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.

We propose a non-perturbative formulation of structure constants of single trace operators in planar N=4 SYM.  We match our results with both weak and strong coupling data available in the literature. Based on work with Benjamin Basso and Pedro Vieira.

I shall demonstrate, under some mild assumptions, that the symmetry group of  extreme, Killing, supergravity horzions contains an sl(2, R) subalgebra.  The proof requires a generalization of the  Lichnerowicz theorem for non-metric connections. The techniques developed can also be applied in the classification
of AdS and Minkowski flux backgrounds.
 

I will discuss how singular fibers in higher codimension in elliptically fibered Calabi-Yau fourfolds can be studied using Coulomb branch phases for d=3 supersymmetric gauge theories. This approach gives an elegent description of the generalized Kodaira fibers in terms of combinatorial/representation-theoretic objects called "box graphs", including the network of flops connecting distinct small resolutions. For physics applications, this approach can be used to constrain the possible matter spectra and possible U(1) charges (models with higher rank Mordell Weil group) for F-theory GUTs.

I will speak about weak Fano 3-folds, K3 surfaces and their Picard lattices, and explain how to solve the matching problem in various situations

Motivated by the study of supersymmetric backgrounds with non-trivial fluxes, we provide a formulation of supergravity in the language of generalised geometry, as first introduced by Hitchin, and its extensions. This description both dramatically simplifies the equations of the theory by making the hidden symmetries manifest, and writes the bosonic sector geometrically as a direct analogue of Einstein gravity. Further, a natural analogue of special holonomy manifolds emerges and coincides with the conditions for supersymmetric backgrounds with flux, thus formulating these systems as integrable geometric structures.
 

The chiral scalar superfield has interesting BRST cohomology, but the relevant cohomology objects all  have spinor indices. So they cannot occur in an action. They need to be coupled to a chiral dotted spinor superfield. Until now, this has been very problematic, since no sensible action for a chiral dotted spinor superfield was known.  The most obvious such action contains higher derivatives and tachyons.

Now,  a sensible  action has been found. When coupled to the cohomology, this action removes the supersymmetry charge from the theory while maintaining the rigidity and power of supersymmetry.The simplest example of this phenomenon has exactly the fermion content of the Leptons or the Quarks.  The mechanism has the potential to get around the cosmological constant problem, and also the problem of the sum rules of spontaneously broken supersymmetry.

This is a report on an ongoing project to construct Calabi-Yau manifolds for which the Hodge numbers $(h^{11}, h^{21})$ are both relatively small. These manifolds are, in a sense, the simplest Calabi-Yau manifolds. I will report on joint work with Volker Braun, Andrei Constantin, Rhys Davies, Challenger Mishra and others.

The planar N=4 SYM is believed to be integrable. Following this thoroughly justified belief, its exact spectrum had been encoded recently into a quantum spectral curve (QSC). We can explicitly solve the QSC in various regimes; in particular, one can perform a highly-efficient weak coupling expansion.

I will explain how QSC looks like for the harmonic oscillator and then, using this analogy, introduce the QSC equations for the SYM spectrum. We will use these equations to compute a particular 6-loop conformal dimension in real time and then discuss explicit results (found up to 10-loop orders) as well as some general statements about the answer at any loop-order.

I will give an overview of ongoing joint work with R. Rubio and C. Tipler, in which we study the moduli problem for the Strominger system of equations. Building on the work of De la Ossa and Svanes and, independently, of Anderson, Gray and Sharpe, we construct an elliptic complex whose first cohomology group is the space of infinitesimal deformations of a solution of the strominger system. I will also discuss an intriguing link between this moduli problem and a moduli problem for holomorphic Courant algebroids over Calabi-Yau threefolds. Finally, we will see how the problem for the Strominger system embeds naturally in generalized geometry, and discuss some perspectives of this approach.

I will describe the computation of the supersymmetric Renyi entropy across an entangling 3-sphere for five-dimensional superconformal field theories. For a class of USp(2N) gauge theories I’ll also construct a holographic dual 1/2 BPS black hole solution of Euclidean Romans F(4) supergravity. The large N limit of the gauge theory results will be shown to agree perfectly with the supergravity computations.

I will discuss the implementation of explicit stabilisation of all closed string moduli in fluxed type IIB Calabi-Yau compactifications with chiral matter.  Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content. We consider Calabi-Yau manifolds with a discrete symmetry that reduces the effective number of complex structure moduli, which allows us to calculate the corresponding periods and find explicit flux vacua. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kaehler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative alpha'-corrections as in the LARGE Volume Scenario.

I will discuss several recent developments regarding the construction of fluxes for F-theory on Calabi-Yau fourfolds. Of particular importance to the effective physics is the structure of the middle (co)homology groups, on which new results are presented. Fluxes dynamically drive the fourfold to Noether-Lefschetz loci in moduli space. While the structure of such loci is generally unknown for Calabi-Yau fourfolds, this problem can be answered in terms of arithmetic for K3 x K3 and a classification is possible.

In this talk, we discuss an analogue of the Hermitian-Einstein equations for generalized Kaehler manifolds proposed by N. Hitchin. We explain in particular how these equations are equivalent to a notion of stability, and that there is a Kobayahsi-Hitchin-type of correspondence between solutions of these equations and stable objects. The correspondence holds even for non-Kaehler manifolds, as long as they are endowed with Gauduchon metrics (which is always the case for generalized Kaehler structures on 4-manifolds).

This is joint work with Shengda Hu and Reza Seyyedali.