theta$
  6. Changing the quantization of the coefficients
  7. Invertible field theories and effective actions
  8. Generalities
  9. Spin manifolds
  10. $\operatorname{\bf Spin}^c$ manifolds
  11. The variation in §4

Key Result

Lemma 5.2

For $N\ge2$ there is an isomorphism $E^4({\mathbb C\mathbb P}^N)\cong {\mathbb Z}$; the homomorphism $H^4({\mathbb C\mathbb P}^N;{\mathbb Z})\longrightarrow E^4({\mathbb C\mathbb P}^N)$ maps a generator to twice a generator. Also, $E^4({\mathbb C\mathbb P}^1)\cong {\mathbb Z}/2{\mathbb Z}$ and a gen

Theorems & Definitions (17)

  • Lemma 5.2
  • proof
  • Definition 5.3
  • Remark 5.5
  • Remark 5.6
  • Remark 5.7
  • Lemma 5.8
  • proof
  • Theorem 5.13
  • proof
  • ...and 7 more