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Register a new userThe Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches
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We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 $mathcal{N}{=}2$ superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special Kahler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the $rm, U(1)_R$ symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.
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We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data. We construct these SCFTs by gauging certain discrete global symmetries of $mathcal N=4$ superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as $mathcal N=4$ sYM theories with disconnected gauge groups.
We study the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries, which potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that this classification is equivalent to the solution of a set of polynomial equations by using an integrability condition for the central charge, scale invariance, constraints coming from demanding single-valuedness of physical quantities on the Coulomb branch, and properties of massless BPS states at singularities. We find solutions corresponding to lagrangian scale invariant theories--including the scale invariant G_2 theory not found before in the literature--as well as many new isolated solutions (having no marginal deformations). All our scale-invariant RSK geometries are consistent with an interpretation as effective theories of N=2 superconformal field theories, and, where we can check, turn out to exist as quantum field theories.
We continue the classification of 2-dimensional scale-invariant rigid special Kahler (RSK) geometries. This classification was begun in [hep-th/0504070] where singularities corresponding to curves of the form y^2=x^6 with a fixed canonical basis of holomorphic one forms were analyzed. Here we perform the analysis for the y^2=x^5 type singularities. (The final maximal singularity type, y^2=x^3(x-1)^3, will be analyzed in a later paper.) These singularities potentially describe the Coulomb branches of N=2 supersymmetric field theories in four dimensions. We show that there are only 13 solutions satisfying the integrability condition (enforcing the RSK geometry of the Coulomb branch) and the Z-consistency condition (requiring massless charged states at singularities). Of these solutions, one has a marginal deformation, and corresponds to the known solution for certain Sp(2) gauge theories, while the rest correspond to isolated strongly interacting conformal field theories.
By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. This opens up several new avenues, which include a new formulation of $q$-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.
We derive closed formulae for the first examples of non-algebraic, elliptic `leading singularities in planar, maximally supersymmetric Yang-Mills theory and show that they are Yangian-invariant.
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