[Paper Review] A BMS-invariant free fermion model

Peng-Xiang Hao, Wei Song+2 more|arXiv (Cornell University)|Nov 13, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper constructs a two-dimensional free fermion model invariant under the BMS₃ group, realized as conformal Carrollian symmetry, using the Cartan formalism to derive Galilean and Carrollian symmetries. Canonical quantization yields highest weight and induced vacua, correlation functions, and a modular-invariant torus partition function, with an N=2 supersymmetric extension achieved by combining with a free scalar model.

ABSTRACT

We used the Cartan formalism to construct fermionic models that are compatible with Galilean or Carrollian symmetry and rigid scaling symmetry. The free Carrollian fermion model exhibits conformal Carrollian symmetry which is isomorphic to the asymptotic symmetries for flat spacetime in three dimensions, namely the BMS$_3$ symmetry. We performed canonical quantization to this free BMS fermion model, discussed both the highest weight vacuum and the induced vacuum, calculated the correlation functions and the torus partition function. Finally we constructed $\mathcal N=2$ supersymmetric theories by combining the free fermion model and the free scalar model.

Motivation & Objective

• To construct explicit quantum field theories with BMS₃ symmetry, addressing the lack of known explicit models beyond symmetry-based properties.
• To explore the realization of BMS₃ symmetry through fermionic systems, particularly in the context of flat holography and asymptotic symmetries in 3D flat spacetime.
• To analyze the quantum structure of the model via canonical quantization, including vacuum states and correlation functions.
• To extend the free fermion model to N=2 supersymmetric theories by coupling with a free scalar field.
• To establish a connection between the BMS₃ algebra and the conformal structure of the model, particularly through the induced vacuum and torus partition function.

Proposed method

• Employed the Cartan formalism to derive field equations invariant under Galilean and Carrollian symmetries, using soldering forms and spin connections.
• Formulated the free fermion action using the BMS₃ algebra, realizing it as a conformal Carrollian symmetry via an ultra-relativistic limit of the Virasoro algebra.
• Perfomed canonical quantization of the free BMS fermion model, identifying both the highest weight vacuum and the induced vacuum states.
• Calculated two-point and higher-point correlation functions using the vacuum states, ensuring consistency with BMS symmetry.
• Derived the torus partition function and demonstrated its modular invariance, a key consistency check for a putative dual CFT.
• Constructed N=2 supersymmetric extensions by coupling the free fermion model with a free scalar model, preserving BMS invariance.

Experimental results

Research questions

• RQ1 Can a free fermion model be constructed that is invariant under the BMS₃ group, and how does this relate to conformal Carrollian symmetry?
• RQ2 What are the quantum vacuum structures (highest weight and induced vacua) in a BMS-invariant fermionic system, and how do they differ?
• RQ3 How do correlation functions and the torus partition function behave in this BMS-fermion model, and is the partition function modular invariant?
• RQ4 Can the BMS-fermion model be extended to an N=2 supersymmetric theory while preserving BMS invariance?
• RQ5 What is the role of the Cartan formalism in systematically deriving BMS-invariant field theories with fermionic degrees of freedom?

Key findings

• The free fermion model constructed is invariant under the BMS₃ group, which is isomorphic to the conformal Carrollian algebra in two dimensions.
• Canonical quantization yields two distinct vacuum states: the highest weight vacuum and the induced vacuum, both consistent with BMS symmetry.
• Two-point correlation functions were computed and shown to respect the BMS symmetry, with explicit expressions derived from the vacuum states.
• The torus partition function was derived and found to be modular invariant, supporting the model's consistency as a candidate for a dual BMSFT.
• An N=2 supersymmetric extension was successfully constructed by coupling the free fermion model with a free scalar model, preserving BMS invariance.
• The action of the supersymmetric model was derived via superfield formalism in superspace, yielding the standard N=2 superfield action with auxiliary fields.