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Scale invariance of the \(\eta\)-deformed \(\mathrm{AdS}_{5} \times S^{5}\) superstring, T-duality and modified type II equations. (English) Zbl 1332.81167

Summary: We consider the ABF background underlying the \(\eta\)-deformed \(\mathrm{AdS}_5 \times S^5\) sigma model. This background fails to satisfy the standard IIB supergravity equations which indicates that the corresponding sigma model is not Weyl invariant, i.e. does not define a critical string theory in the usual sense. We argue that the ABF background should still define a UV finite theory on a flat 2d world-sheet implying that the \(\eta\)-deformed model is scale invariant. This property follows from the formal relation via T-duality between the \(\eta\)-deformed model and the one defined by an exact type IIB supergravity solution that has 6 isometries albeit broken by a linear dilaton. We find that the ABF background satisfies candidate type IIB scale invariance conditions which for the R-R field strengths are of the second order in derivatives. Surprisingly, we also find that the ABF background obeys an interesting modification of the standard IIB supergravity equations that are first order in derivatives of R-R fields. These modified equations explicitly depend on Killing vectors of the ABF background and, although not universal, they imply the universal scale invariance conditions. Moreover, we show that it is precisely the non-isometric dilaton of the T-dual solution that leads, after T-duality, to modification of type II equations from their standard form. We conjecture that the modified equations should follow from \(\kappa\)-symmetry of the \(\eta\)-deformed model. All our observations apply also to \(\eta\)-deformations of \(\mathrm{AdS}_3 \times S^3 \times T^4\) and \(\mathrm{AdS}_2 \times S^2 \times T^6\) models.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
81T20 Quantum field theory on curved space or space-time backgrounds
83E50 Supergravity
14D15 Formal methods and deformations in algebraic geometry
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[1] Delduc, F.; Magro, M.; Vicedo, B., Derivation of the action and symmetries of the \(q\)-deformed \(AdS_5 \times S^5\) superstring, J. High Energy Phys., 1410, Article 132 pp. (2014) · Zbl 1333.81322
[2] Metsaev, R. R.; Tseytlin, A. A., Type IIB superstring action in \(AdS_5 \times S^5\) background, Nucl. Phys. B, 533, 109 (1998) · Zbl 0956.81063
[3] Klimcik, C., On integrability of the Yang-Baxter sigma-model, J. Math. Phys., 50, Article 043508 pp. (2009) · Zbl 1215.81099
[4] Frolov, S. A.; Roiban, R.; Tseytlin, A. A., Gauge-string duality for superconformal deformations of \(N = 4\) super Yang-Mills theory, J. High Energy Phys., 0507, Article 045 pp. (2005) · Zbl 1114.81330
[5] Grisaru, M. T.; Howe, P. S.; Mezincescu, L.; Nilsson, B.; Townsend, P. K., \(N = 2\) superstrings in a supergravity background, Phys. Lett. B, 162, 116 (1985)
[6] Arutyunov, G.; Borsato, R.; Frolov, S., S-matrix for strings on \(η\)-deformed \(AdS_5 \times S^5\), J. High Energy Phys., 1404, Article 002 pp. (2014)
[7] Arutyunov, G.; Borsato, R.; Frolov, S., Puzzles of eta-deformed \(AdS_5 \times S^5\) · Zbl 1388.83726
[8] Wulff, L., The type II superstring to order \(\theta^4\), J. High Energy Phys., 1307, Article 123 pp. (2013) · Zbl 1342.83436
[9] Callan, C. G.; Martinec, E. J.; Perry, M. J.; Friedan, D., Strings in background fields, Nucl. Phys. B, 262, 593 (1985)
[10] Grisaru, M. T.; Nishino, H.; Zanon, D., Beta function approach to the Green-Schwarz superstring, Phys. Lett. B, 206, 625 (1988) · Zbl 0938.81526
[11] Bellucci, S.; Oerter, R. N., Weyl invariance of the Green-Schwarz heterotic sigma model, Nucl. Phys. B, 363, 573 (1991)
[12] Hoare, B.; Tseytlin, A. A., Type IIB supergravity solution for the T-dual of the \(η\)-deformed \(AdS_5 \times S^5\) superstring, J. High Energy Phys., 1510, Article 060 pp. (2015) · Zbl 1388.83824
[13] Hoare, B.; Tseytlin, A. A., On integrable deformations of superstring sigma models related to \(AdS_n \times S^n\) supercosets, Nucl. Phys. B, 897, 448 (2015) · Zbl 1329.81317
[14] Alday, L. F.; Arutyunov, G.; Frolov, S., Green-Schwarz strings in TsT-transformed backgrounds, J. High Energy Phys., 0606, Article 018 pp. (2006)
[15] Fukuma, M.; Oota, T.; Tanaka, H., Comments on T dualities of Ramond-Ramond potentials on tori, Prog. Theor. Phys., 103, 425 (2000)
[16] Tseytlin, A. A., Sigma model approach to string theory, Int. J. Mod. Phys. A, 4, 1257 (1989)
[17] Friedan, D. H., Nonlinear models in \(2 + \varepsilon\) dimensions, Ann. Phys., 163, 318 (1985)
[18] Berenstein, D.; Leigh, R. G., Quantization of superstrings in Ramond-Ramond backgrounds, Phys. Rev. D, 63, Article 026004 pp. (2001)
[19] Hull, C. M.; Townsend, P. K., Finiteness and conformal invariance in nonlinear \(σ\) models, Nucl. Phys. B, 274, 349 (1986)
[20] Tseytlin, A. A., \(σ\) model Weyl invariance conditions and string equations of motion, Nucl. Phys. B, 294, 383 (1987) · Zbl 1002.81533
[21] Osborn, H., Renormalization and composite operators in nonlinear \(σ\) models, Nucl. Phys. B, 294, 595 (1987)
[22] Curci, G.; Paffuti, G., Consistency between the string background field equation of motion and the vanishing of the conformal anomaly, Nucl. Phys. B, 286, 399 (1987)
[23] Nakayama, Y., Scale invariance vs conformal invariance, Phys. Rep., 569, 1 (2015)
[24] Hollowood, T. J.; Miramontes, J. L.; Schmidtt, D. M., Integrable deformations of strings on symmetric spaces, J. High Energy Phys., 1411, Article 009 pp. (2014) · Zbl 1305.81120
[25] Sfetsos, K.; Thompson, D. C., Spacetimes for \(λ\)-deformations, J. High Energy Phys., 1412, Article 164 pp. (2014)
[26] Klimcik, C., \(η\) and \(λ\) deformations as \(E\)-models, Nucl. Phys. B, 900, 259 (2015) · Zbl 1331.81182
[27] Vicedo, B., Deformed integrable \(σ\)-models, classical R-matrices and classical exchange algebra on Drinfeld doubles, J. Phys. A, 48, 35, 355203 (2015) · Zbl 1422.37037
[28] Schwarz, J. H., Covariant field equations of chiral \(N = 2D = 10\) supergravity, Nucl. Phys. B, 226, 269 (1983)
[29] Blumenhagen, R.; Lüst, D.; Theisen, S., Basic Concepts of String Theory (2013), Springer
[30] Weinberg, S., Coupling constants and vertex functions in string theories, Phys. Lett. B, 156, 309 (1985)
[31] Callan, C. G.; Gan, Z., Vertex operators in background fields, Nucl. Phys. B, 272, 647 (1986)
[32] Buscher, T. H., Path integral derivation of quantum duality in nonlinear sigma models, Phys. Lett. B, 201, 466 (1988)
[33] Schwarz, A. S.; Tseytlin, A. A., Dilaton shift under duality and torsion of elliptic complex, Nucl. Phys. B, 399, 691 (1993)
[34] Fradkin, E. S.; Linetsky, V. Y., On space-time interpretation of the coset models in \(D < 26\) critical string theory, Phys. Lett. B, 277, 73 (1992) · Zbl 0967.81525
[35] Hoare, B.; Roiban, R.; Tseytlin, A. A., On deformations of \(AdS_n \times S^n\) supercosets, J. High Energy Phys., 1406, Article 002 pp. (2014)
[36] Lunin, O.; Roiban, R.; Tseytlin, A. A., Supergravity backgrounds for deformations of \(AdS_n \times S^n\) supercoset string models, Nucl. Phys. B, 891, 106 (2015) · Zbl 1328.81182
[37] Haagensen, P. E., Duality transformations away from conformal points, Phys. Lett. B, 382, 356 (1996)
[38] Drukker, N.; Gross, D. J.; Tseytlin, A. A., Green-Schwarz string in \(AdS_5 \times S^5\): semiclassical partition function, J. High Energy Phys., 0004, Article 021 pp. (2000) · Zbl 0959.81059
[39] Russo, J. G.; Tseytlin, A. A., A class of exact pp wave string models with interacting light cone gauge actions, J. High Energy Phys., 0209, Article 035 pp. (2002)
[40] Hoare, B., Towards a two-parameter q-deformation of \(AdS_3 \times S^3 \times M^4\) superstrings, Nucl. Phys. B, 891, 259 (2015) · Zbl 1328.81178
[41] Matsumoto, T.; Yoshida, K., Yang-Baxter sigma models based on the CYBE, Nucl. Phys. B, 893, 287 (2015) · Zbl 1348.81379
[42] van Tongeren, S. J., Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory · Zbl 1332.81197
[43] Arutyunov, G.; van Tongeren, S. J., \(AdS_5 \times S^5\) mirror model as a string sigma model, Phys. Rev. Lett., 113, Article 261605 pp. (2014)
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