Two interesting properties of static curved space QFTs are Casimir Energies, and the Energy Gaps of fluctuations. We investigate what AdS/CFT has to say about these properties by examining holographic CFTs defined on curved but static spatially closed spacetimes. Being holographic, these CFTs have a dual gravitational description under Gauge/Gravity duality, and these properties of the CFT are reflected in the geometry of the dual bulk. We can turn this on its head and ask, what does the existence of the gravitational bulk dual imply about these properties of the CFTs? In this talk we will consider holographic CFTs where the dual vacuum state is described by pure Einstein gravity with negative cosmological constant. We will argue using the bulk geometry first, that if the CFT spacetime's spatial scalar curvature is positive there is a lower bound on the gap for scalar fluctuations, controlled by the minimum value of the boundary Ricci scalar. In fact, we will show that it is precisely the same bound as is satisfied by free scalar CFTs, suggesting that this bound might be something that applies more generally than just in a Holographic context. We will then show, in the case of 2+1 dimensional CFTs, that the Casimir energy is non-positive, and is in fact negative unless the CFT's scalar curvature is constant. In this case, there is no restriction on the boundary scalar curvature, and we can even allow singularities in the bulk, so long as they are 'good' singularities. If time permits, we will also describe some new results about the Hawking-Page transition in this context.

# Past String Theory Seminar

By regarding gravity as the convolution of left and right Yang-Mills theories together with a spectator scalar field in the bi-adjoint representation, we derive in linearised approximation the gravitational symmetries of general covariance, p-form gauge invariance, local Lorentz invariance and local supersymmetry from the flat space Yang-Mills symmetries of local gauge invariance and global super-Poincare. As a concrete example we focus on the new-minimal (12+12) off-shell version of simple four-dimensional supergravity obtained by tensoring the off-shell Yang-Mills multiplets (4+4,NL =1)and(3+0,NR =0).

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.

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.