We review some recent progress in computing massless spectra and moduli in heterotic string compactifications. In particular, it was recently shown that the heterotic Bianchi Identity can be accounted for by the construction of a holomorphic operator. Mathematically, this corresponds to a holomorphic double extension. Moduli can then be computed in terms of cohomologies of this operator. We will see how the same structure can be derived form a Gukov-Vafa-Witten type superpotential. We note a relation between the lifted complex structure and bundle moduli, and cover some examples, and briefly consider obstructions and Yukawa couplings arising from these structures.

# Past String Theory Seminar

Calabi-Yau manifolds without flux are perhaps the best-known

supergravity backgrounds that leave some supersymmetry unbroken. The

supersymmetry conditions on such spaces can be rephrased as the

existence and integrability of a particular geometric structure. When

fluxes are allowed, the conditions are more complicated and the

analogue of the geometric structure is not well understood.

In this talk, I will define the analogue of Calabi-Yau geometry for

generic D=4, N=2 backgrounds with flux in both type II and

eleven-dimensional supergravity. The geometry is characterised by a

pair of G-structures in 'exceptional generalised geometry' that

interpolate between complex, symplectic and hyper-Kahler geometry.

Supersymmetry is then equivalent to integrability of the structures,

which appears as moment maps for diffeomorphisms and gauge

transformations. Similar structures also appear in D=5 and D=6

backgrounds with eight supercharges.

As a simple application, I will discuss the case of AdS5 backgrounds

in type IIB, where deformations of these geometric structures give

exactly marginal deformations of the dual field theories.

The 3d/3d correspondence is about the correspondence between 3d N=2 supersymmetric gauge theories and the 3d complex Chern-Simons theory on a 3-manifold.

In this talk I will describe codimension 2 and 4 supersymmetric defects in this correspondence, by a combination of various existing techniques, such as state-integral models, cluster algebras, holographic dual, and 5d SYM.

In this talk, gauged quiver quantum mechanics will be analysed for BPS state counting. Despite the wall-crossing phenomenon of those countings, an invariant quantity of quiver itself, dubbed quiver invariant, will be carefully defined for a certain class of abelian quiver theories. After that, to get a handle on nonabelian theories, I will overview the abelianisation and the mutation methods, and will illustrate some of their interesting features through a couple of simple examples.

It has been conjectured that the fundamental theory of strings and branes has an $E_{11}$ symmetry. I will explain how this conjecture leads to a generalised space-time, which is automatically equipped with its own geometry, as well as equations of motion for the fields that live on this generalised space-time.

Form factors form a bridge between the purely on-shell amplitudes and the purely off-shell correlation functions. In this talk, we study the form factors of general gauge-invariant local composite operators in N=4 SYM theory via on-shell methods. At tree-level and for a minimalnumber of external fields, the form factor exactly realises the spin-chain picture of N=4 SYM theory in the language of scattering amplitudes. Via generalised unitarity, we obtain the cut-constructible part of the one-loop correction to the minimal form factor of a generic operator. Its UV divergence yields the complete one-loop dilatation operator of the theory. At two-loop order, we employ unitarity to calculate the minimal form factors and thereby the dilatation operator for the Konishi primary operator and all operators in the SU(2) sector. For the former operator as well as other non-protected operators, important subtleties arise which require an extension of the method of unitarity.

Recently, as part of a project to find CY manifolds for which both the Hodge numbers (h^{11}, h^{21}) are small, manifolds have been found with Hodge numbers (4,1) and (1,1). The one-dimensional special geometries of their complex structures are more complicated than those previously studied. I will review these, emphasising the role of the fundamental period and Picard-Fuchs equation. Two arithmetic aspects arise: the first is the role of \zeta(3) in the monodromy matrices and the second is the fact, perhaps natural to a number theorist, that through a study of the CY manifolds over finite fields, modular functions can be associated to the singular manifolds of the family. This is a report on joint work with Volker Braun, Xenia de la Ossa and Duco van Straten.