An Investigation into Supersymmetric Flux Backgrounds and their Moduli via Generalised Geometry

We provide a detailed analysis of flux backgrounds of string and M-theory that preserve minimalsupersymmetry in terms of (exceptional) generalised geometry. The geometry in each case isconveniently described in terms of generalised G-structures, where the integrability conditionsare equivalent to the Killing spinor equations. Interestingly, there seems to be a commonstructure among the G-structures, in that they are described by an involutive complex subbundleof the generalised tangent bundle, and a vanishing moment map. We call these structures‘Exceptional Complex Structures’ (ECS) because of their similarity to (generalised) complexstructures. In analysing the integrability conditions we find interesting links to ‘GeometricInvariant Theory’ (GIT) which may have important consequences for unsolved problems inconventional geometry. The moment map picture also provides a systematic way of studyingthe moduli. We use the relation between symplectic quotients and complexified quotients toanalyse the moduli, giving exact results in a broad range of cases.We start with backgrounds of heterotic string theory with a 4-dimension external Minkowskispace. We show how the Hull-Strominger system can be reinterpreted as an integrable SU(3) ×Spin(6 + n) ⊂ O(6, 6 + n) structure. We provide expressions for the superpotential and theK¨ahler potential in this new language and analyse the moment map involved in the integrabilityconditions. This moment map interpretation of the Hull-Strominger system is an important stepin applying GIT to prove the existence of solutions, given certain constraints. This extensionof Yau’s theorem to particular non-K¨ahler manifolds has been of interest to mathematicians forsome time and our work may indicate possible new approaches to solving it. We also analysethe moduli of the Hull-Strominger system and recover the results of others.The next chapter focuses on M-theory backgrounds with a 5-dimensional external space.While it does not describe the full geometry, we focus on the SU∗(6) ⊂ E6(6) × R+ structurepresent in the supergravity solution. We find the most generic local form for exceptional complexstructures in this case, classifying them as either ‘type 0’ or ‘type 3’. This classification is onlypointwise, as there can be type-changing solutions. Using the general form, we are able tofind the moduli of all constant-type exceptional complex structures, as well as all those thatsatisfy a ‘generalised ∂∂¯-lemma’. Interestingly, these results hold for AdS solutions. We analysethese and show that they are always of constant type 3. Hence, we are able to reinterpret thespectrum of a given CFT4 that is dual to some AdS5 × M6 in terms of cohomology groupsrelated to some integrable distribution ∆ ⊂ TC.We then look at backgrounds of M-theory and type IIB with a 4-dimensional Minkowskiexternal space. We are able to reinterpret both G2 backgrounds and GMPT backgrounds interms of integrable SU(7) ⊂ E7(7) × R+ structures. We are also able to give an expression for5the superpotential and the K¨ahler potential for generic backgrounds using this new language.Once again, we study the implications of the moment map picture and find interesting linkswith GIT. We highlight how this may be used to find a form of stability for G2 structures.Again, we provide a method of systematically finding the moduli of these flux backgrounds andapply it to the G2 and the GMPT cases. For G2 we recover the known results, while for GMPTwe are able to find the exact moduli, extending work that has been done in the past.Finally, we analyse the exceptional complex structures via Hitchin functionals. The K¨ahlerpotentials in each case provide a natural candidate for the extension of Hitchin functionalsto exceptional geometry. Following the work of Pestun and Witten [3], we find the secondvariation of the K¨ahler potentials under complexified generalised diffeomorphisms and quantisethat quadratic action for SU∗(6) and SU(7) structures. We suggest possible applications as1-loop corrections to certain terms in the effective M-theory action in 5 and 4 dimensionsrespectively.

flux: background
string model: heterotic
supergravity: solution
supersymmetry: minimal
moduli
integrability
differential geometry
M-theory
potential: Kaehler
thesis