Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2], [3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2], [3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).

arXiv:2309.16660Nuclear Physics BJun 28, 20245 citations

Anyonic Chains – $\alpha$ -Induction – CFT – Defects – SubfactorsAnyonic Chains – $$\alpha $$-Induction – CFT – Defects – Subfactors

Stefan Hollands

Abstract Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to $1+1$ 1+1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of $1+1$ 1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on $\alpha$ α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.Abstract Given a unitary fusion category, one can define the Hilbert space of a so-called “anyonic spin-chain” and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to $$1+1$$ 1+1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of $$1+1$$ 1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on $$\alpha $$ α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.

arXiv:2205.15243Communications in Mathematical PhysicsDec 19, 20224 citations

Anyonic Chains –

\alpha

-Induction – CFT – Defects – Subfactors

1+1

1+1 -dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bipartite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of

1+1

1+1 CFTs as constructed by Fröhlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called “double triangle algebra” arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by Böckenhauer et al. based on the braided structure of the categories and on

\alpha

α -induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.