THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Off-shell, maximal supersymmetry & exceptional geometry ANNA KARLSSON DepartmentofFundamentalPhysics CHALMERS UNIVERSITY OF TECHNOLOGY Go¨teborg,Sweden2015 Off-shell, maximal supersymmetry & exceptional geometry ANNAKARLSSON ISBN978-91-7597-230-5 c AnnaKarlsson,2015 (cid:13) DoktorsavhandlingarvidChalmerstekniskaho¨gskola Nyserienr3911 ISSN0346-718X DepartmentofFundamentalPhysics ChalmersUniversityofTechnology SE–41296Go¨teborg,Sweden Phone:+46(0)31–7721000 PrintedbyChalmersReproservice ChalmersUniversityofTechnology Go¨teborg,Sweden2015 iii Off-shell, maximal supersymmetry & exceptional geometry ANNAKARLSSON DepartmentofFundamentalPhysics ChalmersUniversityofTechnology Abstract Two lines of research are presented in this thesis, both with a focus on funda- mental properties within the supersymmetric theories of high energy physics. The first features analyses based on the actions for maximal supersymmetric Yang–Mills theory and supergravity, provided by the pure spinor formalism. Thesecondisthedevelopmentofthetheoryofexceptionalgeometry. The analyses within maximally supersymmetric Yang–Mills theory and super- gravity, benefitting from the pure spinor formalism, centres around the inves- tigations of the UV divergences of the amplitude diagrams, where the case of maximal supergravity is subject to ongoing research. There is currently an in- teresting development in connection to the four-dimensional theory, regarding a possible finiteness, indicated in the pure spinor formalism setting of the re- search articles appended to this thesis. The results are contrary to the expected divergence,currentlyofamplitudediagramswithmorethansixloops. Exceptional geometry is an extension of supergravity that is constructed to in- corporate U-duality. It is a geometric formulation with an extended space, in a way similar to in doubled geometry, where T-duality is the symmetry accom- modated for. Constituting a rather recent area of research, the theory is under development with respect to the inherent symmetries, the tensor formalism, etc.,regardingthepropertiesaffectedbytheextendedspace.Itsrecognisedfea- tures, construction and concepts to be investigated are the objects of interest in thisthesis. Keywords:Maximalsupersymmetry,maximalsupergravity,purespinors,loop amplitudes,exceptionalgeometry,U-duality,highenergyparticlephysics v List of appended research papers Thisthesisisbasedontheworkcontainedinthefollowingpublications, referredtobyRomannumeralsinthetext. I. PurespinorsuperfieldsandBorn–Infeldtheory M.CederwallandA.Karlsson J.HighEnergyPhys.11(2011)134 arXiv:1109.0809[hep-th] II. Loopamplitudesinmaximalsupergravitywithmanifestsupersymmetry M.CederwallandA.Karlsson J.HighEnergyPhys.03(2013)114 arXiv:1212.5175[hep-th] III. Exceptionalgeometryandtensorfields M.Cederwall,J.EdlundandA.Karlsson J.HighEnergyPhys.07(2013)028 arXiv:1302.6736[hep-th] IV. Ultravioletdivergencesinmaximalsupergravityfromapurespinorpoint ofview A.Karlsson J.HighEnergyPhys.04(2015)165 arXiv:1412.5983[hep-th] V. Pure spinor indications of ultraviolet finiteness in D = 4 maximal super- gravity A.Karlsson arXiv:1506.07505[hep-th] Other publications I have also written the following conference contribution, not included in this thesis: Loopamplitudediagramsinmanifest,maximalsupergravity • A.Karlsson SpringerProc.Phys.153(2014)95 Ongoingresearchprojectsarementionedinchapter1,4and5. vii Acknowledgements Thankyou, to my supervisor, Martin Cederwall, for being so inspiring and warm-hearted, andhavingsuchagreatsenseofhumour, tomymotherandlategrandmother,forbelievinginme, to family, friends and colleagues that have been there for me in different ways, sometimeslighteningupmyeveryday, and to Johanna and Anton, for your helpful comments on the drafts of this thesis. Anna Tomymother &grandparents