Romans Massive QP Manifolds
Abstract
:1. Introduction
2. QP Manifolds for IIA Branes
2.1. D2 ( Algebroid)
- 1
- For , the identities arising from correspond precisely to Bianchi identities and field equations for the fluxes of massive type IIA with the conventions of [6].
- 2
- The field can naturally be viewed as a gauge connection for a gauge transformation generated by rescalings , . For these transformations to still satisfy (3) then we also need to shift . Putting this into (5), we see that this has the effect of transforming . The constraint (11) then just says that this is a flat connection. Flat connections on gauge bundles are always gauge trivial. That is, we can always pick a gauge where globally. Hence, this is classically equivalent to the massive IIA Bianchi identities.
2.2. D4 ( Algebroid)
- 1
- Setting , we recover the Bianchi identities for massive IIA supergravity.
- 2
- is a flat connection coming from the same rescaling gauge symmetry. The charges of all the fluxes under this are different, however, to the D2 algebroid.
2.3. NS5 ( Algebroid)
- 1
- Setting , we recover the Bianchi identities for massless IIA supergravity. This is always possible as is a flat connection.
- 2
- Note that, unlike the D2 and D4 branes, we do not have the Romans mass appearing here. Viewing the Romans mass as the flux of the D8 brane, we note that the NS5 brane only couples to the D8 brane via a D6 brane [17]. Our construction here, however, does not include any D6 branes since we have no flux. We therefore, do not expect that the QP manifold for NS5 branes would allow for massive deformations and this is indeed what we find.
3. IIA/M-Theory Duality
3.1. M5 to NS5
3.2. M2 to D2
4. Applications
4.1. AKSZ Models
4.1.1. D2 Brane
4.1.2. NS5 Brane
4.2. The (Topologically Massive) D2 Brane Phase Space
5. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
1 | A supermanifold with an extra -grading on its structure sheaf, so all functions f have a well-defined degree ; for us, and equal the Grassmann parity, and this grading is equivalent to the existence of a degree-counting vector field (“Euler”), which we employ later. |
2 | The anchor map is intrinsic to the definition of these higher algebroids; see, e.g., [1]. |
3 | A priori, one could have terms such as and . However, we will assume that is invertible, which is equivalent to saying that the anchor map associated to the algebroid is surjective. Hence, one can always perform a canonical transformation to set and . |
4 | The necessity of performing asymplectomorphism before the symplectic reduction was also observed in [18]. |
5 | In double field theory, Romans IIA theory arises by giving the Ramond–Ramond fields a linear dependence on the dual coordinates [22]. |
References
- Baraglia, D. Leibniz algebroids, twistings and exceptional generalized geometry. J. Geom. Phys. 2012, 62, 903–934. [Google Scholar] [CrossRef]
- Coimbra, A.; Strickland-Constable, C.; Waldram, D. Supergravity as Generalised Geometry I: Type II Theories. J. High Energy Phys. 2011, 11, 091. [Google Scholar] [CrossRef] [Green Version]
- Hull, C.M. Generalised Geometry for M-Theory. J. High Energy Phys. 2007, 7, 79. [Google Scholar] [CrossRef]
- Pires Pacheco, P.; Waldram, D. M-theory, exceptional generalised geometry and superpotentials. J. High Energy Phys. 2008, 9, 123. [Google Scholar] [CrossRef]
- Coimbra, A.; Strickland-Constable, C.; Waldram, D. Supergravity as Generalised Geometry II: Ed(d) × ℝ+ and M theory. J. High Energy Phys. 2014, 3, 19. [Google Scholar] [CrossRef]
- Cassani, D.; de Felice, O.; Petrini, M.; Strickland-Constable, C.; Waldram, D. Exceptional generalised geometry for massive IIA and consistent reductions. J. High Energy Phys. 2016, 8, 74. [Google Scholar] [CrossRef] [Green Version]
- Cattaneo, A.S.; Schaetz, F. Introduction to supergeometry. Rev. Math. Phys. 2011, 23, 669–690. [Google Scholar] [CrossRef] [Green Version]
- Roytenberg, D. Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds; University of California: Berkeley, CA, USA, 1999. [Google Scholar]
- Ševera, P. Letters to Alan Weinstein about Courant algebroids. arXiv 2017, arXiv:1707.00265. [Google Scholar]
- Arvanitakis, A.S. Brane Wess-Zumino terms from AKSZ and exceptional generalised geometry as an L∞-algebroid. Adv. Theor. Math. Phys. 2019, 23, 1159–1213. [Google Scholar] [CrossRef] [Green Version]
- de Wit, B.; Nicolai, H.; Samtleben, H. Gauged Supergravities, Tensor Hierarchies, and M-Theory. J. High Energy Phys. 2008, 2, 44. [Google Scholar] [CrossRef] [Green Version]
- Alexandrov, M.; Schwarz, A.; Zaboronsky, O.; Kontsevich, M. The Geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A 1997, 12, 1405–1429. [Google Scholar] [CrossRef] [Green Version]
- Ikeda, N.; Koizumi, K. Current Algebras and QP Manifolds. Int. J. Geom. Meth. Mod. Phys. 2013, 10, 1350024. [Google Scholar] [CrossRef] [Green Version]
- Alekseev, A.; Strobl, T. Current algebras and differential geometry. J. High Energy Phys. 2005, 3, 35. [Google Scholar] [CrossRef]
- Arvanitakis, A.S. Brane current algebras and generalised geometry from QP manifolds: Or, “when they go high, we go low”. arXiv 2021, arXiv:2103.08608. [Google Scholar] [CrossRef]
- Romans, L.J. Massive N=2a Supergravity in Ten-Dimensions. Phys. Lett. B 1986, 169, 374. [Google Scholar] [CrossRef]
- Bergshoeff, E.; Lozano, Y.; Ortin, T. Massive branes. Nucl. Phys. B 1998, 518, 363–423. [Google Scholar] [CrossRef] [Green Version]
- Arvanitakis, A.S.; Blair, C.D.A.; Thompson, D.C. A QP perspective on topology change in Poisson-Lie T-duality. arXiv 2021, arXiv:2110.08179. [Google Scholar]
- Kokenyesi, Z.; Sinkovics, A.; Szabo, R.J. AKSZ Constructions for Topological Membranes on G2-Manifolds. Fortsch. Phys. 2018, 66, 1800018. [Google Scholar] [CrossRef] [Green Version]
- Hull, C.M. Massive string theories from M theory and F theory. J. High Energy Phys. 1998, 11, 027. [Google Scholar] [CrossRef] [Green Version]
- Ciceri, F.; Guarino, A.; Inverso, G. The exceptional story of massive IIA supergravity. J. High Energy Phys. 2016, 8, 154. [Google Scholar] [CrossRef] [Green Version]
- Hohm, O.; Kwak, S.K. Massive Type II in Double Field Theory. J. High Energy Phys. 2011, 11, 86. [Google Scholar] [CrossRef] [Green Version]
- Ikeda, N. Topological field theories and geometry of Batalin-Vilkovisky algebras. J. High Energy Phys. 2002, 10, 076. [Google Scholar] [CrossRef]
- Roytenberg, D. AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories. Lett. Math. Phys. 2007, 79, 143–159. [Google Scholar] [CrossRef] [Green Version]
- Green, M.B.; Hull, C.M.; Townsend, P.K. D-brane Wess-Zumino actions, t duality and the cosmological constant. Phys. Lett. B 1996, 382, 65–72. [Google Scholar] [CrossRef] [Green Version]
- Lozano, Y. Eleven-dimensions from the massive D2-brane. Phys. Lett. B 1997, 414, 52–57. [Google Scholar] [CrossRef] [Green Version]
- de Boer, J.; Shigemori, M. Exotic Branes in String Theory. Phys. Rept. 2013, 532, 65–118. [Google Scholar] [CrossRef] [Green Version]
- Bergshoeff, E.A.; Riccioni, F. String Solitons and T-duality. J. High Energy Phys. 2011, 5, 131. [Google Scholar] [CrossRef] [Green Version]
- Deser, S.; Jackiw, R.; Templeton, S. Topologically Massive Gauge Theories. Annals Phys. 1982, 140, 372–411. [Google Scholar] [CrossRef]
- Howe, P.S.; Lambert, N.D.; West, P.C. A New massive type IIA supergravity from compactification. Phys. Lett. B 1998, 416, 303–308. [Google Scholar] [CrossRef] [Green Version]
- Lavrinenko, I.V.; Lu, H.; Pope, C.N. Fiber bundles and generalized dimensional reduction. Class. Quant. Grav. 1998, 15, 2239–2256. [Google Scholar] [CrossRef] [Green Version]
- Riccioni, F. Local E(11) and the gauging of the trombone symmetry. Class. Quant. Grav. 2010, 27, 125009. [Google Scholar] [CrossRef] [Green Version]
- Arvanitakis, A.S.; Blair, C.D.A. Unifying Type-II Strings by Exceptional Groups. Phys. Rev. Lett. 2018, 120, 211601. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Arvanitakis, A.S.; Blair, C.D.A. The Exceptional Sigma Model. J. High Energy Phys. 2018, 4, 64. [Google Scholar] [CrossRef] [Green Version]
- Bergshoeff, E.; Cowdall, P.M.; Townsend, P.K. Massive IIA supergravity from the topologically massive D2-brane. Phys. Lett. B 1997, 410, 13–21. [Google Scholar] [CrossRef] [Green Version]
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).
Share and Cite
Arvanitakis, A.S.; Malek, E.; Tennyson, D. Romans Massive QP Manifolds. Universe 2022, 8, 147. https://doi.org/10.3390/universe8030147
Arvanitakis AS, Malek E, Tennyson D. Romans Massive QP Manifolds. Universe. 2022; 8(3):147. https://doi.org/10.3390/universe8030147
Chicago/Turabian StyleArvanitakis, Alex S., Emanuel Malek, and David Tennyson. 2022. "Romans Massive QP Manifolds" Universe 8, no. 3: 147. https://doi.org/10.3390/universe8030147