Intersection cohomology of Drinfeld's compactifications

Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification \widetilde{\on{Bun}}_P of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of \widetilde{\on{Bun}}_P. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.Comment: An erratum adde

Quantum affine Gelfand-Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We construct the action of the quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the affine version of Laumon spaces. We write down explicit formulae for this action in the affine Gelfand-Tsetlin base, corresponding to the fixed point base in the localized equivariant K-theory.Comment: v2: multiple typos fixed, proofs of Theorems 4.13 and 4.19 expanded, 23 pages. v3: formulas of Theorems 4.9 and 4.13 corrected, resulting minor changes added. arXiv admin note: text overlap with arXiv:0812.4656, arXiv:math/0503456, arXiv:0806.0072 by other author

Intersection cohomology of Drinfeld‚s compactifications

Let X be a smooth complete curve, G be a reductive group and a parabolic. Following Drinfeld, one defines a (relative) compactification of the moduli stack of P-bundles on X. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of . The description is given in terms of the combinatorics of the Langlands dual Lie algebra

A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P^1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra U(g,e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of arXiv:math/0401409 when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of certain shifted Yangians.Comment: minor change

On a class of representations of the Yangian and moduli space of monopoles

A new class of infinite dimensional representations of the Yangians $Y(\frak{g})$ and $Y(\frak{b})$ corresponding to a complex semisimple algebra $\frak{g}$ and its Borel subalgebra $\frak{b}\subset\frak{g}$ is constructed. It is based on the generalization of the Drinfeld realization of $Y(\frak{g})$, $\frak{g}=\frak{gl}(N)$ in terms of quantum minors to the case of an arbitrary semisimple Lie algebra $\frak{g}$. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of $G$-monopoles defined as the components of the space of based maps of $\mathbb{P}^1$ into the generalized flag manifold $X=G/B$. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.Comment: 16 pages, LaTex2e, some misprints are fixe

Multiplicative slices, relativistic Toda and shifted quantum affine algebras

We introduce the shifted quantum affine algebras. They map homomorphically into the quantized $K$-theoretic Coulomb branches of $3d\ {\mathcal N}=4$ SUSY quiver gauge theories. In type $A$, they are endowed with a coproduct, and they act on the equivariant $K$-theory of parabolic Laumon spaces. In type $A_1$, they are closely related to the open relativistic quantum Toda lattice of type $A$.Comment: 125 pages. v2: references updated; in section 11 the third local Lax matrix is introduced. v3: references updated. v4=v5: 131 pages, minor corrections, table of contents added, Conjecture 10.25 is now replaced by Theorem 10.25 (whose proof is based on the shuffle approach and is presented in a new Appendix). v6: Final version as published, references updated, footnote 4 adde

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of $SL_q(2)$ and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.Comment: 50 pages, 7 figures, minor revisions, typos corrected, Communications in Mathematical Physics, in pres

The Impact of Non-Equipartition on Cosmological Parameter Estimation from Sunyaev-Zel'dovich Surveys

The collisionless accretion shock at the outer boundary of a galaxy cluster should primarily heat the ions instead of electrons since they carry most of the kinetic energy of the infalling gas. Near the accretion shock, the density of the intracluster medium is very low and the Coulomb collisional timescale is longer than the accretion timescale. Electrons and ions may not achieve equipartition in these regions. Numerical simulations have shown that the Sunyaev-Zel'dovich observables (e.g., the integrated Comptonization parameter Y) for relaxed clusters can be biased by a few percent. The Y-mass relation can be biased if non-equipartition effects are not properly taken into account. Using a set of hydrodynamical simulations, we have calculated three potential systematic biases in the Y-mass relations introduced by non-equipartition effects during the cross-calibration or self-calibration when using the galaxy cluster abundance technique to constraint cosmological parameters. We then use a semi-analytic technique to estimate the non-equipartition effects on the distribution functions of Y (Y functions) determined from the extended Press-Schechter theory. Depending on the calibration method, we find that non-equipartition effects can induce systematic biases on the Y functions, and the values of the cosmological parameters Omega_8, sigma_8, and the dark energy equation of state parameter w can be biased by a few percent. In particular, non-equipartition effects can introduce an apparent evolution in w of a few percent in all of the systematic cases we considered. Techniques are suggested to take into account the non-equipartition effect empirically when using the cluster abundance technique to study precision cosmology. We conclude that systematic uncertainties in the Y-mass relation of even a few percent can introduce a comparable level of biases in cosmological parameter measurements.Comment: 10 pages, 3 figures, accepted for publication in the Astrophysical Journal, abstract abridged slightly. Typos corrected in version

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as Kazhdan--Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.Comment: 27pp., amsart++, xy. v2: references added, some other minor addition