Deformations of colored sl(N) link homologies via foams

We generalize results of Lee, Gornik and Wu on the structure of deformed colored sl(N) link homologies to the case of non-generic deformations. To this end, we use foam technology to give a completely combinatorial construction of Wu's deformed colored sl(N) link homologies. By studying the underlying deformed higher representation theoretic structures and generalizing the Karoubi envelope approach of Bar-Natan and Morrison we explicitly compute the deformed invariants in terms of undeformed type A link homologies of lower rank and color.Comment: 64 pages, many figure

HOMFLYPT homology for links in handlebodies via type A Soergel bimodules

We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.Comment: 25 pages, lots of figures, comments welcom

The sl n foam 2-category: A combinatorial formulation of Khovanov–Rozansky homology via categorical skew Howe duality

Abstract We give an elementary construction of colored sln link homology. The invariant takes values in a 2-category where 2-morphisms are given by foams, singular cobordisms between sln webs; applying a (TQFT-like) representable functor recovers (colored) Khovanov–Rozansky homology. Novel features of the theory include the introduction of “enhanced” foam facets which fix sign issues associated with the original matrix factorization formulation and the use of skew Howe duality to show that (enhanced) closed foams can be evaluated in a completely combinatorial manner. The latter answers a question posed in [42]

Optical angular momentum: Multipole transitions and photonics

The premise that multipolar decay should produce photons uniquely imprinted with a measurably corresponding angular momentum is shown in general to be untrue. To assume a one-to-one correlation between the transition multipoles involved in source decay and detector excitation is to impose a generally unsupportable one-to-one correlation between the multipolar form of emission transition and a multipolar character for the detected field. It is specifically proven impossible to determine without ambiguity, by use of any conventional detector, and for any photon emitted through the nondipolar decay of an atomic excited state, a unique multipolar character for the transition associated with its generation. Consistent with the angular quantum uncertainty principle, removal of a detector from the immediate vicinity of the source produces a decreasing angular uncertainty in photon propagation direction, reflected in an increasing range of integer values for the measured angular momentum. In such a context it follows that when the decay of an electronic excited state occurs by an electric quadrupolar transition, for example, any assumption that the radiation so produced is conveyed in the form of “quadrupole photons” is experimentally unverifiable. The results of the general proof based on irreducible tensor analysis invite experimental verification, and they signify certain limitations on quantum optical data transmission