14508627
PERVERSE COHERENT SHEAVES ON BLOW-UP. II. WALL-CROSSING AND BETTI NUMBERS FORMULA
This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up p: b X → X of a projective surface X at a point 0. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle O(C) associated with the exceptional divisor C. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in (24): c) The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to (C). d) The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by O(−mC) for sufficiently largem. Therefore usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.
56216283
K-theoretic Donaldson invariants via instanton counting
In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms.
873508
t-analogs of q-characters of Kirillov-Reshetikhin modules of quantm affine algebras
We prove the Kirillov-Reshetikhin conjecture concerning certain finite dimensional representations of a quantum affine algebra $\Ua$ when $\hat\g$ is an untwisted affine Lie algebra of type $ADE$. We use $t$--analog of $q$--characters introduced by the author in an essential way.
10853977
Lectures on instanton counting
These notes have two parts. The first is a study of Nekrasov's deformed partition functions $Z(\ve_1,\ve_2,\vec{a};\q,\vec{\tau})$ of N=2 SUSY Yang-Mills theories, which are generating functions of the integration in the equivariant cohomology over the moduli spaces of instantons on $\mathbb R^4$. The second is review of geometry of the Seiberg-Witten curves and the geometric engineering of the gauge theory, which are physical backgrounds of Nekrasov's partition functions. 
The first part is continuation of math.AG/0306198, where we identified the Seiberg-Witten prepotential with $Z(0,0,\vec{a};\q,0)$. 
We put higher Casimir operators to the partition function and clarify their relation to the Seiberg-Witten $u$-plane. We also determine the coefficients of $\ve_1\ve_2$ and $(\ve_1^2+\ve_2^2)/3$ (the genus 1 part) of the partition function, which coincide with two measure factors $A$, $B$ appeared in the $u$-plane integral. 
The proof is based on the blowup equation which we derived in the previous paper.
119132882
Refined Chern-Simons theory and Hilbert schemes of points on the plane
Dedicated to Professor Igor Frenkel on the occasion of his sixtieth birthday Abstract. Aganagic and Shakirov propose a refinement of the SU(N) Chern- Simons theory for links in three manifolds with S 1 -symmetry, such as torus knots in S 3 , based on deformation of the S and T matrices, originally found by Kirillov and Cherednik. We relate the large N limit of the S matrix to the Hilbert schemes of points on the affine plane. As an application, we find an explicit formula for the Euler characteristics of the universal sheaf, applied arbitrary Schur functor.
8936511
Instantons and affine Lie algebras
Various constructions of the affine Lie algebra action on the moduli space of instantons on 4-manifolds are discussed. The analogy between the local-global principle and the role of mass is also explained. The detailed proofs are given in separated papers [16,17].
53532682
Effects of the quark field on the ghost propagator of lattice Landau gauge QCD
Infrared features of the ghost propagator of color-diagonal and color antisymmetric ghost propagator of quenched SU(2) and quenched SU(3) are compared with those of unquenched Kogut-Susskind fermion SU(3) lattice Landau gauge. We compare (i) the fluctuation of the ghost propagator (ii) the ghost condensate parameter v of the local composite operator (LCO) approach, and (iii) the Binder cumulant of color antisymmetric ghost propagator between quenched and unquenched configurations. The color-diagonal SU(3) ghost dressing function of unquenched configurations has weaker singularity than the quenched configurations. In both cases fluctuations become large in q /( ){sup 2}), where {phi}-vector(q) is the color antisymmetric ghost propagator measured by the sample average of gauge fixed configurations via parallel tempering method, becomes {approx}4/9 in all the momentum region. The Binder cumulant of the color antisymmetric ghost propagator of quenched SU(2) can be explained by the 3D Gaussian distribution, but that of the unquenched MILC{sub c} deviates slightly from that of the eight-dimensional Gaussian distribution. The strongermore » singularity and large fluctuation in the quenched configuration could be the cause of the deviation of the Kugo-Ojima confinement parameter c from 1, and the presence of ordering in the ghost propagator of unquenched configurations makes it closer to 1.« less
123022148
Structure of the Gauge Transformation Group in the Square Integrable Space and Gribov's Ambiguity in the Coulomb Gauge
It is found that a well-defined gauge transformation group and the well-known discon­ nected structure of it emerge only from the square integrability with some natural assump­ tion of the boundedness on the gradient of the gauge field. The geometrical formulation of the Coulomb gauge fixing degeneracy is restated with more careful treatment to the dual space considered as space of distribution, than in our previous paper. A comment on the suitably restricted space version of the formulation is made. As such an example, '0 (3) symmetric' case is treated. Together with the discon­ nected structure of the gauge transformation group, it is found that there exist a lot of Coulomb gauge fixing degeneracies such that "topological number" jumps by any integer value between transverse fields.
119481701
Instanton Equations for the Supersymmetric CPN−1 Sigma Model on Non(anti)commutative Superspace
We study the instanton equation of the supersymmetric CP N −1 sigma model on non(anti)commutative superspace in two dimensions. We show that the undeformed instanton equation is consistent with the deformed equations of motion. Then we conclude that the instanton equation is not deformed by superspace non(anti)commutativity. 1. Introduction Studies of strings with non-trivial classical values of its field components have brought about new development in quantum field theories. If one allows some of the field components to possess background values on D-branes, one obtains new classes of quantum field theories in the low energy limits of string theory. They have novel features due to the non(anti)commutativity (NC) properties of spacetime coordinates or superspace coordinates. Superstring theory with its graviphoton field in the R-R sector having a self-dual background value gives rise to supersymmetric Yang-Mills theories with N =1 /2 supersymmetry as an effective field theory, 1) providing a good realization of the NC superspace considered in different contexts some time ago. 2) NC field theories and SUSY field theories on NC superspace (field theories with deformed SUSY) have properties quite different from the usual field theories, and they need to be examined, in comparison with their commutative partners. Perturbative properties of SUSY field theories on NC superspace have been studied extensively in the last few years. Positive results have been obtained concerning the renormalizability of the model. 3)–5) One non-perturbative issue in deformed SUSY field theories is whether and how its instanton equation is modified from that of the usual SUSY field theories. This question has been studied in the case of super Yang-Mills theories in d = 4, and the modified self-dual equation has been derived, 6) and is extended to the NC deformed N = 2 SUSY case. 7) The solutions to this new self-dual equation have also been studied. 8) N = 2 SUSY CP N −1 models in d = 2 share many nice properties of super Yang-Mills theories and it allows exact treatment. From this point of view, we have constructed extension of the SUSY CP N −1 model to NC superspace. 5),9) The usual
118709116
Horizon function in Landau gauge QCD revisited -- Free boundary case from viewpoint of network QCD --
In Gribov-Zwanziger scenario of color confinement, Zwanziger proposed two types of horizon function in lattice Landau gauge, the first one and the second one of which foundation was discussed in detail. The second type horizon function is focussed on in the present study. Its derivation and the horizon condition are briefly reviewed along the line of Zwanziger, and it is also reviewed that this horizon condition and Kugo-Ojima color confinement criterion coincide in the continuum limit. In case of free boundary condition in contrast to periodic boundary condition, it was shown that the horizon condition holds for each gauge fixed configuration in Landau gauge. It is clarified that this fact can be derived from some identity relation which holds on an arbitrary network of links in Landau gauge. Thus the origin of the pointwise validity of horizon condition is highlighted.
17703115
X-Ray Observation on the Monoceros R2 Star-Forming Region with the Chandra ACIS-I Array
We report on the results of the Chandra ACIS-I observation on the central region of the Monoceros R2 cloud (Mon R2), a high-mass star-forming region (SFR) at a distance of 830pc. With a deep exposure of ∼ 100ks, we detected 368 X-ray sources, ∼80% of which were identified with the near-infrared (NIR) counterparts. We systematically analyzed the spectra and time variability of most of the X-ray emitting sources and provided a comprehensive X-ray source catalog for the first time. Using the J-, H-, and K-band magnitudes of the NIR counterparts, we estimated the evolutionary phase (classical T Tauri stars and weak-lined T Tauri stars) and the mass of the X-ray emitting sources, and analyzed the X-ray properties as a function of the age and mass. We found a marginal hint that classical T Tauri stars have a slightly higher temperature (2.4keV) than that of weak-lined T Tauri stars (2.0keV). A significant fraction of the high- and intermediate-mass sources have a time variability and high plasma temperatures (2.7keV) similar to those of low-mass sources (2.0keV). We performed the same analysis for other SFRs, the Orion Nebula Cluster and Orion Molecular Cloud-2/3, and obtained similar results to Mon R2. This supports the earlier results of this observation obtained by Kohno et al. (2002, ApJ, 567, 423) and Preibisch et al. (2002, A&A, 392, 945) that high- and intermediate-mass young stellar objects emit X-rays via magnetic activity. We also found a significant difference in the spatial distribution between X-ray and NIR sources.