119569923
Instanton counting on blowup. II. K-theoretic partition
function
AbstractWe study Nekrasov's deformed partition function $Z(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$ of 5-dimensional supersymmetric
Yang-Mills theory compactified on a circle. Mathematically it is the
generating function of the characters of the coordinate rings of the moduli
spaces of instantons on $\mathbb R^4$. We show that it satisfies a system of
functional equations, called blowup equations, whose solution is unique.
As applications, we prove (a) $F(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta) = \varepsilon_1\varepsilon_2 \log Z(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$ is regular at $\varepsilon_1 = \varepsilon_2 = 0$ (a part of Nekrasov's conjecture), and (b) the genus $1$ parts, which are first several Taylor coefficients of $F(\varepsilon_1,\varepsilon_2,\vec{a};\mathfrak q,\boldsymbol\beta)$, are written explicitly in terms of $\tau = d^2 F(0,0,\vec{a};\mathfrak q,\boldsymbol\beta)/da^2$ in rank $2$ case.
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.
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].
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.
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.
121927270
Spontaneous breaking of chiral symmetry in a vector-gluon model
Solutions of the self-consistent equation for fermion propagator in a vector-gluon model are fully examined. The equation is characterized by a set of parameters, i.e., the coupling constant g, the bare mass of the fermion mo and the cutoff A. It is proved that with a suitable gauge chosen, the equation without ~utoff has solutions ·only in the case mo=O, It is then shown that, if g'/4n Bn, the "normal-state" solution for the equation without cutoff, even if it existed, should necessarily have an unphysical singularity. This fact implies that the "normal-state" solution becomes unstable for a su.fficiently large value of g'. § l. Introduction The. reason for success of low energy theorems obtained by current algebra and PCAC-treatment may be well understood in terms of chiral symmetry in which the pseudoscalar mesons play a special role among hadtons as the Nambu­ Goldstone (NG) bosons1l transforming nonlinearly under chiral transformation. On the other hand, in the composite hadron models based on SU(3) or SU(6), it is clear that there is no vital distinction between pseudoscalar mesons and others such as vector- and tensor-mesons. It is therefore worth expecting that the pseudoscalar NG bosons can also be interpreted as the bound states in con­ formity with the viewpoint of the composite model. . This homogeneity and· heterogeneity ot the pseudoscalar mesons to other mesons should rather be in­ vestigated by comparing their internal structure with composite particles· . . In this respect it would be meaningful to study a model in which spontaneous breaking of chiral symmetry is realized by a composite NG boson: The Nambu­ Jona-Lasinio model2l is known as such an example. However, the use of chain approximation and the momentum cutoff inevitable for the local four-fermion
8772231
Instantons in Deformed Super Yang-Mills Theories
We study the instanton effective action in deformed four-dimensional N=2 and N=4 super Yang-Mills (SYM) theories. These deformed gauge theories are defined on the D-brane world-volume in the presence of constant, self-dual Ramond-Ramond (R-R) 3-form field strength background $\mathcal{F}$ which is scaled as $(2 \pi \alpha')^{1/2} \mathcal{F} = fixed$ in the zero-slope limit $\alpha' \to 0$. The instanton effective action is obtained by solving equations of motion of the deformed N=2 SYM action. We show that this effective action correctly reproduces the string theory result derived from D3/D(-1)-branes system in the lowest order of the backgrounds and gauge coupling constant.
118415383
Instanton Effective Action in Deformed Super Yang-Mills Theories
We study the ADHM construction of instantons in N = 2 supersymmetric Yang‐Mills theory deformed in constant Ramond‐Ramond (R‐R) 3‐form field strength background in type IIB superstrings. We compare the deformed instanton effective action with the effective action of fractional D3/D(−1) branes at the orbifold singularity of C2/Z2 in the same R‐R background. We find discrepancy between them at the second order in deformation parameters, which comes from the coupling of the translational zero modes of the D(−1)‐branes to the R‐R background. We improve the deformed action by adding a term with spacetime dependent gauge coupling such that the action reproduces the effective action of the fractional branes.
118929109
A Systematic Study of X-Ray Flares from Low-Mass Young Stellar Objects in the $\rho$ Ophiuchi Star-Forming Region with Chandra
We report on the results of a systematic study of X-ray flares from low-mass young stellar objects, using two deep exposure Chandra observations of the main region of the ρ Ophiuchi star-forming cloud. From 195 X-ray sources, including class I-III sources and some young brown dwarfs, we detected a total of 71 X-ray flares. Most of the flares have the typical profile of solar and stellar flares, fast rise and slow decay, while some bright flares show unusually long rise timescales. We derived the time-averaged temperature (� kT � ), luminosity (� LX� ), rise and decay timescales (τr and τd) of the flares, finding that (1) class I-II sources tend to have a highkT � , which sometimes exceeds 5keV, (2) the distribution ofLXduring flares is nearly the same for all classes from ∼10 29.5 to ∼10 31.5 ergs −1 , although there is a marginal hint of a higherLXdistribution for class I than class II-III, and (3) positive and negative log-linear correlations are found between τr and τd ,a ndkTand τr. In order to explain these relations, we used the framework of magnetic reconnection model with heat conduction and chromospheric evaporation to formulate the observational parameters (τr, τd ,a ndkT� ) as a function of the pre-flare (coronal) electronic density (nc), the half-length of the reconnected magnetic loop (L), and magnetic field strength (B). The observed correlations are well reproduced if loop lengths are nearly the same for all classes, regardless of the existence of an accretion disk. The estimated loop length is almost comparable to the typical stellar radius of these objects (10 10 -10 11 cm), which indicates that the observed flares are triggered by solar-type loops, rather than larger ones (∼ 10 12 cm) connecting the star with its inner accretion disk. The higherkTobserved for class I sources may be explained by a slightly higher magnetic field strength (≈ 500G) than for class II-III sources (200-300G).