Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author

Long-lasting effects of chronic exposure to chemical pollution on the hologenome of the Manila clam

Chronic exposure to pollutants affects natural populations, creating specific molecular and biochemical signatures. In the present study, we tested the hypothesis that chronic exposure to pollutants might have substantial effects on the Manila clam hologenome long after removal from contaminated sites. To reach this goal, a highly integrative approach was implemented, combining transcriptome, genetic and microbiota analyses with the evaluation of biochemical and histological profiles of the edible Manila clam Ruditapes philippinarum, as it was transplanted for 6 months from the polluted area of Porto Marghera (PM) to the clean area of Chioggia (Venice lagoon, Italy). One month post‐transplantation, PM clams showed several modifications to its resident microbiota, including an overrepresentation of the opportunistic pathogen Arcobacter spp. This may be related to the upregulation of several immune genes in the PM clams, potentially representing a host response to the increased abundance of deleterious bacteria. Six months after transplantation, PM clams demonstrated a lower ability to respond to environmental/physiological stressors related to the summer season, and the hepatopancreas‐associated microbiota still showed different compositions among PM and CH clams. This study confirms that different stressors have predictable effects in clams at different biological levels and demonstrates that chronic exposure to pollutants leads to long‐lasting effects on the animal hologenome. In addition, no genetic differentiation between samples from the two areas was detected, confirming that PM and CH clams belong to a single population. Overall, the obtained responses were largely reversible and potentially related to phenotypic plasticity rather than genetic adaptation. The results here presented will be functional for the assessment of the environmental risk imposed by chemicals on an economically important bivalve species

Dorey's Rule and the q-Characters of Simply-Laced Quantum Affine Algebras

Let Uq(ghat) be the quantum affine algebra associated to a simply-laced simple Lie algebra g. We examine the relationship between Dorey's rule, which is a geometrical statement about Coxeter orbits of g-weights, and the structure of q-characters of fundamental representations V_{i,a} of Uq(ghat). In particular, we prove, without recourse to the ADE classification, that the rule provides a necessary and sufficient condition for the monomial 1 to appear in the q-character of a three-fold tensor product V_{i,a} x V_{j,b} x V_{k,c}.Comment: 30 pages, latex; v2, to appear in Communications in Mathematical Physic

The sl_3 web algebra

In this paper we use Kuperberg’s sl3-webs and Khovanov’s sl3-foams to define a new algebra KS, which we call the sl3-web algebra. It is the sl3 analogue of Khovanov’s arc algebra. We prove that KS is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of q-skew Howe duality, which allows us to prove that KS is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group K0 (WS )Q(q) , to show that its center is isomorphic to the cohomology ring of a certain Spaltenstein variety, and to prove that KS is a graded cellular algebra.info:eu-repo/semantics/publishedVersio

Composite GUTs: models and expectations at the LHC

We investigate grand unified theories (GUTs) in scenarios where electroweak (EW) symmetry breaking is triggered by a light composite Higgs, arising as a Nambu-Goldstone boson from a strongly interacting sector. The evolution of the standard model (SM) gauge couplings can be predicted at leading order, if the global symmetry of the composite sector is a simple group G that contains the SM gauge group. It was noticed that, if the right-handed top quark is also composite, precision gauge unification can be achieved. We build minimal consistent models for a composite sector with these properties, thus demonstrating how composite GUTs may represent an alternative to supersymmetric GUTs. Taking into account the new contributions to the EW precision parameters, we compute the Higgs effective potential and prove that it realizes consistently EW symmetry breaking with little fine-tuning. The G group structure and the requirement of proton stability determine the nature of the light composite states accompanying the Higgs and the top quark: a coloured triplet scalar and several vector-like fermions with exotic quantum numbers. We analyse the signatures of these composite partners at hadron colliders: distinctive final states contain multiple top and bottom quarks, either alone or accompanied by a heavy stable charged particle, or by missing transverse energy.Comment: 55 pages, 13 figures, final version to be published in JHE

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants $\int_S\mathcal A$ of a surface $S$, determined by the choice of a braided tensor category $\mathcal A$, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a {\em braided module category} for $\mathcal A$, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called {\em quantum moment maps}. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided $\mathcal A$-modules are objects of the torus category $\int_{T^2}\mathcal A$. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of $\mathcal A=\operatorname{Rep_q} G$ with the category $\mathcal D_q(G/G)-\operatorname{mod}$ of equivariant quantum $\mathcal D$-modules. When $G=GL_n$, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra (DAHA) $\mathbb{SH}_{q,t}$.Comment: 33 pages, 5 figures. Final version, to appear in Sel. Math. New Se

Dynamics of Ferrofluid Drops on Magnetically Patterned Surfaces

The motion of liquid drops on solid surfaces is attracting a lot of attention because of its fundamental implications and wide technological applications. In this article, we present a comprehensive experimental study of the interaction between gravity-driven ferrofluid drops on very slippery oil-impregnated surfaces and a patterned magnetic field. The drop speed can be accurately tuned by the magnetic interaction, and more interestingly, drops are found to undergo a stick–slip motion whose contrast and phase can be easily tuned by changing either the strength of the magnetic field or the ferrofluid concentration. This motion is the result of the periodic modulation of the external magnetic field and can be accurately analyzed because the intrinsic pinning due to chemical defects is negligible on oil-impregnated surfaces