R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of $N$ D-brane probes for both $N \to \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.

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Counting BPS operators in gauge theories: quivers, syzygies and plethystics

This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar{DT}{}^\alpha(\tau)$ are defined for all classes $\alpha$, and are equal to $DT^\alpha(\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. To prove all this, the authors study the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\mathfrak M$ may be written locally as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.

A theory of generalized Donaldson-Thomas invariants

Introduction Hilbert scheme of points Framed moduli space of torsion free sheaves on $\mathbb{P}^2$ Hyper-Kahler metric on $(\mathbb{C}^2)^{[n]}$ Resolution of simple singularities Poincare polynomials of the Hilbert schemes (1) Poincare polynomials of Hilbert schemes (2) Hilbert scheme on the cotangent bundle of a Riemann surface Homology group of the Hilbert schemes and the Heisenberg algebra Symmetric products of an embedded curve, symmetric functions and vertex operators Bibliography Index.

Lectures on Hilbert schemes of points on surfaces

An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

Toric degenerations of Gelfand–Cetlin systems and potential functions

We propose a new method to find gravity duals to a large class of three-dimensional Chern-Simons-matter theories, using techniques from dimer models. The gravity dual is given by M-theory on AdS4 × Y7, where Y7 is an arbitrary seven-dimensional toric Sasaki-Einstein manifold. The cone of Y7 is a toric Calabi-Yau 4-fold, which coincides with a branch of the vacuum moduli space of Chern-Simons-matter theories.

https://iopscience.iop.org/article/10.1088/1126-6708/2008/12/045/pdf

Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of $A_n$-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in A. Ishii and H. Uehara, "Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces", J. Differential Geom., 71(3):385–435, 2005, and the homological mirror symmetry for $A_n$-singularities.

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Stability Conditions on An-Singularities

We study the behavior of a dimer model under the operation of removing a corner from the lattice polygon and taking the convex hull of the rest. This refines an operation of Gulotta, and the special McKay correspondence plays an essential role in this refinement. As a corollary, we show that for any lattice polygon, there is a dimer model such that the derived category of finitely-generated modules over the path algebra of the corresponding quiver with relations is equivalent to the derived category of coherent sheaves on a toric Calabi-Yau 3-fold determined by the lattice polygon. Our proof is based on a detailed study of relationship between combinatorics of dimer models and geometry of moduli spaces, and does not depend on the result of math/9908027.

Dimer models and the special McKay correspondence

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus Open image in new window We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.

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Kazushi Ueda

Papers

Toric degenerations of Gelfand–Cetlin systems and potential functions

Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories

Stability Conditions on An-Singularities

Dimer models and the special McKay correspondence

Homological Mirror Symmetry for Toric del Pezzo Surfaces