A fusion for the periodic Temperley-Lieb algebra and its continuum limit

Abstract

The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT.

We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish — using, in particular, Frobenius reciprocity — the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the continuum limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    A.M. Gainutdinov, D. Ridout and I. Runkel, Special issue on Logarithmic conformal field theory, J. Phys. A 46 (49) (2013).

  2. [2]

    N. Read and H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B 777 (2007) 316 [hep-th/0701117] [INSPIRE].

  3. [3]

    A.M. Gainutdinov and R. Vasseur, Lattice fusion rules and logarithmic operator product expansions, Nucl. Phys. B 868 (2013) 223 [arXiv:1203.6289] [INSPIRE].

  4. [4]

    M.R. Gaberdiel and H.G. Kausch, Indecomposable fusion products, Nucl. Phys. B 477 (1996) 293 [hep-th/9604026] [INSPIRE].

  5. [5]

    A.M. Gainutdinov, J.L. Jacobsen, H. Saleur and R. Vasseur, A physical approach to the classification of indecomposable Virasoro representations from the blob algebra, Nucl. Phys. B 873 (2013) 614 [arXiv:1212.0093] [INSPIRE].

  6. [6]

    P.A. Pearce, J. Rasmussen and E. Tartaglia, Logarithmic Superconformal Minimal Models, J. Stat. Mech. 1405 (2014) P05001 [arXiv:1312.6763] [INSPIRE].

  7. [7]

    A.M. Gainutdinov, N. Read and H. Saleur, Continuum Limit and Symmetries of the Periodic gl(1|1) Spin Chain, Nucl. Phys. B 871 (2013) 245 [arXiv:1112.3403] [INSPIRE].

  8. [8]

    A.M. Gainutdinov, N. Read and H. Saleur, Bimodule Structure in the Periodic gl(1|1) Spin Chain, Nucl. Phys. B 871 (2013) 289 [arXiv:1112.3407] [INSPIRE].

  9. [9]

    A.M. Gainutdinov, N. Read and H. Saleur, Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the \( \mathfrak{g}\mathfrak{l}\left(1\Big|1\right) \) periodic spin chain, Howe duality and the interchiral algebra, Commun. Math. Phys. 341 (2016) 35 [arXiv:1207.6334] [INSPIRE].

  10. [10]

    A.M. Gainutdinov, J.L. Jacobsen, N. Read, H. Saleur and R. Vasseur, Logarithmic Conformal Field Theory: a Lattice Approach, J. Phys. A 46 (2013) 494012 [arXiv:1303.2082] [INSPIRE].

  11. [11]

    A.M. Gainutdinov, N. Read, H. Saleur and R. Vasseur, The periodic sℓ(2|1) alternating spin chain and its continuum limit as a bulk logarithmic conformal field theory at c = 0, JHEP 05 (2015) 114 [arXiv:1409.0167] [INSPIRE].

  12. [12]

    M.S. Zini and Z. Wang, Conformal Field Theories as Scaling Limit of Anyonic Chains, Commun. Math. Phys. 363 (2018) 877 [arXiv:1706.08497] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  13. [13]

    P. di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer (1997).

  14. [14]

    A.M. Gainutdinov and H. Saleur, Fusion and braiding in finite and affine Temperley-Lieb categories, arXiv:1606.04530 [HAMBURGER-BEITRAGE-ZUR-MATHEMATIK-596] [INSPIRE].

  15. [15]

    J.J. Graham and G.I. Lehrer, The representation theory of affine Temperley-Lieb algebras, Enseign. Math. 44 (1998) 173.

    MathSciNet  MATH  Google Scholar 

  16. [16]

    P.P. Martin and H. Saleur, The Blob algebra and the periodic Temperley-Lieb algebra, Lett. Math. Phys. 30 (1994) 189 [hep-th/9302094] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    P.P. Martin and H. Saleur, On an Algebraic approach to higher dimensional statistical mechanics, Commun. Math. Phys. 158 (1993) 155 [hep-th/9208061] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. [18]

    G.W. Mackey, Induced representations of groups, Am. J. Math. 73 (1951) 576.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    P.A. Pearce, J. Rasmussen and J.-B. Zuber, Logarithmic minimal models, J. Stat. Mech. 0611 (2006) P11017 [hep-th/0607232] [INSPIRE].

  20. [20]

    F.C. Alcaraz, U. Grimm and V. Rittenberg, The XXZ Heisenberg Chain, Conformal Invariance and the Operator Content of c < 1 Systems, Nucl. Phys. B 316 (1989) 735 [INSPIRE].

  21. [21]

    V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, Nucl. Phys. B 330 (1990) 52.

  22. [22]

    N. Read and H. Saleur, Enlarged symmetry algebras of spin chains, loop models and S-matrices, Nucl. Phys. B 777 (2007) 263 [cond-mat/0701259] [INSPIRE].

  23. [23]

    W.M. Koo and H. Saleur, Representations of the Virasoro algebra from lattice models, Nucl. Phys. B 426 (1994) 459 [hep-th/9312156] [INSPIRE].

  24. [24]

    V. Jones, A quotient of the affine Hecke algebra in the Brauer algebra, Enseign. Math. 40 (1994) 313.

  25. [25]

    V. Chari and A. Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996) 295.

    MathSciNet  Article  MATH  Google Scholar 

  26. [26]

    J. Belletête, A.M. Gainutdinov, J.L. Jacobsen, H. Saleur and R. Vasseur, On the correspondence between boundary and bulk lattice models and (logarithmic) conformal field theories, J. Phys. A 50 (2017) 484002 [arXiv:1705.07769] [INSPIRE].

  27. [27]

    D. Ridout and Y. Saint-Aubin, Standard modules, induction and the structure of the Temperley-Lieb algebra, Adv. Theor. Math. Phys. 18 (2014) 957 [arXiv:1204.4505] [INSPIRE].

  28. [28]

    J. Belletête, D. Ridout and Y. Saint-Aubin, Restriction and induction of indecomposble modules over the Temperley-Lieb algebra, J. Phys. A 51 (2018) 045201 [arXiv:1605.05159].

  29. [29]

    S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].

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Correspondence to Azat M. Gainutdinov.

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Gainutdinov, A.M., Jacobsen, J.L. & Saleur, H. A fusion for the periodic Temperley-Lieb algebra and its continuum limit. J. High Energ. Phys. 2018, 117 (2018). https://doi.org/10.1007/JHEP11(2018)117

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Keywords

  • Lattice Integrable Models
  • Conformal Field Theory
  • Conformal and W Symmetry
  • Quantum Groups