The Bethe Ansatz equations were initially conceived as a method to solve some particular Quantum Integrable Models (IM), but are nowadays a central tool of investigation in a variety of physical and mathematical theories such as string theory, supersymmetric gauge theories, and Donaldson-Thomas invariants. Surprisingly, it has been observed, in several examples, that the solutions of the same Bethe Ansatz equations are provided by the monodromy data of some ordinary differential operators with an irregular singularity (ODE/IM correspondence).

In this talk I will present the results of my investigation on the ODE/IM correspondence in quantum $g$-KdV models, where $g$ is an untwisted affine Kac-Moody algebra. I will construct solutions of the corresponding Bethe Ansatz equations, as the (irregular) monodromy data of a meromorphic $L(g)$-oper, where $L(g)$ denotes the Langlands dual algebra of $g$.

The talk is based on:

D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case. Comm. Math. Phys. (2016)

D Masoero, A Raimondo, D Valeri, Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections II: The nonsimply-laced case. Comm. Math. Phys. (2017)

D Masoero, A Raimondo, Opers corresponding to Higher States of the $g$-Quantum KdV model. arXiv 2018.

Non-relativistic string theory is described by a sigma model that maps a two dimensional string worldsheet to a non-relativistic spacetime geometry. We discuss recent developments in understanding the spacetime geometry of non-relativistic string theory trying to provide several new insights. We show that the non-relativistic string action admits a surprisingly large number of symmetries. We introduce a non-relativistic limit to obtain the non-relativistic string action which also provides us the non-relativistic T-duality transformation rules and spacetime equations of motion.

I will review basic notions and results in topological quantum field theory and discuss its orbifolds, with the aim to apply them in the context of topological quantum computation.

Identifying patterns in data enables us to formulate questions that can lead to exact results. Since many of the patterns are subtle, machine learning has emerged as a useful tool in discovering these relationships. We show that topological features of Calabi–Yau geometries are machine learnable. We indicate the broad applicability of our methods to existing large data sets by finding relations between knot invariants, in particular, the hyperbolic volume of the knot complement and the Jones polynomial.

The apparently random pattern of the non-trivial zeroes of the Riemann zeta function (all on the critical line, according to the Riemann hypothesis) has led to the suggestion that they may be related to the spectrum of an operator. It has also been known for some time that the statistical properties of the eigenvalue distribution of an ensemble of random matrices resemble those of the zeroes of the zeta function. With the objective to identify a suitable operator, we start by assuming the Riemann hypothesis and construct a unitary matrix model (UMM) for the zeta function. Our approach, however, could be termed piecemeal, in the sense that, we consider each factor (in the Euler product representation) of the zeta function to get a UMM for each prime, and then assemble these to get a matrix model for the full zeta function. This way we can write the partition function as a trace of an operator. Similar construction works for a family of related zeta functions.

I will discuss BPS invariants associated with quantum line operators in certain supersymmetric quantum field theories. Such operators can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by representation theory data. I will discuss the associated BPS spectral problem and the relevant indices.

Since the 1980s, the study of invariants of 3-dimensional manifolds has benefited from the connections between topology, physics and number theory. Motivated by the recent discovery of a new homological invariant (corresponding to the half-index of certain $3d$ $N=2$ theories), in this talk I describe the role of quantum modular forms, mock and false theta functions in the study of $3$-manifold invariants. The talk is based on 1809.10148 and work in progress with Cheng, Chun, Feigin, Gukov, and Harrison.

Extremal black holes show the presence of an $\operatorname{AdS}_2$ factor as an universal feature. This fact provides a strong motivation for getting a deeper understanding of $\operatorname{AdS}_2$ spacetimes. In this talk, I will argue how $\operatorname{AdS}_2$ systems have a natural description in terms of $1d$ Calogero-type models and, in turn to SLE curves, which describe the geodesic motion of particles in $\operatorname{AdS}_2$. This treatment allows to compute the dimension of the phase space of these geodesics, linking it to the leading Bekenstein-Hawking black hole entropy and the black hole degeneracy.

In this talk, I am going to review some aspects of the current state of the art of Integrability in the AdS/CFT correspondence and beyond. I will first review a general nonperturbative approach to compute multipoint correlation functions of local operators in the $N=4$ SYM theory which allows us to explore the theory even beyond the planar level. In the second part, I will describe my recent work about exploring deformations of $N=4$ SYM by irrelevant operators, which revives an old attempt of generalizing the AdS/CFT correspondence. Here integrability seems to also play an important role and opens the door for its application for non-conformal field theories.