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On the boundary Carrollian conformal algebra

Lucas Buzaglo, Xiao He, Tuan Anh Pham, Haijun Tan, Girish S Vishwa, Kaiming Zhao

TL;DR

This work develops a representation-theoretic framework for the boundary Carrollian conformal algebra (BCCA), a non-graded infinite-dimensional symmetry arising in Carrollian physics. It simultaneously analyzes restrictions of Virasoro/BMS$_3$ modules to the subalgebras $\mathcal{O}$ and $\widehat{\mathfrak{b}}$ and constructs intrinsic BCCA modules via a new basis and a decreasing filtration, culminating in the construction and irreducibility criteria for Whittaker modules over $\mathcal{O}$ and $\mathfrak{b}$. Key results include precise irreducibility conditions for Verma and massive modules under restriction, a decomposition of certain massive modules into two $\widehat{\mathfrak{b}}$-submodules, and a complete intrinsic Whittaker theory for the BCCA built on the new basis and orbit-method perspective. The paper thus provides foundational tools for non-graded infinite-dimensional representation theory with physical relevance to tensionless open strings and Carrollian holography, and outlines multiple fruitful directions, including contractions from Virasoro and supersymmetric extensions. The formal development of filtrations, alternate bases, and explicit irreducibility criteria opens a path to further spectral analysis and potential connections to non-Lorentzian geometries in high-energy physics.

Abstract

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call $\mathcal{O}$, by restriction of well-known modules of the BMS$_3$ and Witt algebras respectively. Along the way, we prove the irreducibility criteria for the so-called ``induced modules'' of the BMS$_3$ algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS$_3$ algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When this criteria holds, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be irreducible. Meanwhile, restricting Verma modules to the BCCA and $\mathcal{O}$ leads to free or ``almost free'' modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra $\mathcal{O}$ and prove criteria for their irreducibility.

On the boundary Carrollian conformal algebra

TL;DR

This work develops a representation-theoretic framework for the boundary Carrollian conformal algebra (BCCA), a non-graded infinite-dimensional symmetry arising in Carrollian physics. It simultaneously analyzes restrictions of Virasoro/BMS modules to the subalgebras and and constructs intrinsic BCCA modules via a new basis and a decreasing filtration, culminating in the construction and irreducibility criteria for Whittaker modules over and . Key results include precise irreducibility conditions for Verma and massive modules under restriction, a decomposition of certain massive modules into two -submodules, and a complete intrinsic Whittaker theory for the BCCA built on the new basis and orbit-method perspective. The paper thus provides foundational tools for non-graded infinite-dimensional representation theory with physical relevance to tensionless open strings and Carrollian holography, and outlines multiple fruitful directions, including contractions from Virasoro and supersymmetric extensions. The formal development of filtrations, alternate bases, and explicit irreducibility criteria opens a path to further spectral analysis and potential connections to non-Lorentzian geometries in high-energy physics.

Abstract

We initiate the mathematical study of the boundary Carrollian conformal algebra (BCCA), an infinite-dimensional Lie algebra recently discovered in the context of Carrollian physics. The BCCA is an intriguing object from both physical and mathematical perspectives, since it is a filtered but not graded Lie algebra. In this paper, we first construct some modules for the BCCA and one of its subalgebras, which we call , by restriction of well-known modules of the BMS and Witt algebras respectively. Along the way, we prove the irreducibility criteria for the so-called ``induced modules'' of the BMS algebra (which we prefer to call massive modules to avoid ambiguity) and show that this is the same criteria for the irreducibility of the Verma modules of the BMS algebra. Interestingly, the modules generated by the action of the BCCA on the generating vector of the massive modules are also irreducible under the same criteria. When this criteria holds, every massive module decomposes into a direct sum of two BCCA-submodules, each of which we conjecture to be irreducible. Meanwhile, restricting Verma modules to the BCCA and leads to free or ``almost free'' modules, which are not particularly interesting from a representation-theoretic viewpoint. This motivates the construction of BCCA modules intrinsically. To do this, we go through some structure theory on the BCCA to define a new basis and a decreasing filtration on the algebra, using which we construct Whittaker modules over the BCCA and the subalgebra and prove criteria for their irreducibility.

Paper Structure

This paper contains 23 sections, 74 theorems, 204 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The boundary Carrollian conformal algebra
  4. Whittaker modules
  5. Representations of O and b by restriction
  6. O-modules via restriction of Verma modules
  7. b-modules via restriction of Verma modules
  8. b-modules via restriction of massive modules
  9. The O-module Omega(lambda,a)
  10. Alternate basis of the BCCA
  11. Subalgebras of the Witt algebra
  12. Change of basis for O
  13. Change of basis for Pb
  14. Filtering the BCCA
  15. Defining the filtration
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $h_L, h_M, c_L, c_M, a, b, \mathbf{M}, \mathbf{s} \in \Bbbk$, and $\lambda \in \Bbbk^*$. Then the following hold.

Figures (1)

  • Figure 7.1: Diagram summarising the relations between $\mathop{\mathrm{Vir}}\nolimits$, $\mathfrak{bms}$ and $\widehat{\mathfrak{b}}$. The vertical dashed arrows denote restrictions while the horizontal solid ones denote Lie algebra contractions.

Theorems & Definitions (179)

  • Theorem 1.1: Lemma \ref{['lem:tensor density direct sum']}, Propositions \ref{['prop:V(h,c)_free_rank1']}, \ref{['prop:BMS Verma is almost free over b']}, \ref{['prop:massive module over b']}, and \ref{['Omega-simplicity']}
  • Theorem 1.2: Propositions \ref{['prop:P and Phat acting on Ohat_n']} and \ref{['pre-formula']}, Corollary \ref{['cor:bms_massive_module_decomposable']}
  • Conjecture 1.3: Conjecture \ref{['conj:Ind']}
  • Theorem 1.4: Theorem \ref{['thm:bms_massive_modules_irreducibility']}
  • Theorem 1.5: Theorem \ref{['thm:O lambda = L(s)']}, Proposition \ref{['prop:g lambda isomorphism']}, and Corollary \ref{['cor:two_O_subalgebras']}
  • Theorem 1.6: Corollary \ref{['cor:new_basis_for_b']}
  • Proposition 1.7: Proposition \ref{['prop:filtration for BCCA']}
  • Theorem 1.8: Theorems \ref{['theo: simplicity']} and \ref{['thm:bcca Whittaker']}
  • Definition 2.1
  • Definition 2.2
  • ...and 169 more