Integrable Deformations of Sine-Liouville Conformal Field Theory and Duality

Vladimir A. Fateev

Symmetry, integrability and geometry: methods and applications, Tome 13 (2017)

Résumé

We study integrable deformations of sine-Liouville conformal field theory. Every integrable perturbation of this model is related to the series of quantum integrals of motion (hierarchy). We construct the factorized scattering matrices for different integrable perturbed conformal field theories. The perturbation theory, Bethe ansatz technique, renormalization group and methods of perturbed conformal field theory are applied to show that all integrable deformations of sine-Liouville model possess non-trivial duality properties.

Détail

Keywords: integrability; duality; Ricci flow.

Bibliographie

[1] Ahn C., Balog J., Ravanini F., NLIE for the sausage model, arXiv: 1701.08933

[2] Arinshtein A. E., Fateev V. A., Zamolodchikov A. B., “Quantum $S$-matrix of the $(1+1)$-dimensional Todd chain”, Phys. Lett. B, 87 (1979), 389–392 | DOI

[3] Arutyunov G., Borsato R., Frolov S., “$S$-matrix for strings on $\eta$-deformed ${\rm AdS}_5\times {\rm S}^5$”, J. High Energy Phys., 2014:4 (2014), 002, 24 pp., arXiv: 1312.3542 | DOI | MR

[4] Bazhanov V. V., Kotousov G. A., Lukyanov S. L., Quantum transfer-matrices for the sausage model, arXiv: 1706.09941

[5] Bonneau G., Delduc F., “$S$-matrix properties versus renormalizability in two-dimensional ${\rm O}(N)$ symmetric models”, Nuclear Phys. B, 250 (1985), 561–592 | DOI | MR

[6] Braden H. W., Corrigan E., Dorey P. E., Sasaki R., “Affine Toda field theory and exact $S$-matrices”, Nuclear Phys. B, 338 (1990), 689–746 | DOI | MR | Zbl

[7] Coleman S., “Quantum sine-Gordon equation as the massive Thirring model”, Phys. Rev. D, 11 (1975), 2088–2097 | DOI

[8] de Vega H. J., Maillet J. M., “Renormalization character and quantum $S$-matrix for a classically integrable theory”, Phys. Lett. B, 101 (1981), 302–306 | DOI | MR

[9] Delius G. W., Grisaru M. T., Zanon D., “Exact $S$-matrices for non-simply-laced affine Toda theories”, Nuclear Phys. B, 382 (1992), 365–406, arXiv: hep-th/9201067 | DOI | MR | Zbl

[10] Dijkgraaf R., Verlinde H., Verlinde E., “String propagation in a black hole geometry”, Nuclear Phys. B, 371 (1992), 269–314 | DOI | MR

[11] Fateev V. A., “Integrable deformations in $Z_N$-symmetrical models of the conformal quantum field theory”, Internat. J. Modern Phys. A, 6 (1991), 2109–2132 | DOI | MR

[12] Fateev V. A., “The exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories”, Phys. Lett. B, 324 (1994), 45–51 | DOI | MR

[13] Fateev V. A., “The duality between two-dimensional integrable field theories and sigma models”, Phys. Lett. B, 357 (1995), 397–403 | DOI | MR

[14] Fateev V. A., “Integrable deformations of affine Toda theories and duality”, Nuclear Phys. B, 479 (1996), 594–618 | DOI | MR | Zbl

[15] Fateev V. A., “The sigma model (dual) representation for a two-parameter family of integrable quantum field theories”, Nuclear Phys. B, 473 (1996), 509–538 | DOI | MR | Zbl

[16] Fateev V. A., Onofri E., “An eigenvalue problem related to the non-linear $\sigma$-model: analytical and numerical results”, J. Phys. A: Math. Gen., 36 (2003), 11881–11899, arXiv: math-ph/0307010 | DOI | MR | Zbl

[17] Fateev V. A., Onofri E., Zamolodchikov A. B., “Integrable deformations of the ${\rm O}(3)$ sigma model. The sausage model”, Nuclear Phys. B, 406 (1993), 521–565 | DOI | MR | Zbl

[18] Fateev V. A., Zamolodchikov A. B., “Self-dual solutions of the star-triangle relations in $Z_N$-models”, Phys. Lett. A, 92 (1982), 37–39 | DOI | MR

[19] Friedan D., “Nonlinear models in $2+\varepsilon $ dimensions”, Phys. Rev. Lett., 45 (1980), 1057–1060 | DOI | MR

[20] Ganin M. P., “On a Fredholm integral equation whose kernel depends on the difference of the arguments”, Izv. Vyssh. Uchebn. Zaved. Mat., 2:2 (1963), 31–43 | MR | Zbl

[21] Goddard P., Nuyts J., Olive D., “Gauge theories and magnetic charge”, Nuclear Phys. B, 125 (1977), 1–28 | DOI | MR

[22] Gukov S., Klebanov I. R., Polyakov A. M., “Dynamics of $(n,1)$ strings”, Phys. Lett. B, 423 (1998), 64–70, arXiv: hep-th/9711112 | DOI | MR

[23] Hasenfratz P., Maggiore M., Niedermayer F., “The exact mass gap of the ${\rm O}(3)$ and ${\rm O}(4)$ nonlinear $\sigma$-models in $d=2$”, Phys. Lett. B, 245 (1990), 522–528 | DOI | MR

[24] Hoare B., Roiban R., Tseytlin A. A., “On deformations of ${\rm AdS}_n \times {\rm S}^n$ supercosets”, J. High Energy Phys., 2014:6 (2014), 002, 34 pp., arXiv: 1403.5517 | DOI | MR

[25] Kiritsis E. B., “Duality in gauged WZW models”, Modern Phys. Lett. A, 6 (1991), 2871–2879 | DOI | MR | Zbl

[26] Klimčík C., “Integrability of the bi-Yang–Baxter $\sigma$-model”, Lett. Math. Phys., 104 (2014), 1095–1106, arXiv: 1402.2105 | DOI | MR | Zbl

[27] Lund F., Regge T., “Unified approach to strings and vortices with soliton solutions”, Phys. Rev. D, 14 (1976), 1524–1535 | DOI | MR | Zbl

[28] Maldacena J., “The large $N$ limit of superconformal field theories and supergravity”, Adv. Theor. Math. Phys., 2 (1998), 231–252, arXiv: hep-th/9711200 | DOI | MR | Zbl

[29] Maldacena J., “The large-$N$ limit of superconformal field theories and supergravity”, Internat. J. Theoret. Phys., 38 (1999), 1113–1133 | DOI | MR | Zbl

[30] Maldacena J., Moore G., Seiberg N., “Geometrical interpretation of D-branes in gauged WZW models”, J. High Energy Phys., 2001:7 (2001), 046, 63 pp., arXiv: hep-th/0105038 | DOI | MR

[31] Mandelstam S., “Soliton operators for the quantized sine-Gordon equation”, Phys. Rev. D, 11 (1975), 3026–3030 | DOI | MR

[32] Montonen C., Olive D., Magnetic monopoles as gauge particles?, Phys. Lett. B, 125 (1977), 117–120 | DOI

[33] Nemeschansky D., “Feĭgin–Fuchs representation of $\widehat{\rm su}(2)_k$ Kac–Moody algebra”, Phys. Lett. B, 224 (1989), 121–124 | DOI | MR

[34] Perelman G., The entropy formula for the Ricci flow and its geometric applications, arXiv: math.DG/0211159

[35] Pohlmeyer K., “Integrable Hamiltonian systems and interactions through quadratic constraints”, Comm. Math. Phys., 46 (1976), 207–221 | DOI | MR | Zbl

[36] Seiberg N., Witten E., “Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang–Mills theory”, Nuclear Phys. B, 426 (1994), 19–52, arXiv: hep-th/9407087 | DOI | MR | Zbl

[37] Witten E., “String theory and black holes”, Phys. Rev. D, 44 (1991), 314–324 | DOI | MR | Zbl

[38] Witten E., “Anti de Sitter space and holography”, Adv. Theor. Math. Phys., 2 (1998), 253–291, arXiv: hep-th/9802150 | DOI | MR | Zbl

[39] Zamolodchikov A. B., “Exact two-particle $S$-matrix of quantum sine-Gordon solitons”, Comm. Math. Phys., 55 (1977), 183–186 | DOI | MR

[40] Zamolodchikov A. B., “Mass scale in the sine-Gordon model and its reduction”, Internat. J. Modern Phys. A, 10 (1995), 1125–1150 | DOI

[41] Zamolodchikov A. B., Fateev V. A., “A model factorized $S$-matrix and an integrable spin-1 Heisenberg chain”, Soviet J. Nuclear Phys., 32 (1980), 298–303 | MR

[42] Zamolodchikov A. B., Fateev V. A., “Nonlocal (parafermion) currents in two-dimensional conformal quantum field theory and self-dual critical points in $Z_N$-symmetric statistical systems”, Sov. Phys. JETP, 62 (1985), 215–225 | MR

[43] Zamolodchikov A. B., Zamolodchikov A. B., “Massless factorized scattering and sigma models with topological terms”, Nuclear Phys. B, 379 (1992), 602–623 | DOI | MR

[44] Zamolodchikov A. B., Zamolodchikov A. B., “Conformal bootstrap in Liouville field theory”, Nuclear Phys. B, 477 (1996), 577–605, arXiv: hep-th/9506136 | DOI | MR | Zbl