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Equivariant homology and K-theory of affine Grassmannians and Toda lattices
For an almost simple complex algebraic group G with affine Grassmannian , we consider the equivariant homology and K-theory . They both have a commutative ring structure with respect to convolution. We identify the spectrum of homology ring with the universal group-algebra centralizer of the Langlands dual group , and we relate the spectrum of K-homology ring to the universal group-group centralizer of G and of . If we add the loop-rotation equivariance, we obtain a noncommutative deformation of the (K-)homology ring, and thus a Poisson structure on its spectrum. We relate this structure to the standard one on the universal centralizer. The commutative subring of -equivariant homology of the point gives rise to a polarization which is related to Kostant\u27s Toda lattice integrable system. We also compute the equivariant K-ring of the affine Grassmannian Steinberg variety. The equivariant K-homology of GrG is equipped with a canonical basis formed by the classes of simple equivariant perverse coherent sheaves. Their convolution is again perverse and is related to the FeiginâLoktev fusion product of -modules
A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories
We give an explicit differential equation which is expected to determine the
instanton partition function in the presence of the full surface operator in
N=2 SU(N) gauge theory. The differential equation arises as a quantization of a
certain Hamiltonian system of isomonodromy type discovered by Fuji, Suzuki and
Tsuda.Comment: 15 pages, v2: typos corrected and references added, v3: discussion,
appendix and references adde
Autoimmune Pancreatitis Associated with High Levels of Chromogranin A, Serotonin and 5-Hydroxyindoleacetic Acid
We report a case of a male patient with autoimmune pancreatitis in whom biochemical examination revealed high plasma chromogranin A concentrations, histological demonstration of a small lymphocytic infiltrate and rapid decrease in size of the pancreatic mass following short-lasting therapy with methylprednisolone. To our knowledge, this is the first patient with autoimmune pancreatitis who had a simultaneous increase of serum chromogranin A levels, circulating and urinary serotonin concentrations and urine 5-hydroxyindoleacetic acid concentrations. This is one of the few cases of mass forming pancreatitis with small lymphocytic infiltrate found in a Caucasian patient and rapid decrease in size of the pancreatic mass following short-lasting therapy with methylprednisolone
Multiplicative slices, relativistic Toda and shifted quantum affine algebras
We introduce the shifted quantum affine algebras. They map homomorphically
into the quantized -theoretic Coulomb branches of SUSY
quiver gauge theories. In type , they are endowed with a coproduct, and they
act on the equivariant -theory of parabolic Laumon spaces. In type ,
they are closely related to the open relativistic quantum Toda lattice of type
.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
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
Fusion in the entwined category of Yetter--Drinfeld modules of a rank-1 Nichols algebra
We rederive a popular nonsemisimple fusion algebra in the braided context,
from a Nichols algebra. Together with the decomposition that we find for the
product of simple Yetter-Drinfeld modules, this strongly suggests that the
relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in
the (p,1) logarithmic models of conformal field theory. For this, the category
of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category
(the one with monodromy, but not with braiding).Comment: 36 pages, amsart++, times, xy. V3: references added, an unnecessary
assumption removed, plus some minor change
Lattice fusion rules and logarithmic operator product expansions
The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing
over the last few years thanks to recent developments coming from various
approaches. A particularly fruitful point of view consists in considering
lattice models as regularizations for such quantum field theories. The
indecomposability then encountered in the representation theory of the
corresponding finite-dimensional associative algebras exactly mimics the
Virasoro indecomposable modules expected to arise in the continuum limit. In
this paper, we study in detail the so-called Temperley-Lieb (TL) fusion functor
introduced in physics by Read and Saleur [Nucl. Phys. B 777, 316 (2007)]. Using
quantum group results, we provide rigorous calculations of the fusion of
various TL modules. Our results are illustrated by many explicit examples
relevant for physics. We discuss how indecomposability arises in the "lattice"
fusion and compare the mechanisms involved with similar observations in the
corresponding field theory. We also discuss the physical meaning of our lattice
fusion rules in terms of indecomposable operator-product expansions of quantum
fields.Comment: 54pp, many comments adde
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
An Introduction to Conformal Field Theory
A comprehensive introduction to two-dimensional conformal field theory is
given.Comment: 69 pages, LaTeX; references adde
Autoimmune Pancreatitis Exhibiting Multiple Mass Lesions
Our case is a first report of autoimmune pancreatitis with multiple masses within the pancreas which was pathologically diagnosed by endoscopic ultrasound-guided fine needle aspiration and treated by steroid. The masses disappeared by steroid therapy. Our case is informative to know that autoimmune pancreatitis sometimes exhibits multiple masses within the pancreas and to diagnose it without unnecessary surgery
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