We present a general class of natural theories in which the Higgs is a pseudo-goldstone boson in an orbifolded gauge theory. The symmetry protecting the Higgs at low energies is an accidental global symmetry of the quadratic action, rather than a full continuous symmetry. The lightest degrees of freedom protecting the weak scale carry no Standard Model (SM) quantum numbers and interact with visible matter principally through the Higgs portal. This opens the door to the systematic study of "neutral naturalness": natural theories with SM-neutral states that are as yet untested by the LHC.

We show that the BPS spectrum of pure SU (3) four-dimensional super YangMills with N = 2 supersymmetry exhibits a surprising phenomenon: there are regions of the Coulomb branch where the growth of the BPS degeneracies with the charge is exponential. We show this using spectral networks and independently using wall-crossing formulae and quiver methods. The computations using spectral networks provide a very nontrivial example of how these networks determine the four-dimensional BPS spectrum. We comment on some physical implications of the wild spectrum: for example, exponentially many field-theoretic BPS states with large charge are gigantic. Finally, we exhibit some surprising, thus far unexplained, regularities of the BPS spectrum.arXiv:1305.5454v1 [hep-th]

We introduce and systematically study an expansive class of "orbifold Higgs" theories in which the weak scale is protected by accidental symmetries arising from the orbifold reduction of continuous symmetries. The protection mechanism eliminates quadratic sensitivity of the Higgs mass to higher scales at one loop (or more) and does not involve any new states charged under the Standard Model. The structures of the Higgs and top sectors are universal and determined exclusively by group theoretical considerations. The twin Higgs model fits within our framework as the simplest example of an orbifold Higgs. Our models admit UV completions as geometric orbifolds in higher dimensions, and fit naturally within frameworks of low scale gauge coupling unification.

The BPS spectrum of d=4 N=2 field theories in general contains not only hyper- and vector-multipelts but also short multiplets of particles with arbitrarily high spin. This paper extends the method of spectral networks to give an algorithm for computing the spin content of the BPS spectrum of d=4 N=2 field theories of class S. The key new ingredient is an identification of the spin of states with the writhe of paths on the Seiberg-Witten curve. Connections to quiver representation theory and to Chern-Simons theory are briefly discussed.Comment: 45 pages + appendi

We introduce a new perspective and a generalization of spectral networks for 4d N = 2 theories of class S associated to Lie algebras g = A n , D n , E 6 , and E 7 . Spectral networks directly compute the BPS spectra of 2d theories on surface defects coupled to the 4d theories. A Lie algebraic interpretation of these spectra emerges naturally from our construction, leading to a new description of 2d-4d wall-crossing phenomena. Our construction also provides an efficient framework for the study of BPS spectra of the 4d theories. In addition, we consider novel types of surface defects associated with minuscule representations of g.

We develop geometric techniques for counting BPS states in five-dimensional gauge theories engineered by M theory on a toric Calabi-Yau threefold. The problem is approached by studying framed 3d-5d wall-crossing in presence of a single M5 brane wrapping a special Lagrangian submanifold L. The spectrum of 3d-5d BPS states is encoded by the geometry of the manifold of vacua of the 3d-5d system, which further coincides with the mirror curve describing moduli of the Lagrangian brane. The information about the BPS spectrum is extracted from the geometry of the mirror curve by construction of a nonabelianization map for the exponential networks. For the simplest Calabi-Yau, C 3 we reproduce the count of 5d BPS states encoded by the Mac Mahon function in the context of topological strings, and match predictions of 3d tt * geometry for the count of 3d-5d BPS states. We comment on applications of our construction to the study of enumerative invariants of toric Calabi-Yau threefolds.

The recently conjectured knots-quivers correspondence [1,2] relates gauge theoretic invariants of a knot K in the 3-sphere to representation theory of a quiver Q K associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of large N duality of Ooguri and Vafa [3], that relates knot invariants to counts of holomorphic curves with boundary on L K , the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d N = 2 theory T [Q K ] whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of Q K . From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on L K is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver Q K and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to Q K and (doubly) refined version of LMOV invariants [3][4][5][6][7].

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of gl 1. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the N = 2 affine Yangian, which is isomorphic to the universal enveloping algebra of u(1)⊕W N =2 ∞ [λ]. All algebras that we construct have natural representations in terms of "twin plane partitions", a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.