We provide a detailed analysis of flux backgrounds of string and M-theory that preserve minimal
supersymmetry in terms of (exceptional) generalised geometry. The geometry in each case is
conveniently described in terms of generalised G-structures, where the integrability conditions
are equivalent to the Killing spinor equations. Interestingly, there seems to be a common
structure among the G-structures, in that they are described by an involutive complex subbundle
of the generalised tangent bundle, and a vanishing moment map. We call these structures
‘Exceptional Complex Structures’ (ECS) because of their similarity to (generalised) complex
structures. In analysing the integrability conditions we find interesting links to ‘Geometric
Invariant Theory’ (GIT) which may have important consequences for unsolved problems in
conventional geometry. The moment map picture also provides a systematic way of studying
the moduli. We use the relation between symplectic quotients and complexified quotients to
analyse the moduli, giving exact results in a broad range of cases.
We start with backgrounds of heterotic string theory with a 4-dimension external Minkowski
space. 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 the
K¨ahler potential in this new language and analyse the moment map involved in the integrability
conditions. This moment map interpretation of the Hull-Strominger system is an important step
in applying GIT to prove the existence of solutions, given certain constraints. This extension
of Yau’s theorem to particular non-K¨ahler manifolds has been of interest to mathematicians for
some time and our work may indicate possible new approaches to solving it. We also analyse
the 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
+ structure
present in the supergravity solution. We find the most generic local form for exceptional complex
structures in this case, classifying them as either ‘type 0’ or ‘type 3’. This classification is only
pointwise, as there can be type-changing solutions. Using the general form, we are able to
find the moduli of all constant-type exceptional complex structures, as well as all those that
satisfy a ‘generalised ∂∂¯-lemma’. Interestingly, these results hold for AdS solutions. We analyse
these and show that they are always of constant type 3. Hence, we are able to reinterpret the
spectrum of a given CFT4 that is dual to some AdS5 × M6 in terms of cohomology groups
related to some integrable distribution ∆ ⊂ TC.
We then look at backgrounds of M-theory and type IIB with a 4-dimensional Minkowski
external space. We are able to reinterpret both G2 backgrounds and GMPT backgrounds in
terms of integrable SU(7) ⊂ E7(7) × R
+ structures. We are also able to give an expression for
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the 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 links
with 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 and
apply it to the G2 and the GMPT cases. For G2 we recover the known results, while for GMPT
we 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¨ahler
potentials in each case provide a natural candidate for the extension of Hitchin functionals
to exceptional geometry. Following the work of Pestun and Witten [3], we find the second
variation of the K¨ahler potentials under complexified generalised diffeomorphisms and quantise
that quadratic action for SU∗
(6) and SU(7) structures. We suggest possible applications as
1-loop corrections to certain terms in the effective M-theory action in 5 and 4 dimensions
respectively.