There are various proposals as to what initial state can give rise to an inflationary cosmology. The two most popular ones are the no boundary (Hartle-Hawking) and the tunneling (Vilenkin) proposals. Both of them explain only part of the observations and lead to some paradoxes. I will review these proposals and I will propose a novel initial state (wavefunction) of the universe which in the far past has asymptotically AdS boundary conditions. In the semiclassical limit it is a Euclidean wormhole solution that can give rise to an expanding universe upon analytic continuation to Lorentzian signature. This proposal evades some of the issues that plagued the no boundary and the tunneling proposals. Moreover, the asymptotic AdS conditions in the Euclidean past could in principle allow for the description of inflationary cosmologies and their perturbations within the context of holography, leading to microscopic models.

Finding the right degrees of freedom is a key step in solving strongly interacting field theories. Considerable progress in this direction was made in the last decades for higher dimensional supersymmetric gauge theories, guided by dualities with string theory. Two complementary approaches have been particularly useful in the exploration of the quantum structure of these theories: one is supersymmetric localisation and the other is integrability. Very recently it was noticed that common structures appear in both approaches when they are applied to the N=2 super-conformal theory in four dimensions which is obtained as a Z_K orbifold of the N=4 super Yang-Mills (SYM) theory. This hints at a deeper structure underlying both localisation and integrability. A special class of three-point functions in the N=2 SYM theory were computed first by localisation, in terms of a Fredholm determinant of a Bessel operator generalising the Tracy-Widom distribution. The talk, based on a recently completed work in collaboration with Gwenaël Ferrando, Shota Komatsu and Gabriel Lefundes, will explain how this result can be derived from the integrability approach to correlation functions, which was developed initially for the N=4 SYM theory and where the Fredholm determinant appears naturally.

I will discuss some interesting results on instanton partition functions of Vafa Witten theory on CP^2 with gauge group SU(N). These results can be related to counting of sheaves on CP^2. In 2301.06711 paper we observed the growth of certain mock cusp forms associated with these mock modular forms. Interestingly the growth of these coefficients remain higher than those expected from Deligne bound. In our ongoing work we show that the holomorphic anomaly in theories are associated with these partition functions corresponding to gauge groups of lower ranks.

How well do we understand string theory? As one indicator can be thought of the degree of our understanding of its spectrum. Yet, other than comprising infinitely many physical states of arbitrarily high spin and mass, what does the string spectrum look like? Traditional methodologies can yield its state content on a level-by-level basis, a straightforward procedure which however becomes cumbersome as the level increases. In this seminar, we will discuss a new, covariant and efficient technology with which entire physical trajectories can be excavated. It is based on the observation that the Virasoro constraints in fact encode the generators of a bigger algebra, that is a symplectic algebra, which commutes with the spacetime Lorentz algebra. This enables constructing trajectories deeper inside the spectrum as “clones” of simpler ones, upon suitably dressing the latter, depending on the “depth” of the trajectory we aim to reach.