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Division Algebras, Supersymmetry and Higher Gauge Theory UNIVERSITY OF CALIFORNIA RIVERSIDE Division Algebras, Supersymmetry and Higher Gauge Theory A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics by John Gmerek Huerta June 2011 arXiv:1106.3385v1 [math-ph] 17 Jun 2011 Dissertation Committee: Professor John Baez, Chairperson Professor Vyjayanthi Chari Professor Stefano Vidussi Copyright by John Gmerek Huerta 2011 Acknowledgements A dissertation is the capstone to a doctoral program, and the acknowledgements provide a use- ful place to thank the countless people who have helped out along the way, both personally and professionally. First, of course, I thank my advisor, John Baez. It is hard to imagine a better advisor, and no one deserves more credit for my mathematical and professional growth during this program. “Thanks” does not seem sufficient, but it is all I have to give. Also deserving special mention is John’s collaborator, James Dolan. I am convinced there is no subject in mathematics for which Jim does not have some deep insight, and I thank him for sharing a few of these insights with me. Together, John and Jim are an unparalleled team: there are no two better people with whom to talk about mathematics, and no two people more awake to the joy of mathematics. I would also like to thank Geoffrey Dixon, Tevian Dray, Robert Helling, Corinne Manogue, Chris Rogers, Hisham Sati, James Stasheff, and Riccardo Nicoletti for helpful conversations and correspondence. I especially thank Urs Schreiber for many discussions of higher gauge theory and L1-superalgebras, smooth 1-groups, and supergeometry. In addition, I thank Vyjayanthi Chari and Stefano Vidussi for serving on my committee. I thank Mike Stay for providing some beautiful diagrams formatted in TikZ. I thank An Huang, Theo Johnson-Freyd and David Speyer for catching some errors. This work was partially supported by the FQXi grant RFP2-08-04, and by a 2008–2009 Research Mentorship Fellowship from UC Riverside. I owe FQXi and UC Riverside thanks for their support. Many of the people I name above are not just colleagues, but close personal friends. Besides those above, I would also like to thank Jack Bennett, Alex Hoffnung, Felix Marhuenda-Donate, Rob Niemeyer, Tim Ridenour and Jonathan Sarhad, among many others, for making graduate school so much fun. Finally, I would like to thank my girlfriend, Jen Crowley, for being a wonderful, loving companion. iii For my parents, Eliseo and Marion Huerta. iv ABSTRACT OF THE DISSERTATION Division Algebras, Supersymmetry and Higher Gauge Theory by John Gmerek Huerta Doctor of Philosophy, Graduate Program in Mathematics University of California, Riverside, June 2011 Professor John Baez, Chairperson From the four normed division algebras—the real numbers, complex numbers, quaternions and octonions, of dimension k = 1, 2, 4 and 8, respectively—a systematic procedure gives a 3-cocycle on the Poincare´ superalgebra in dimensions k + 2=3, 4, 6 and 10, and a 4-cocycle on the Poincare´ superalgebra in dimensions k+3=4, 5, 7 and 11. The existence of these cocycles follow from spinor identities that hold only in these dimensions, and which are closely related to the existence of the superstring in dimensions k + 2, and the super-2-brane in dimensions k + 3. In general, an (n + 1)-cocycle on a Lie superalgebra yields a ‘Lie n-superalgebra’: that is, roughly, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalge- bra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincare´ superalgebra in dimensions k + 2, and Lie 3-superalgebras extending the Poincare´ superalgebra in dimensions k + 3. We present evidence, based on the work of Sati, Schreiber and Stasheff, that these Lie n-superalgebras describe infinitesimal ‘higher symmetries’ of the superstring and 2-brane. Generically, integrating a Lie n-superalgebra to a Lie n-supergroup yields a ‘Lie n-supergroup’ that is hugely infinite-dimensional. However, when the Lie n-superalgebra is obtained from an (n + 1)-cocycle on a nilpotent Lie superalgebra, there is a geometric procedure to integrate the cocycle to one on the corresponding nilpotent Lie supergroup. In general, a smooth (n + 1)-cocycle on a supergroup yields a ‘Lie n-supergroup’: that is, a weak n-group internal to supermanifolds. Using our geometric procedure to integrate the 3- cocycle in dimensions k + 2, we obtain a Lie 2-supergroup extending the Poincare´ supergroup in those dimensions, and similarly integrating the 4-cocycle in dimensions k + 3, we obtain a Lie 3-supergroup extending the Poincare´ supergroup in those dimensions. v Contents List of Tables viii 1 Introduction 9 1.1 Overview . .9 1.2 Introduction . 10 1.3 Plan of the thesis . 18 1.4 Prior work . 19 2 Spacetime geometry from division algebras 21 2.1 Normed division algebras . 23 2.2 Spacetime geometry in k+2 dimensions . 25 2.3 Spacetime geometry in k+3 dimensions . 31 2.4 The spinor identities . 35 3 Supertranslation algebras and their cohomology 41 3.1 Superalgebra . 41 3.2 Cohomology of Lie superalgebras . 42 4 An application: super-Yang–Mills theory 47 5 Lie n-superalgebras from Lie superalgebra cohomology 53 5.1 Examples of slim Lie n-superalgebras . 57 5.1.1 The string Lie 2-algebra . 57 5.1.2 The Heisenberg Lie 2-algebra . 57 5.1.3 The supertranslation Lie n-superalgebras . 58 5.1.4 Superstring Lie 2-superalgebras, 2-brane Lie 3-superalgebras . 58 6 Lie n-groups from group cohomology 62 6.1 Lie 2-groups . 65 6.2 Lie 3-groups . 74 7 Integrating nilpotent Lie n-algebras 83 7.1 Integrating Lie algebra cochains . 86 7.2 Examples: Explicitly integrating 0-, 1-, 2- and 3-cochains . 95 7.3 The Heisenberg Lie 2-group . 104 vi 8 Supergeometry and supergroups 106 8.1 Supermanifolds . 108 8.1.1 Super vector spaces as supermanifolds . 108 8.1.2 Supermanifolds in general . 113 8.2 Supergroups from nilpotent Lie superalgebras . 114 9 Lie n-supergroups from supergroup cohomology 118 10 Integrating nilpotent Lie n-superalgebras 125 11 Superstring Lie 2-supergroups, 2-brane Lie 3-supergroups 134 11.1 Outlook . 138 Bibliography 139 vii List of Tables 2.1 Division algebra notation vs. index notation . 31 viii Chapter 1 Introduction 1.1 Overview There is a deep relationship between supersymmetry, division algebras, and higher gauge theory. In this thesis, we begin to tell this story: how division algebras give rise to higher infinitesimal symmetries of strings and membranes, modeled by a generalization of a Lie algebra called a ‘Lie n- algebra’, and how this infinitesimal picture can be integrated to global one, with higher symmetries modeled by a ‘Lie n-group’. In this overview, we want to take the opportunity to explain the big picture, postponing references until the next section. From a physical perspective, gauge theory is the geometric language which allows us to describe how point particles change as they move through spacetime. Higher gauge theory is a generalization which describes how strings and membranes change as they move through spacetime. We can view higher gauge theory as a categorification of gauge theory, which is intuitively clear from the diagrams we use to describe higher categories: as a particle moves through spacetime from point x to point y, it sweeps out a worldline γ that we can view as a morphism from x to y in a certain category: γ x •• y : The job of a connection in gauge theory is to assign to γ an element hol(γ) in the gauge group which describes how the state of our particle changes as it moves along γ. Boosting up a dimension, when a string moves through spacetime, it sweeps out a worldsheet Σ, which we can view as a 2-morphism in a certain 2-category: ••Σ : D The job of a ‘2-connection’ in ‘higher gauge theory’ is to assign to Σ an element hol(Σ) in the ‘higher gauge group’ which describes how the state of our string changes as it moves along Σ. In practice, the strings and membranes of interest in physics are supersymmetric, so they are called superstrings and supermembranes. This also leads to higher gauge theory, but it goes through the normed division algebras. There is a mysterious connection between supersymmetry and the 9 four normed division algebras: the real numbers, complex numbers, quaternions and octonions. This can be seen in super-Yang–Mills theory, in superstring theory, and in theories of supermembranes and supergravity. Most simply, the connection is visible from the fact that the normed division algebras have dimensions 1, 2, 4 and 8, while classical superstring theories and minimal super- Yang–Mills theories live in spacetimes of dimension two higher: 3, 4, 6 and 10. The simplest classical super-2-brane theories make sense in spacetimes of dimensions three higher: 4, 5, 7 and 11. Classical supergravity makes sense in all of these dimensions, but the octonionic cases are the most important: in 10 dimensions supergravity is a low-energy limit of superstring theory, while in 11 dimensions it is believed to be a low-energy limit of ‘M-theory’, which incorporates the 2-brane. These numerical relationships are far from coincidental. They arise because we can use the normed division algebras to construct the spacetimes in question, as well as their associated spinors. A certain spinor identity that holds in dimensions 3, 4, 6 and 10 is an easy consequence of this construction, as is a related identity that holds in dimensions 4, 5, 7 and 11. These identities are fundamental to the physical theories just listed.
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