Approximation of functions in a certain Banach space by some generalized singular integrals

We introduced the $q$-Picard, the $q$-Picard-Cauchy, the $q$-Gauss-Weierstrass, and the $q$-truncated Picard singular integrals. Using the last three mentioned integrals, the orders of approximation for functions from a generalized Hölder space were determined, both in the $L^{p}$-norm and in the generalized Hölder-norm

Traintrack Calabi-Yaus from Twistor Geometry

We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in $\mathbb{P}^{3}$. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in $\mathbb{P}^{1} \times \mathbb{P}^{1}$. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction.Comment: 23 pages, 5 figure

Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals

Lattice statistical mechanics, often provides a natural (holonomic) framework to perform singularity analysis with several complex variables that would, in a general mathematical framework, be too complex, or could not be defined. Considering several Picard-Fuchs systems of two-variables "above" Calabi-Yau ODEs, associated with double hypergeometric series, we show that holonomic functions are actually a good framework for actually finding the singular manifolds. We, then, analyse the singular algebraic varieties of the n-fold integrals $\chi^{(n)}$, corresponding to the decomposition of the magnetic susceptibility of the anisotropic square Ising model. We revisit a set of Nickelian singularities that turns out to be a two-parameter family of elliptic curves. We then find a first set of non-Nickelian singularities for $\chi^{(3)}$ and $\chi^{(4)}$, that also turns out to be rational or ellipic curves. We underline the fact that these singular curves depend on the anisotropy of the Ising model. We address, from a birational viewpoint, the emergence of families of elliptic curves, and of Calabi-Yau manifolds on such problems. We discuss the accumulation of these singular curves for the non-holonomic anisotropic full susceptibility.Comment: 36 page

Holomorphic Anomaly in Gauge Theories and Matrix Models

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.Comment: 34 pages, 2 eps figures, expansion at the monopole point corrected and interpreted, and references adde

Nonperturbative Effective Actions of N=2 Supersymmetric Gauge Theories

We elaborate on our previous work on N=2 supersymmetric Yang-Mills theory. In particular, we show how to explicitly determine the low energy quantum effective action for $G=SU(3)$ from the underlying hyperelliptic Riemann surface, and calculate the leading instanton corrections. This is done by solving Picard-Fuchs equations and asymptotically evaluating period integrals. We find that the dynamics of the $SU(3)$ theory is governed by an Appell system of type $F_4$, and compute the exact quantum gauge coupling explicitly in terms of Appell functions.Comment: 57p, harvmac with hyperlinks, 9 uuencoded ps figure

Type II/F-theory Superpotentials with Several Deformations and N=1 Mirror Symmetry

We present a detailed study of D-brane superpotentials depending on several open and closed-string deformations. The relative cohomology group associated with the brane defines a generalized hypergeometric GKZ system which determines the off-shell superpotential and its analytic properties under deformation. Explicit expressions for the N=1 superpotential for families of type II/F-theory compactifications are obtained for a list of multi-parameter examples. Using the Hodge theoretic approach to open-string mirror symmetry, we obtain new predictions for integral disc invariants in the A model instanton expansion. We study the behavior of the brane vacua under extremal transitions between different Calabi-Yau spaces and observe that the web of Calabi-Yau vacua remains connected for a particular class of branes.Comment: 62 pages; v2: typos corrected and references adde