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## ‣ Stringy K-theory and the Chern character

Jarvis, Tyler J.; Kaufmann, Ralph; Kimura, Takashi
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.6424%
For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.; Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematicae

## ‣ Symmetric homotopy theory for operads

Dehling, Malte; Vallette, Bruno
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.649126%

## ‣ Idempotent (Asymptotic) Mathematics and the Representation Theory

Litvinov, Grigori; Maslov, Viktor; Shpiz, Grigori
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.67829%
A brief survey of some basic ideas of the so-called Idempotent Mathematics is presented; an "idempotent" version of the representation theory is discussed. The Idempotent Mathematics can be treated as a result of a dequantization of the traditional mathematics over numerical fields in the limit of the vanishing "imaginary Planck constant"; there is a correspondence, in the spirit of N. Bohr's correspondence principle, between constructions and results in traditional mathematics over the fields of real and complex numbers and similar constructions and results over idempotent semirings. In particular, there is an "idempotent" version of the theory of linear representations of groups. Some basic concepts and results of the "idempotent" representation theory are presented. In the framework of this theory the well-known Legendre transform can be treated as an idempotent version of the traditional Fourier transform. Some unexpected versions of the Engel theorem are given.; Comment: 10 pages

## ‣ Parametrized K-Theory

Michel, Nicolas
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.627627%
In nature, one observes that a K-theory of an object is defined in two steps. First a "structured" category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a general framework that deals with the first step of this process. The K-theory of an object is defined via a category of "locally trivial" objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.; Comment: 31 pages

## ‣ Knot Homology from Refined Chern-Simons Theory

Aganagic, Mina; Shakirov, Shamil
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.625454%
We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold via the refined topological string and the (2,0) theory on N M5 branes. The refined Chern-Simons theory is defined on any three-manifold with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. We obtain new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the knot invariants we compute are the Poincare polynomials of the sl(n) knot homology theory. The latter includes the Khovanov-Rozansky knot homology, as a special case. The conjecture passes a number of nontrivial checks. We show that, for a large number of torus knots colored with the fundamental representation of SU(N), our knot invariants agree with the Poincare polynomials of Khovanov-Rozansky homology. As a byproduct, we show that our theory on S^3 has a large-N dual which is the refined topological string on X=O(-1)+O(-1)->P^1; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to sl(n) knot homology. We also provide a matrix model description of some amplitudes of the refined Chern-Simons theory on S^3.; Comment: 73 pages...

## ‣ Twisted and untwisted K-theory quantization, and symplectic topology

Savelyev, Yasha; Shelukhin, Egor