[Paper Review] The $\mathsf{CP^{n-1}}$-model with fermions: a new look

Dmitri Bykov|arXiv (Cornell University)|Sep 9, 2020

Black Holes and Theoretical Physics5 citations

TL;DR

This paper reinterprets the CP^{n-1} model with fermions as a gauged Gross-Neveu model, identifying its super phase space as a supersymplectic quotient. It shows that chiral gauge anomalies arise from both bosonic and fermionic contributions and are canceled when the supergroup representations satisfy specific geometric constraints—linking anomaly cancellation to the integrability of the model through moment maps and Yangian symmetry.

ABSTRACT

We elaborate the formulation of the $\mathsf{CP^{n-1}}$ sigma model with fermions as a gauged Gross-Neveu model. This approach allows to identify the super phase space of the model as a supersymplectic quotient. Potential chiral gauge anomalies are shown to receive contributions from bosons and fermions alike and are related to properties of this phase space. Along the way we demonstrate that the worldsheet supersymmetric model is a supersymplectic quotient of a model with target space supersymmetry. Possible generalizations to other quiver supervarieties are briefly discussed.

Motivation & Objective

• To provide a geometric framework for CP^{n-1} models with fermions using supersymplectic quotients.
• To clarify the origin of chiral gauge anomalies in fermionic CP^{n-1} models and show their cancellation is tied to supergroup representation constraints.
• To demonstrate that worldsheet supersymmetry arises naturally from target-space supersymmetry via gauging supergroups.
• To unify the description of interactions in such models through moment maps and Tr(μμ̄) terms.
• To generalize the construction to quiver supervarieties satisfying anomaly cancellation conditions.

Proposed method

• Formulates the CP^{n-1} model with fermions as a gauged Gross-Neveu model, extending the βγ-system and GLSM approaches.
• Identifies the super phase space as a supersymplectic quotient of a supergroup action on a complex symplectic manifold.
• Uses moment maps μ_a = ∑(V_a)^i p_i to describe interactions via Tr(μμ̄), encoding the Hamiltonian structure.
• Analyzes chiral anomalies via superdeterminant contributions and shows cancellation depends on representation invariants.
• Applies the moment map evolution equation ∂̄μ = κ[μ̄, μ] to derive the principal chiral model dynamics.
• Demonstrates that supersymmetry is preserved when the supergroup action satisfies anomaly cancellation conditions.

Experimental results

Research questions

• RQ1 How can the CP^{n-1} model with fermions be consistently formulated as a gauged Gross-Neveu model?
• RQ2 What is the geometric origin of chiral gauge anomalies in this model, and how are they canceled?
• RQ3 How does the supersymplectic quotient construction relate to the emergence of worldsheet supersymmetry?
• RQ4 What role do moment maps play in encoding interactions and preserving integrability?
• RQ5 Can the framework be generalized to other quiver supervarieties with anomaly-free representations?

Key findings

• The super phase space of the CP^{n-1} model with fermions is geometrically realized as a supersymplectic quotient of a supergroup action.
• Chiral gauge anomalies receive contributions from both bosons and fermions and are canceled when the supergroup representations satisfy a specific anomaly cancellation condition.
• The interaction term is universally expressed as Tr(μμ̄), where μ are moment maps for the symmetry group action.
• The model's integrability is linked to anomaly cancellation, as the cancellation of Yangian anomalies follows from the same representation constraints.
• The moment map evolution equation ∂̄μ = κ[μ̄, μ] is derived independently of homogeneity, confirming the model's connection to the principal chiral model.
• The framework naturally extends to general quiver supervarieties, provided the anomaly cancellation condition is satisfied.