On topological M-theory

We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on $M\times S^1$ the theory naturally reduces to the standard A-model on Calabi-Yau manifold and to a membrane theory localized on special Lagrangian submanifolds. We discuss some properties of topological membrane theory on $G_2$-manifolds. We also generalize our construction to topological $p$--branes on special manifolds by exploring a relation between vector cross product structures and TFTs.Comment: 20 page

tt∗tt^* Toda equations for surface defects in N=2{\mathcal N}=2 SYM and instanton counting for classical Lie groups

The partition function of $\mathcal{N}=2$ super Yang-Mills theories with arbitrary simple gauge group coupled to a self-dual $\Omega$-background is shown to be fully determined by studying the renormalization group equations relevant to the surface operators generating its one-form symmetries. The corresponding system of equations results in a ${\it non-autonomous}$ Toda chain on the root system of the Langlands dual, the evolution parameter being the RG scale. A systematic algorithm computing the full multi-instanton corrections is derived in terms of recursion relations whose gauge theoretical solution is obtained just by fixing the perturbative part of the IR prepotential as its asymptotic boundary condition for the RGE. We analyse the explicit solutions of the $\tau$-system for all the classical groups at the diverse levels, extend our analysis to affine twisted Lie algebras and provide conjectural bilinear relations for the $\tau$-functions of linear quiver gauge theory.Comment: 34 pages + appendices, comments welcom

On the convergence of Nekrasov functions

In this note we present some results on the convergence of Nekrasov partition functions as power series in the instanton counting parameter. We focus on $U(N)$ ${\mathcal N}=2$ gauge theories in four dimensions with matter in the adjoint and in the fundamental representations of the gauge group respectively and find rigorous lower bounds for the convergence radius in the two cases: if the theory is {\it conformal}, then the series has at least a {\it finite} radius of convergence, while if it is {\it asymptotically free} it has {\it infinite} radius of convergence. Via AGT correspondence, this implies that the related irregular conformal blocks of $W_N$ algebrae admit a power expansion in the modulus converging in the whole plane. By specifying to the $SU(2)$ case, we apply our results to analyse the convergence properties of the corresponding Painlev\'e $\tau$-functions.Comment: 1+25 pages, comments welcome. v2 published versio

Defects, nested instantons and comet shaped quivers

We introduce and study a surface defect in four dimensional gauge theories supporting nested instantons with respect to the parabolic reduction of the gauge group at the defect. This is engineered from a D3/D7-branes system on a non compact Calabi-Yau threefold $X$. For $X=T^2 imes T^*{mathcal C}_{g,k}$, the product of a two torus $T^2$ times the cotangent bundle over a Riemann surface ${mathcal C}_{g,k}$ with marked points, we propose an effective theory in the limit of small volume of ${mathcal C}_{g,k}$ given as a comet shaped quiver gauge theory on $T^2$, the tail of the comet being made of a flag quiver for each marked point and the head describing the degrees of freedom due to the genus $g$. Mathematically, we obtain for a single D7-brane conjectural explicit formulae for the virtual equivariant elliptic genus of a certain bundle over the moduli space of the nested Hilbert scheme of points on the affine plane. A connection with elliptic cohomology of character varieties and an elliptic version of modified Macdonald polynomials naturally arises

Electrochemical loading of hydrogen in Mg thin films

The science and technology of hydrogen production from renewable energy, hydrogen sensing and hydrogen storage are central to the realization of a hydrogen-based society, in which me may be living one day. The focus of my Master thesis will be the solid-state hydrogen storage in nanostructured metallic hydrides. In fact, beside the physical storage of hydrogen through pressurized tanks, it is also possible to store hydrogen in the chemical bonds of the metallic hydrides such as MgH2, LaNi5H6,FeTiH2, just to name a few. The problem is to achieve these goals at temperature and pressure as close as possible to ambient conditions. Knowledge-based design of advanced materials can help to solve issues related to unfavourable kinetics and thermodynamics of hydrogen storage. In particular, size effects and interface engineering in nanomaterials are exciting tools to modify and improving hydrogen sorption properties beyond the current state of the art. This is the subject of the present project

Wild Quiver Gauge Theories

We study N=2 supersymmetric SU(2) gauge theories coupled to non-Lagrangian superconformal field theories induced by compactifying the six dimensional A_1 (2,0) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We propose that the prepotential of these gauge theories on the Omega-background can be obtained from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.Comment: 34 pages; v2. typos correcte

Quantum Hitchin Systems via β -Deformed Matrix Models

We study the quantization of Hitchin systems in terms of β-deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov–Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the ϵ1-deformation of the chiral observables of the corresponding N= 2 four dimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions