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Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations. (English) Zbl 1390.83426
Summary: We determine the constraints imposed on the 10d target superspace geometry by the requirement of classical kappa-symmetry of the Green-Schwarz superstring. In the type I case we find that the background must satisfy a generalization of type I supergravity equations. These equations depend on an arbitrary vector $$X_a$$ and imply the one-loop scale invariance of the GS sigma model. In the special case when $$X_a$$ is the gradient of a scalar $$\phi$$ (dilaton) one recovers the standard type I equations equivalent to the 2d Weyl invariance conditions of the superstring sigma model. In the type II case we find a generalized version of the 10d supergravity equations the bosonic part of which was introduced in [G. Arutyunov et al., Nucl. Phys., B 903, 262–303 (2016; Zbl 1332.81167)]. These equations depend on two vectors $$\mathrm{X}_a$$ and $$K_a$$ subject to 1st order differential relations (with the equations in the NS-NS sector depending only on the combination $$X_a=\mathrm{X}_a+K_a$$). In the special case of $$K_a=0$$ one finds that $$\mathrm{X}_a=\partial_a\phi$$ and thus obtains the standard type II supergravity equations. New generalized solutions are found if $$K_a$$ is chosen to be a Killing vector (and thus they exist only if the metric admits an isometry). Non-trivial solutions of the generalized equations describe $$K$$-isometric backgrounds that can be mapped by T-duality to type II supergravity solutions with dilaton containing a linear isometry-breaking term. Examples of such backgrounds appeared recently in the context of integrable $$\eta$$-deformations of $$\mathrm{AdS}_n\times S^n$$ sigma models. The classical kappa-symmetry thus does not, in general, imply the 2d Weyl invariance conditions for the GS sigma model (equivalent to type II supergravity equations) but only weaker scale invariance type conditions.

##### MSC:
 8.3e+51 Supergravity
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##### References:
 [1] Witten, E., Twistor-like transform in ten-dimensions, Nucl. Phys., B 266, 245, (1986) · Zbl 0608.53068 [2] Grisaru, MT; Howe, PS; Mezincescu, L.; Nilsson, B.; Townsend, PK, $$N$$ = 2 superstrings in a supergravity background, Phys. Lett., B 162, 116, (1985) [3] Howe, PS; West, PC, the complete N = 2, D = 10 supergravity, Nucl. Phys., B 238, 181, (1984) [4] Shapiro, JA; Taylor, CC, Superspace supergravity from the superstring, Phys. Lett., B 186, 69, (1987) [5] Berkovits, N.; Howe, PS, Ten-dimensional supergravity constraints from the pure spinor formalism for the superstring, Nucl. Phys., B 635, 75, (2002) · Zbl 0996.81075 [6] Berkovits, N., Super Poincaré covariant quantization of the superstring, JHEP, 04, 018, (2000) · Zbl 0959.81065 [7] Bergshoeff, E.; Sezgin, E.; Townsend, PK, Supermembranes and eleven-dimensional supergravity, Phys. Lett., B 189, 75, (1987) · Zbl 1156.81434 [8] Howe, PS, Weyl superspace, Phys. Lett., B 415, 149, (1997) [9] Arutyunov, G.; Frolov, S.; Hoare, B.; Roiban, R.; Tseytlin, AA, scale invariance of the η-deformed AdS_{5} × $$S$$\^{}{5}superstring, T-duality and modified type-II equations, Nucl. Phys., B 903, 262, (2016) · Zbl 1332.81167 [10] Callan, CG; Martinec, EJ; Perry, MJ; Friedan, D., Strings in background fields, Nucl. Phys., B 262, 593, (1985) [11] Hull, CM; Townsend, PK, Finiteness and conformal invariance in nonlinear σ models, Nucl. Phys., B 274, 349, (1986) [12] Tseytlin, AA, Conformal anomaly in two-dimensional σ-model on curved background and strings, Phys. Lett., B 178, 34, (1986) [13] Shore, GM, A local renormalization group equation, diffeomorphisms and conformal invariance in σ models, Nucl. Phys., B 286, 349, (1987) [14] Nilsson, BEW, Simple ten-dimensional supergravity in superspace, Nucl. Phys., B 188, 176, (1981) [15] Wulff, L., the type-II superstring to order θ\^{}{4}, JHEP, 07, 123, (2013) · Zbl 1342.83436 [16] Borsato, R.; Tseytlin, AA; Wulff, L., supergravity background of λ-deformed model for AdS_{2} × $$S$$\^{}{2}supercoset, Nucl. Phys., B 905, 264, (2016) · Zbl 1332.81170 [17] C. Klimčík, On integrability of the Yang-Baxter σ-model, J. Math. Phys.50 (2009) 043508 [arXiv:0802.3518] [INSPIRE]. [18] Delduc, F.; Magro, M.; Vicedo, B., an integrable deformation of the AdS_{5} × $$S$$\^{}{5}superstring action, Phys. Rev. Lett., 112, 051601, (2014) · Zbl 1333.81322 [19] Arutyunov, G.; Borsato, R.; Frolov, S., puzzles of η-deformed AdS_{5} × $$S$$\^{}{5}, JHEP, 12, 049, (2015) · Zbl 1388.83726 [20] Hollowood, TJ; Miramontes, JL; Schmidtt, DM, an integrable deformation of the AdS_{5} × $$S$$\^{}{5}superstring, J. Phys., A 47, 495402, (2014) · Zbl 1305.81120 [21] H. Kyono and K. Yoshida, Supercoset construction of Yang-Baxter deformed AdS_{5} × $$S$$\^{}{5}backgrounds, arXiv:1605.02519 [INSPIRE]. · Zbl 1361.81128 [22] B. Hoare and S.J. van Tongeren, On Jordanian deformations of AdS_{5}and supergravity, arXiv:1605.03554 [INSPIRE]. · Zbl 1352.81050 [23] Demulder, S.; Sfetsos, K.; Thompson, DC, integrable λ-deformations: squashing coset CFTs and AdS_{5} × $$S$$\^{}{5}, JHEP, 07, 019, (2015) · Zbl 1388.83790 [24] Sfetsos, K.; Thompson, DC, Spacetimes for λ-deformations, JHEP, 12, 164, (2014) [25] Hoare, B.; Tseytlin, AA, type IIB supergravity solution for the T-dual of the η-deformed AdS_{5} × $$S$$\^{}{5}superstring, JHEP, 10, 060, (2015) · Zbl 1388.83824 [26] Hoare, B.; Tseytlin, AA, on integrable deformations of superstring σ-models related to AdS_{$$n$$} × $$S$$\^{}{$$n$$}supercosets, Nucl. Phys., B 897, 448, (2015) · Zbl 1329.81317 [27] Kulik, B.; Roiban, R., T duality of the Green-Schwarz superstring, JHEP, 09, 007, (2002) [28] Sorokin, DP, Superbranes and superembeddings, Phys. Rept., 329, 1, (2000) · Zbl 1006.83056 [29] Dragon, N., Torsion and curvature in extended supergravity, Z. Phys., C 2, 29, (1979) [30] Wulff, L., On integrability of strings on symmetric spaces, JHEP, 09, 115, (2015) · Zbl 1388.81611 [31] Howe, PS; Umerski, A., On superspace supergravity in ten-dimensions, Phys. Lett., B 177, 163, (1986) [32] Mikhailov, A., Cornering the unphysical vertex, JHEP, 11, 082, (2012) [33] Mikhailov, A., Vertex operators of ghost number three in type IIB supergravity, Nucl. Phys., B 907, 509, (2016) · Zbl 1336.81074
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