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1. Pairing symmetry and fermion projective symmetry groups

   https://doi.org/10.21468/SciPostPhys.17.6.161  

   Yang, Xu; Biswas, Sayak; Lu, Shuangyuan; Randeria, Mohit; Lu, Yuan-Ming (January 2024, SciPost Physics)

   The Ginzburg-Landau (GL) theory is very successful in describing the pairing symmetry, a fundamental characterization of the broken symmetries in a paired superfluid or superconductor. However, GL theory does not describe fermionic excitations such as Bogoliubov quasiparticles or Andreev bound states that are directly related to topological properties of the superconductor. In this work, we show that the symmetries of the fermionic excitations are captured by a Projective Symmetry Group (PSG), which is a group extension of the bosonic symmetry group in the superconducting state. We further establish a correspondence between the pairing symmetry and the fermion PSG. When the normal and superconducting states share the same spin rotational symmetry, there is a simpler correspondence between the pairing symmetry and the fermion PSG, which we enumerate for all 32 crystalline point groups. We also discuss the general framework for computing PSGs when the spin rotational symmetry is spontaneously broken in the superconducting state. This PSG formalism leads to experimental consequences, and as an example, we show how a given pairing symmetry dictates the classification of topological superconductivity. 

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2. Algebraic higher symmetry and categorical symmetry: A holographic and entanglement view of symmetry

   https://doi.org/10.1103/PhysRevResearch.2.043086  

   Kong, Liang; Lan, Tian; Wen, Xiao-Gang; Zhang, Zhi-Hao; Zheng, Hao (October 2020, Physical Review Research)
3. Anomaly inflow, accidental symmetry, and spontaneous symmetry breaking

   https://doi.org/10.1007/JHEP01(2020)117  

   Bah, Ibrahima; Bonetti, Federico (January 2020, Journal of High Energy Physics)
4. Emergent generalized symmetry and maximal symmetry topological order

   https://doi.org/10.1103/jzfv-ygmr  

   Chatterjee, Arkya; Ji, Wenjie; Wen, Xiao-Gang (September 2025, Physical Review B)

   A characteristic property of a gapless liquid state is its emergent symmetry and dual symmetry, associated with the conservation laws of symmetry charges and symmetry defects respectively. These conservation laws, considered on an equal footing, can't be described simply by the representation theory of a group (or a higher group). They are best described in terms of {\it a topological order (TO) with gappable boundary in one higher dimension}; we call this the {\it symTO} of the gapless state. The symTO can thus be considered a fingerprint of the gapless state. We propose that a largely complete characterization of a gapless state, up to local-low-energy equivalence, can be obtained in terms of its {\it maximal} emergent symTO. In this paper, we review the symmetry/topological-order (Symm/TO) correspondence and propose a definition of {\it maximal symTO}. We discuss various examples to illustrate these ideas. We find that the 1+1D Ising critical point has a maximal symTO described by the 2+1D double-Ising topological order. We provide a derivation of this result using symmetry twists in an exactly solvable model of the Ising critical point. The critical point in the 3-state Potts model has a maximal symTO of double (6,5)-minimal-model topological order. As an example of a noninvertible symmetry in 1+1D, we study the possible gapless states of a Fibonacci anyon chain with emergent double-Fibonacci symTO. We find the Fibonacci-anyon chain without translation symmetry has a critical point with unbroken double-Fibonacci symTO. In fact, such a critical theory has a maximal symTO of double (5,4)-minimal-model topological order. We argue that, in the presence of translation symmetry, the above critical point becomes a stable gapless phase with no symmetric relevant operator. 

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5. Love symmetry

   https://doi.org/10.1007/JHEP10(2022)175  

   Charalambous, Panagiotis; Dubovsky, Sergei; Ivanov, Mikhail M. (October 2022, Journal of High Energy Physics)

   A bstract Perturbations of massless fields in the Kerr-Newman black hole background enjoy a (“Love”) SL(2 , ℝ) symmetry in the suitably defined near zone approximation. We present a detailed study of this symmetry and show how the intricate behavior of black hole responses in four and higher dimensions can be understood from the SL(2 , ℝ) representation theory. In particular, static perturbations of four-dimensional black holes belong to highest weight SL(2 , ℝ) representations. It is this highest weight properety that forces the static Love numbers to vanish. We find that the Love symmetry is tightly connected to the enhanced isometries of extremal black holes. This relation is simplest for extremal charged spherically symmetric (Reissner-Nordström) solutions, where the Love symmetry exactly reduces to the isometry of the near horizon AdS 2 throat. For rotating (Kerr-Newman) black holes one is lead to consider an infinite-dimensional SL(2 , ℝ) ⋉ $$ \hat{\textrm{U}}{(1)}_{\mathcal{V}} $$ U ̂ 1 V extension of the Love symmetry. It contains three physically distinct subalgebras: the Love algebra, the Starobinsky near zone algebra, and the near horizon algebra that becomes the Bardeen-Horowitz isometry in the extremal limit. We also discuss other aspects of the Love symmetry, such as the geometric meaning of its generators for spin weighted fields, connection to the no-hair theorems, non-renormalization of Love numbers, its relation to (non-extremal) Kerr/CFT correspondence and prospects of its existence in modified theories of gravity. 

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