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arXiv:2410.24142 · Journal of High Energy Physics

Quantum groups as global symmetries

Barak Gabai, Victor Gorbenko, Jiaxin Qiao, Bernardo Zan, Aleksandr Zhabin

Published Aug 12, 2025• 6 citations
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Abstract

A bstract We study quantum field theories which have quantum groups as global internal symmetries. We show that in such theories operators are generically non-local, and should be thought as living at the ends of topological lines. We describe the general constraints of the quantum group symmetry, given by Ward identities, that correlation functions of the theory should satisfy. We also show that generators of the symmetry can be represented by topological lines with some novel properties. We then discuss a particular example of U q ( sl 2 ) symmetric CFT, which we solve using bootstrap techniques and relying on the symmetry. We finally show strong evidence that for a special value of q a subsector of this theory reproduces the fermionic formulation of the Ising model. This suggests that a quantum group can act on local operators as well, however, it generically transforms them into non-local ones.

DOI
10.1007/jhep08(2025)087

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A bstract We study quantum field theories which have quantum groups as global internal symmetries. We show that in such theories operators are generically non-local, and should be thought as living at the ends of topological lines. We describe the general constraints of the quantum group symmetry, given by Ward identities, that correlation functions of the theory should satisfy. We also show that generators of the symmetry can be represented by topological lines with some novel properties. We then discuss a particular example of U q ( sl 2 ) symmetric CFT, which we solve using bootstrap techniques and relying on the symmetry. We finally show strong evidence that for a special value of q a subsector of this theory reproduces the fermionic formulation of the Ising model. This suggests that a quantum group can act on local operators as well, however, it generically transforms them into non-local ones.
This thesis studies Quantum Field Theories (QFTs) and many-body systems that have quantum groups as global internal symmetries. We establish that quantum groups can act as genuine global symmetries, constraining the spectrum, correlation functions and dynamics. Operators that transform naturally under this symmetry are shown to be non-local, living at the ends of topological lines that carry some novel properties. These properties are illustrated by explicit examples including spin chains and their field-theoretic continuum limit. We solve the continuum theory in two ways: first, by using constraints coming from quantum group symmetry together with conformal bootstrap techniques; second, by applying the Coulomb gas formalism. We further show that this symmetry is not an artifact of integrability: it survives as a global symmetry even when integrability is broken and the system becomes chaotic. Finally, we observe that it governs late-time hydrodynamic transport, as manifest in the slow decay of the transverse spin autocorrelation function. Altogether, these results position quantum groups as a new and robust class of symmetry governing the physics of QFTs and many-body systems.
We study a conformal field theory that arises in the infinite-volume limit of a spin chain with U_q(sl_2) U q ( s l 2 ) global symmetry. Most operators in the theory are defect-ending operators which allows U_q(sl_2) U q ( s l 2 ) symmetry transformations to act on them in a consistent way. We use Coulomb gas techniques to construct correlation functions and compute all OPE coefficients of the model in closed form, as well as to prove that the properties imposed by the quantum group symmetry are indeed satisfied by the correlation functions. In particular, we treat the non-chiral operators present in the theory. Free boson realization elucidates the origin of the defects attached to the operators. We also comment on the role of quantum group in generalized minimal models.
It has been discussed earlier that ( weak quasi-) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
Non-invertible dualities/symmetries have become an important tool in the study of quantum field theories and quantum lattice models in recent years. One of the most studied examples is non-invertible dualities obtained by gauging a discrete group. When the physical system has more global symmetries than the gauged symmetry, it has not been thoroughly investigated how those global symmetries transform under non-invertible duality. In this paper, we study the change of global symmetries under non-invertible duality of gauging a discrete group G G in the context of (1+1)-dimensional quantum lattice models. We obtain the global symmetries of the dual model by focusing on different Hilbert space sectors determined by the \mathrm{Rep}(G) R e p ( G ) symmetry. We provide general conjectures of global symmetries of the dual model forming an algebraic ring of the double cosets. We present concrete examples of the XXZ models and the duals, providing strong evidence for the conjectures.
A bstract Recent studies by Copetti, Córdova and Komatsu have revealed that when non-invertible symmetries are spontaneously broken, the conventional crossing relation of the S-matrix is modified by the effects of the corresponding topological quantum field theory (TQFT). In this paper, we extend these considerations to (1 + 1)-dimensional quantum field theories (QFTs) with boundaries. In the presence of a boundary, one can define not only the bulk S-matrix but also the boundary S-matrix, which is subject to a consistency condition known as the boundary crossing relation. We show that when the boundary is weakly-symmetric under the non-invertible symmetry, the conventional boundary crossing relation also receives a modification due to the TQFT effects. As a concrete example of the boundary scattering, we analyze kink scattering in the gapped theory obtained from the Φ (1,3) -deformation of a minimal model. We explicitly construct the boundary S-matrix that satisfies the Ward-Takahashi identities associated with non-invertible symmetries.
A bstract We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.