Related papers: The rank 2 classification problem I: scale invaria…

Four dimensional N=2 generalized superconformal field theory can be defined by compactifying six dimensional (0,2) theory on a Riemann surface with regular punctures. In previous studies, gauge coupling constant space is identified with the…

Let $G\subset SO(4)$ denote a finite subgroup containing the Heisenberg group. In these notes we classify all these groups, we find the dimension of the spaces of $G$-invariant polynomials and we give equations for the generators whenever…

We refine our previous proposal for systematically classifying 4d rank-1 $\mathcal N=2$ SCFTs by constructing their possible Coulomb branch geometries. Four new recently discussed rank-1 theories, including novel $\mathcal{N}=3$ SCFTs, sit…

We define and study a new class of 4d N=1 superconformal quiver gauge theories associated with a planar bipartite network. While UV description is not unique due to Seiberg duality, we can classify the IR fixed points of the theory by a…

A basic datum of a rank-$r$ $\mathcal{N}{=}2$ superconformal field theory (SCFT) is the $r$-tuple of its Coulomb branch scaling dimensions, i.e., the scaling dimensions of a set of special protected scalar operators whose vevs generate the…

We consider 4d field theories obtained by reducing the 6d (1,0) SCFT of $N$ M5-branes probing a $\mathbb C^2/\mathbb Z_k$ singularity on a Riemann surface with fluxes. We follow two different routes. On the one hand, we consider the…

New examples of N=2 supersymmetric conformal field theories are found as fixed points of SU(2) N=2 supersymmetric QCD. Relations among the scaling dimensions of their relevant chiral operators, global symmetries, and Higgs branches are…

We consider the problem of classifying (possibly noncommutative) R-algebras of low rank over an arbitrary base ring R. We first classify algebras by their degree, and we relate the class of algebras of degree 2 to algebras with a standard…

The N=2 superconformal Ward identities and their anomalies are discussed in N=2 superspace (including N=2 harmonic superspace), at the level of the low-energy effective action (LEEA) in four-dimensional N=2 supersymmetric field theories.…

We study rank-2 wobbly bundles on a Riemann surface $C$ of genus $g\geq 2$, i.e. semi-stable bundles admitting nonzero nilpotent Higgs fields, in terms of direct images of line bundles on smooth spectral curves $\tilde{C}…

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, there are…

Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain 2-morphisms in the 2-category of `2-tangles in 4…

We prove a general uniformization theorem for N=2 superconformal and N=1 superanalytic DeWitt super-Riemann surfaces, showing that in general an N=2 superconformal (resp. N=1 superanalytic) DeWitt super-Riemann surface is N=2…

We solve for the effective actions on the Coulomb branches of a class of N=2 supersymmetric theories by finding the complex structure of an M5 brane in an appropriate background hyperkahler geometry corresponding to the lift of two O6^-…

We construct a large class of Argyres-Douglas type theories by compactifying six dimensional (2,0) A_N theory on a Riemann surface with irregular singularities. We give a complete classification for the choices of Riemann surface and the…

We derive a family of matrix models which encode solutions to the Seiberg-Witten theory in 4 and 5 dimensions. Partition functions of these matrix models are equal to the corresponding Nekrasov partition functions, and their spectral curves…

We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the…

A specialization of a K3 surface with Picard rank one to a K3 with rank two defines a vanishing class of order two in the Brauer group of the general K3 surface. We give the B-field invariants of this class. We apply this to the K3 double…

We "solve" the Freed-Witten anomaly equation, i.e., we find a geometrical classification of the B-field and A-field configurations in the presence of D-branes that are anomaly-free. The mathematical setting being provided by the geometry of…

We consider a generalisation of the Seiberg-Witten invariant to the families Seiberg-Witten invariants of a smooth family of 4-manifolds with fibres diffeomorphic to a 4-manifold $X$. Of particular interest is the special case when the…