Stokes matrices for the quantum differential equations of some Fano varieties

The classical Stokes matrices for the quantum differential equation of projective n-space are computed, using multisummation and the so-called monodromy identity. Thus, we recover the results of D. Guzzetti that confirm Dubrovin's conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of the quantum differential equations of smooth Fano hypersurfaces in projective n-space and for weighted projective spaces.Comment: 20 pages. Introduction has been changed. Small corrections in the tex

On Duality Walls in String Theory

Following the RG flow of an N=1 quiver gauge theory and applying Seiberg duality whenever necessary defines a duality cascade, that in simple cases has been understood holographically. It has been argued that in certain cases, the dualities will pile up at a certain energy scale called the duality wall, accompanied by a dramatic rise in the number of degrees of freedom. In string theory, this phenomenon is expected to occur for branes at a generic threefold singularity, for which the associated quiver has Lorentzian signature. We here study sequences of Seiberg dualities on branes at the C_3/Z_3 orbifold singularity. We use the naive beta functions to define an (unphysical) scale along the cascade. We determine, as a function of initial conditions, the scale of the wall as well as the critical exponent governing the approach to it. The position of the wall is piecewise linear, while the exponent appears to be constant. We comment on the possible implications of these results for physical walls.Comment: 22 pages, 2 figures. v2: physical interpretation rectified, reference adde

A categorification of Morelli's theorem

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety. Specifically, let $X$ be a proper toric variety of dimension $n$ and let M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n be the Lie algebra of the compact dual (real) torus T_\bR^\vee\cong U(1)^n. Then there is a corresponding conical Lagrangian \Lambda \subset T^*M_\bR and an equivalence of triangulated dg categories \Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda), where \Perf_T(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and \Sh_{cc}(M_\bR;\Lambda) is the triangulated dg category of complex of sheaves on M_\bR with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on M_\bR.Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v

Minimal Family Unification

Absract It is proposed that there exist, within a new $SU(2)^{'}$, a gauged discrete group $Q_6$ (the order 12 double dihedral group) acting as a family symmetry. This nonabelian finite group can explain hierarchical features of families, using an assignment for quarks and leptons dictated by the requirements of anomaly cancellation and of no additional quarks.Comment: 10 pages, IFP-701-UNC;VAND-TH-94-

Seiberg Duality is an Exceptional Mutation

The low energy gauge theory living on D-branes probing a del Pezzo singularity of a non-compact Calabi-Yau manifold is not unique. In fact there is a large equivalence class of such gauge theories related by Seiberg duality. As a step toward characterizing this class, we show that Seiberg duality can be defined consistently as an admissible mutation of a strongly exceptional collection of coherent sheaves.Comment: 32 pages, 4 figures; v2 refs added, "orbifold point" discussion refined; v3 version to appear in JHEP, discussion of torsion sheaves improve

Topological Orbifold Models and Quantum Cohomology Rings

We discuss the toplogical sigma model on an orbifold target space. We describe the moduli space of classical minima for computing correlation functions involving twisted operators, and show, through a detailed computation of an orbifold of ${\bf CP}^1$ by the dihedral group $D_{4},$ how to compute the complete ring of observables. Through this procedure, we compute all the rings from dihedral ${\bf CP}^1$ orbifolds; we note a similarity with rings derived from perturbed $D-$series superpotentials of the $A-D-E$ classification of $N = 2$ minimal models. We then consider ${\bf CP}^2/D_4,$ and show how the techniques of topological-anti-topological fusion might be used to compute twist field correlation functions for nonabelian orbifolds.Comment: 48 pages, harvmac, HUTP-92/A06

Quantum-mechanical model for particles carrying electric charge and magnetic flux in two dimensions

We propose a simple quantum mechanical equation for $n$ particles in two dimensions, each particle carrying electric charge and magnetic flux. Such particles appear in (2+1)-dimensional Chern-Simons field theories as charged vortex soliton solutions, where the ratio of charge to flux is a constant independent of the specific solution. As an approximation, the charge-flux interaction is described here by the Aharonov-Bohm potential, and the charge-charge interaction by the Coulomb one. The equation for two particles, one with charge and flux ($q, \Phi/Z$) and the other with ($-Zq, -\Phi$) where $Z$ is a pure number is studied in detail. The bound state problem is solved exactly for arbitrary $q$ and $\Phi$ when $Z>0$. The scattering problem is exactly solved in parabolic coordinates in special cases when $q\Phi/2\pi\hbar c$ takes integers or half integers. In both cases the cross sections obtained are rather different from that for pure Coulomb scattering.Comment: 12 pages, REVTeX, no figur

Dibaryons from Exceptional Collections

We discuss aspects of the dictionary between brane configurations in del Pezzo geometries and dibaryons in the dual superconformal quiver gauge theories. The basis of fractional branes defining the quiver theory at the singularity has a K-theoretic dual exceptional collection of bundles which can be used to read off the spectrum of dibaryons in the weakly curved dual geometry. Our prescription identifies the R-charge R and all baryonic U(1) charges Q_I with divisors in the del Pezzo surface without any Weyl group ambiguity. As one application of the correspondence, we identify the cubic anomaly tr R Q_I Q_J as an intersection product for dibaryon charges in large-N superconformal gauge theories. Examples can be given for all del Pezzo surfaces using three- and four-block exceptional collections. Markov-type equations enforce consistency among anomaly equations for three-block collections.Comment: 47 pages, 11 figures, corrected ref

A Geometric Unification of Dualities

We study the dynamics of a large class of N=1 quiver theories, geometrically realized by type IIB D-brane probes wrapping cycles of local Calabi-Yau threefolds. These include N=2 (affine) A-D-E quiver theories deformed by superpotential terms, as well as chiral N=1 quiver theories obtained in the presence of vanishing 4-cycles inside a Calabi-Yau. We consider the various possible geometric transitions of the 3-fold and show that they correspond to Seiberg-like dualities (represented by Weyl reflections in the A-D-E case or `mutations' of bundles in the case of vanishing 4-cycles) or large N dualities involving gaugino condensates (generalized conifold transitions). Also duality cascades are naturally realized in these classes of theories, and are related to the affine Weyl group symmetry in the A-D-E case.Comment: 94 pages, 18 figures. Added referenc

NC Calabi-Yau Orbifolds in Toric Varieties with Discrete Torsion

Using the algebraic geometric approach of Berenstein et {\it al} (hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with discrete torsion. We first develop a new way of getting complex $d$ mirror Calabi-Yau hypersurfaces $H_{\Delta}^{\ast d}$ in toric manifolds $M_{\Delta }^{\ast (d+1)}$ with a $C^{\ast r}$ action and analyze the general group of the discrete isometries of $H_{\Delta}^{\ast d}$. Then we build a general class of $d$ complex dimension NC mirror Calabi-Yau orbifolds where the non commutativity parameters $\theta_{\mu \nu}$ are solved in terms of discrete torsion and toric geometry data of $M_{\Delta}^{(d+1)}$ in which the original Calabi-Yau hypersurfaces is embedded. Next we work out a generalization of the NC algebra for generic $d$ dimensions NC Calabi-Yau manifolds and give various representations depending on different choices of the Calabi-Yau toric geometry data. We also study fractional D-branes at orbifold points. We refine and extend the result for NC $T^{2})/(\mathbf{{Z_{2}}\times {Z_{2})}}$ to higher dimensional torii orbifolds in terms of Clifford algebra.Comment: 38 pages, Late