Abstract
We explore type II supersymmetric double field theory in superspace. The double supervielbein is an element of the orthosymplectic group OSp(10, 10|64), which also governs the structure of generalized superdiffeomorphisms. Unlike bosonic double field theory, the local tangent space must be enhanced from the double Lorentz group in order to eliminate unphysical components of the supervielbein and to define covariant torsion and curvature tensors. This leads to an infinite hierarchy of local tangent space symmetries, which are connected to the super-Maxwell∞ algebra. A novel feature of type II is the Ramond-Ramond sector, which can be encoded as an orthosymplectic spinor (encoding the complex of super p-forms in conventional superspace). Its covariant field strength bispinor itself appears as a piece of the supervielbein. We provide a concise discussion of the superspace Bianchi identities through dimension two and show how to recover the component supersymmetry transformations of type II DFT. In addition, we show how the democratic formulation of type II superspace may be recovered by gauge-fixing.
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References
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 09 (2009) 090 [arXiv:0908.1792] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 07 (2010) 016 [arXiv:1003.5027] [INSPIRE].
O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 08 (2010) 008 [arXiv:1006.4823] [INSPIRE].
O. Hohm, S.K. Kwak and B. Zwiebach, Unification of Type II Strings and T-duality, Phys. Rev. Lett. 107 (2011) 171603 [arXiv:1106.5452] [INSPIRE].
O. Hohm, S.K. Kwak and B. Zwiebach, Double Field Theory of Type II Strings, JHEP 09 (2011) 013 [arXiv:1107.0008] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Ramond-Ramond Cohomology and O(D, D) T-duality, JHEP 09 (2012) 079 [arXiv:1206.3478] [INSPIRE].
I. Jeon, K. Lee, J.-H. Park and Y. Suh, Stringy Unification of Type IIA and IIB Supergravities under N = 2 D = 10 Supersymmetric Double Field Theory, Phys. Lett. B 723 (2013) 245 [arXiv:1210.5078] [INSPIRE].
P. West, E11, generalised space-time and IIA string theory, Phys. Lett. B 696 (2011) 403 [arXiv:1009.2624] [INSPIRE].
A. Rocen and P. West, E11, generalised space-time and IIA string theory: the R-R sector, in Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer, A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger, eds., World Scientific (2010), pg. 403, https://doi.org/10.1142/9789814412551_0020 [arXiv:1012.2744] [INSPIRE].
D. Brace, B. Morariu and B. Zumino, T duality and Ramond-Ramond backgrounds in the matrix model, Nucl. Phys. B 549 (1999) 181 [hep-th/9811213] [INSPIRE].
M. Fukuma, T. Oota and H. Tanaka, Comments on T dualities of Ramond-Ramond potentials on tori, Prog. Theor. Phys. 103 (2000) 425 [hep-th/9907132] [INSPIRE].
S.F. Hassan, T duality, space-time spinors and RR fields in curved backgrounds, Nucl. Phys. B 568 (2000) 145 [hep-th/9907152] [INSPIRE].
S.F. Hassan, SO(d, d) transformations of Ramond-Ramond fields and space-time spinors, Nucl. Phys. B 583 (2000) 431 [hep-th/9912236] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Supergravity as Generalised Geometry I: Type II Theories, JHEP 11 (2011) 091 [arXiv:1107.1733] [INSPIRE].
A. Coimbra, C. Strickland-Constable and D. Waldram, Generalised Geometry and type II Supergravity, Fortsch. Phys. 60 (2012) 982 [arXiv:1202.3170] [INSPIRE].
D. Butter, Notes on Ramond-Ramond spinors and bispinors in double field theory, arXiv:2208.11162 [INSPIRE].
D. Butter, Exploring the geometry of supersymmetric double field theory, JHEP 01 (2022) 152 [arXiv:2101.10328] [INSPIRE].
J. Gomis and A. Kleinschmidt, On free Lie algebras and particles in electro-magnetic fields, JHEP 07 (2017) 085 [arXiv:1705.05854] [INSPIRE].
J. Gomis, A. Kleinschmidt and J. Palmkvist, Symmetries of M-theory and free Lie superalgebras, JHEP 03 (2019) 160 [arXiv:1809.09171] [INSPIRE].
M. Poláček and W. Siegel, Natural curvature for manifest T-duality, JHEP 01 (2014) 026 [arXiv:1308.6350] [INSPIRE].
O. Hohm and S.K. Kwak, \( \mathcal{N} \) = 1 supersymmetric double field theory, JHEP 03 (2012) 080 [arXiv:1111.7293] [INSPIRE].
I. Jeon, K. Lee and J.-H. Park, Supersymmetric Double Field Theory: Stringy Reformulation of Supergravity, Phys. Rev. D 85 (2012) 081501 [arXiv:1112.0069] [Erratum ibid. 86 (2012) 089903] [INSPIRE].
M. Hatsuda, K. Kamimura and W. Siegel, Superspace with manifest T-duality from type II superstring, JHEP 06 (2014) 039 [arXiv:1403.3887] [INSPIRE].
M. Hatsuda, K. Kamimura and W. Siegel, Ramond-Ramond gauge fields in superspace with manifest T-duality, JHEP 02 (2015) 134 [arXiv:1411.2206] [INSPIRE].
M. Cederwall, Double supergeometry, JHEP 06 (2016) 155 [arXiv:1603.04684] [INSPIRE].
C.M. Hull, Timelike T duality, de Sitter space, large N gauge theories and topological field theory, JHEP 07 (1998) 021 [hep-th/9806146] [INSPIRE].
L. Wulff and A.A. Tseytlin, Kappa-symmetry of superstring sigma model and generalized 10d supergravity equations, JHEP 06 (2016) 174 [arXiv:1605.04884] [INSPIRE].
R. D’Auria, P. Fré, P.K. Townsend and P. van Nieuwenhuizen, Invariance of actions, rheonomy, and the new minimal N = 1 supergravity in the group manifold approach, Annals Phys. 155 (1984) 423.
L. Castellani, R. D’Auria and P. Fré, Supergravity and Superstrings: A Geometric Perspective. Vol. 2: Supergravity, World Scientific, Singapore (1991).
S.J. Gates, Jr., Ectoplasm has no topology: The Prelude, in 2nd International Seminar on Supersymmetries and Quantum Symmetries: Dedicated to the Memory of Victor I. Ogievetsky, Dubna, Russia (1997), pg. 46 [hep-th/9709104] [INSPIRE].
S.J. Gates, Jr., M.T. Grisaru, M.E. Knutt-Wehlau and W. Siegel, Component actions from curved superspace: Normal coordinates and ectoplasm, Phys. Lett. B 421 (1998) 203 [hep-th/9711151] [INSPIRE].
A.H. Chamseddine, N = 4 supergravity coupled to N = 4 matter and hidden symmetries, Nucl. Phys. B 185 (1981) 403.
E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Ten-Dimensional Maxwell-Einstein Supergravity, Its Currents, and the Issue of Its Auxiliary Fields, Nucl. Phys. B 195 (1982) 97 [INSPIRE].
E. Cremmer, B. Julia and J. Scherk, Supergravity theory in 11 dimensions, Phys. Lett. B 76 (1978) 409 [INSPIRE].
L.J. Romans, Massive N = 2a Supergravity in Ten-Dimensions, Phys. Lett. B 169 (1986) 374 [INSPIRE].
J.H. Schwarz and P.C. West, Symmetries and Transformations of Chiral N = 2 D = 10 Supergravity, Phys. Lett. B 126 (1983) 301 [INSPIRE].
P.S. Howe and P.C. West, The Complete N = 2, D = 10 Supergravity, Nucl. Phys. B 238 (1984) 181 [INSPIRE].
B.E.W. Nilsson, Simple 10-dimensional supergravity in superspace, Nucl. Phys. B 188 (1981) 176 [INSPIRE].
J.L. Carr, S.J. Gates, Jr. and R.N. Oerter, D = 10, N = 2a Supergravity in Superspace, Phys. Lett. B 189 (1987) 68 [INSPIRE].
L. Wulff, The type II superstring to order θ4, JHEP 07 (2013) 123 [arXiv:1304.6422] [INSPIRE].
E. Bergshoeff, R. Kallosh, T. Ortin, D. Roest and A. Van Proeyen, New formulations of D = 10 supersymmetry and D8-O8 domain walls, Class. Quant. Grav. 18 (2001) 3359 [hep-th/0103233] [INSPIRE].
P. Howe and J. Palmkvist, Forms and algebras in (half-)maximal supergravity theories, JHEP 05 (2015) 032 [arXiv:1503.00015] [INSPIRE].
E.A. Bergshoeff, M. de Roo, S.F. Kerstan and F. Riccioni, IIB supergravity revisited, JHEP 08 (2005) 098 [hep-th/0506013] [INSPIRE].
E. Bergshoeff, P.S. Howe, S. Kerstan and L. Wulff, Kappa-symmetric SL(2, R) covariant D-brane actions, JHEP 10 (2007) 050 [arXiv:0708.2722] [INSPIRE].
A. Kleinschmidt, I. Schnakenburg and P.C. West, Very extended Kac-Moody algebras and their interpretation at low levels, Class. Quant. Grav. 21 (2004) 2493 [hep-th/0309198] [INSPIRE].
G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A.A. Tseytlin, Scale invariance of the η-deformed AdS5 × S5 superstring, T-duality and modified type II equations, Nucl. Phys. B 903 (2016) 262 [arXiv:1511.05795] [INSPIRE].
Y. Sakatani, S. Uehara and K. Yoshida, Generalized gravity from modified DFT, JHEP 04 (2017) 123 [arXiv:1611.05856] [INSPIRE].
J.-H. Park, Green-Schwarz superstring on doubled-yet-gauged spacetime, JHEP 11 (2016) 005 [arXiv:1609.04265] [INSPIRE].
I. Bandos, Superstring in doubled superspace, Phys. Lett. B 751 (2015) 408 [arXiv:1507.07779] [INSPIRE].
J.-I. Sakamoto and Y. Sakatani, Local β-deformations and Yang-Baxter sigma model, JHEP 06 (2018) 147 [arXiv:1803.05903] [INSPIRE].
K. Morand and J.-H. Park, Classification of non-Riemannian doubled-yet-gauged spacetime, Eur. Phys. J. C 77 (2017) 685 [arXiv:1707.03713] [Erratum ibid. 78 (2018) 901] [INSPIRE].
K. Cho and J.-H. Park, Remarks on the non-Riemannian sector in Double Field Theory, Eur. Phys. J. C 80 (2020) 101 [arXiv:1909.10711] [INSPIRE].
J.-H. Park and S. Sugimoto, String Theory and non-Riemannian Geometry, Phys. Rev. Lett. 125 (2020) 211601 [arXiv:2008.03084] [INSPIRE].
D.S. Berman, C.D.A. Blair and R. Otsuki, Non-Riemannian geometry of M-theory, JHEP 07 (2019) 175 [arXiv:1902.01867] [INSPIRE].
J. Gomis and H. Ooguri, Nonrelativistic closed string theory, J. Math. Phys. 42 (2001) 3127 [hep-th/0009181] [INSPIRE].
P.C. West, E11, SL(32) and central charges, Phys. Lett. B 575 (2003) 333 [hep-th/0307098] [INSPIRE].
G. Bossard, A. Kleinschmidt, J. Palmkvist, C.N. Pope and E. Sezgin, Beyond E11, JHEP 05 (2017) 020 [arXiv:1703.01305] [INSPIRE].
G. Bossard, A. Kleinschmidt and E. Sezgin, A master exceptional field theory, JHEP 06 (2021) 185 [arXiv:2103.13411] [INSPIRE].
G. Bossard, A. Kleinschmidt and E. Sezgin, On supersymmetric E11 exceptional field theory, JHEP 10 (2019) 165 [arXiv:1907.02080] [INSPIRE].
J.-H. Park, Lecture note on Clifford algebra, J. Korean Phys. Soc. 81 (2022) 1 [arXiv:2205.09509] [INSPIRE].
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Butter, D. Type II double field theory in superspace. J. High Energ. Phys. 2023, 187 (2023). https://doi.org/10.1007/JHEP02(2023)187
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DOI: https://doi.org/10.1007/JHEP02(2023)187