Related papers: N=1 Curve
In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes",…
We define and study the harmonic curves on domains in $\mathbb{R}^n$ into the first Heisenberg group $\mathbb{H}^1$. These are the $C^2$-regular mappings which are critical points of the second Dirichlet energy and satisfy the weak…
We consider N=2 supersymmetric Yang-Mills theories in four dimensions with gauge group SU(N) for N larger than two. Using the cubic curve for a matter hypermultiplet transforming in the symmetric representation, obtained from M-theory by…
The paper consists of two parts. In the first part Schroedinger's equation for a charged quantum particle in a Galilei-Newton curved space-time is derived in a fully geometrical way. Gravitational and electromagnetic fields are coded into…
We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this…
The purpose of this note is to prove that there is an algebraic stack U parameterizing all curves. The curves that appear in the algebraic stack U are allowed to be arbitrarily singular, non-reduced, disconnected, and reducible. We also…
We consider configurations of Neveu-Schwarz fivebranes, Dirichlet fourbranes and an orientifold sixplane in type IIA string theory. Upon lifting the configuration to M-theory and proposing a description of how to include the effects of the…
In mathematics curves are typically defined as the images of continuous real functions (parametrizations) defined on a closed interval. They can also be defined as connected one-dimensional compact subsets of points. For simple curves of…
We construct families of hyper-elliptic curves which describe the quantum moduli spaces of vacua of $N=2$ supersymmetric $SU(N_c)$ gauge theories coupled to $N_f$ flavors of quarks in the fundamental representation. The quantum moduli…
A typical solution of an integrable system is described in terms of a holomorphic curve and a line bundle over it. The curve provides the action variables while the time evolution is a linear flow on the curve's Jacobian. Even though the…
We analyze the Seiberg-Witten curve of the six-dimensional N=(1,1) gauge theory compactified on a torus to four dimensions. The effective theory in four dimensions is a deformation of the N=2* theory. The curve is naturally holomorphically…
The strongly coupled vacua of an N=1 supersymmetric gauge theory can be described by imposing quantization conditions on the periods of the gauge theory resolvent, or equivalently by imposing factorization conditions on the associated N=2…
For a smooth plane cubic $B$, we count curves $C$ of degree $d$ such that the normalizations of $C\backslash B$ are isomorphic to $\Bbb A^1$, for $d\leq7$ (for $d=7$ under some assumption). We also count plane rational quartic curves…
Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety. Let $\alpha$ a class of…
Motivated by Vafa's model, we study the $tt^{*}$ geometry of a degenerate class of FQHE models with an abelian group of symmetry acting transitively on the classical vacua. Despite it is not relevant for the phenomenology of the FQHE, this…
We derive a set of necessary and sufficient conditions for obtaining N=1 backgrounds of M-theory and type IIA strings in the presence of fluxes. Our metrics are warped products of four-dimensional Minkowski space-time with a curved internal…
We examine space-filling curves, which are surjective continuous maps from $[0,1]$ to some higher-dimensional space, usually the unit square $[0,1]^2$. In particular, we define Peano's curve and Lebesgue's curve, and state some of their…
A bi-Hamiltonian hierarchy of complex vector soliton equations is derived from geometric flows of non-stretching curves in the Lie groups $G=SO(N+1),SU(N)\subset U(N)$, generalizing previous work on integrable curve flows in Riemannian…
The dimension of spaces of global sections for line bundles on semistable curves parametrized by the compactified Picard scheme is studied. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following…
A framework allowing for perturbative calculations to be carried out for quantum field theories with arbitrary smoothly curved boundaries is described. It is based on an expansion of the heat kernel derived earlier for arbitrary mixed…