arXiv subjects

Ron Donagi

Publications and source records attributed to Ron Donagi.

84 records · Page 5Linked to original sources

arXiv2000

Standard Model Vacua in Heterotic M-Theory

We present a class of N=1 supersymmetric ``standard'' models of particle physics, derived directly from heterotic M-theory, that contain three families of chiral quarks and leptons coupled to the gauge group SU(3)_C X SU(2)_L X U(1)_Y. These models are a fundamental form of ``brane world'' theories, with an observable and hidden sector each confined, after compactification on a Calabi--Yau threefold, to a BPS three-brane separated by a higher dimensional bulk space with size of the order of the intermediate scale. The requirement of three families, coupled to the fundamental conditions of anomaly freedom and supersymmetry, constrains these models to contain additional five-branes located in the bulk space and wrapped around holomorphic curves in the Calabi--Yau threefold.

hep-th↗

arXiv1999

Moduli Spaces of Fivebranes on Elliptic Calabi-Yau Threefolds

We present a general method for calculating the moduli spaces of fivebranes wrapped on holomorphic curves in elliptically fibered Calabi-Yau threefolds, in particular, in the context of heterotic M theory. The cases of fivebranes wrapped purely on a fiber curve, purely on a curve in the base and, generically, on a curve with components both in the fiber and the base are each discussed in detail. The number of irreducible components of the fivebrane and their properties, such as their intersections and phase transitions in moduli space, follow from the analysis. Even though generic curves have a large number of moduli, we show that there are isolated curves that have no moduli associated with the Calabi-Yau threefold. We present several explicit examples, including cases which correspond to potentially realistic three family models with grand unified gauge group SU(5).

hep-th↗

arXiv1999

Holomorphic Vector Bundles and Non-Perturbative Vacua in M-Theory

We review the spectral cover formalism for constructing both U(n) and SU(n) holomorphic vector bundles on elliptically fibered Calabi-Yau three-folds which admit a section. We discuss the allowed bases of these three-folds and show that physical constraints eliminate Enriques surfaces from consideration. Relevant properties of the remaining del Pezzo and Hirzebruch surfaces are presented. Restricting the structure group to SU(n), we derive, in detail, a set of rules for the construction of three-family particle physics theories with phenomenologically relevant gauge groups. We show that anomaly cancellation generically requires the existence of non-perturbative vacua containing five-branes. We illustrate these ideas by constructing four explicit three-family non-perturbative vacua.

hep-th↗

arXiv1998

Non-Perturbative Vacua and Particle Physics in M-Theory

In this letter, we introduce a general theory for the construction of particle physics theories, with three families and realistic gauge groups, within the context of heterotic M-theory. This is achieved using semi-stable holomorphic gauge bundles over elliptically fibered Calabi-Yau three-folds. Construction of realistic theories is facilitated by the appearance of non-perturbative five-branes in the vacuum. The complete moduli space of these five-branes is computed and their worldvolume gauge theory discussed. It is shown, within the context of holomorphic gauge bundles, how grand unified gauge groups can be spontaneously broken to the gauge group of the standard model. These ideas are illustrated in an explicit SU(5) three-family example.

hep-th↗

arXiv1997

The arithmetic-geometric mean and isogenies for curves of higher genus

Computation of Gauss's arithmetic-geometric mean involves iteration of a simple step, whose algebro-geometric interpretation is the construction of an elliptic curve isogenous to a given one, specifically one whose period is double the original period. A higher genus analogue should involve the explicit construction of a curve whose jacobian is isogenous to the jacobian of a given curve. The doubling of the period matrix means that the kernel of the isogeny should be a lagrangian subgroup of the group of points of order 2 in the jacobian. In genus 2 such a construction was given classically by Humbert and was studied more recently by Bost and Mestre. In this article we give such a construction for general curves of genus 3. We also give a similar but simpler construction for hyperelliptic curves of genus 3. We show that the hyperelliptic construction is a degeneration of the general one, and we prove that the kernel of the induced isogeny on jacobians is a lagrangian subgroup of the points of order 2. We show that for g at least 4 no similar construction exists, and we also reinterpret the genus 2 case in our setup. Our construction of these correspondences uses the bigonal and the trigonal constructions, familiar in the theory of Prym varieties.

alg-geom↗

arXiv1995

Supersymmetric Yang-Mills Systems And Integrable Systems

The Coulomb branch of $N=2$ supersymmetric gauge theories in four dimensions is described in general by an integrable Hamiltonian system in the holomorphic sense. A natural construction of such systems comes from two-dimensional gauge theory and spectral curves. Starting from this point of view, we propose an integrable system relevant to the $N=2$ $SU(n)$ gauge theory with a hypermultiplet in the adjoint representation, and offer much evidence that it is correct. The model has an $SL(2,{\bf Z})$ $S$-duality group (with the central element $-1$ of $SL(2,{\bf Z})$ acting as charge conjugation); $SL(2,{\bf Z})$ permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.

hep-th↗

arXiv1995

Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles

This is the expanded text of a series of CIME lectures. We present an algebro-geometric approach to integrable systems, starting with those which can be described in terms of spectral curves. The prototype is Hitchin's system on the cotangent bundle of the moduli space of stable bundles on a curve. A variant involving meromorphic Higgs bundles specializes to many familiar systems of mathematics and mechanics, such as the geodesic flow on an ellipsoid and the elliptic solitons. We then describe some systems in which the spectral curve is replaced by various higher dimensional analogues: a spectral cover of an arbitrary variety, a Lagrangian subvariety in an algebraically symplectic manifold, or a Calabi-Yau manifold. One peculiar feature of the CY system is that it is integrable analytically, but not algebraically: the Liouville tori (on which the system is linearized) are the intermediate Jacobians of a family of Calabi-Yau manifolds. Most of the results concerning these three types of non-curve-based systems are quite recent. Some of them, as well as the compatibility between spectral systems and the KP hierarchy, are new, while other parts of the story are scattered through several recent preprints. As best we could, we tried to maintain the survey style of this article, starting with some basic notions in the field and building gradually to the recent developments.

alg-geom↗

arXiv1995

Spectral Covers

This is a survey of various results about spectral covers and their relationship to Higgs bundles. To a G-principal Higgs bundle on a variety S corresponds a cameral cover \widetilde{S} of S (a W-Galois cover, where W is the Weyl group of G) together with a sheaf on \widetilde{S} which in simple cases is a line bundle, and is W-equivariant up to certain twists and shifts. Various other types of spectral covers, depending on the choice of a representation or weight of G, arise as associated objects of \widetilde{S}. We focus on the decomposition of the Picards of these spectral covers into Pryms (this includes various well-known Prym identities as special cases) and on the interpretation, in the spirit of Hitchin's abelianization program, of a distinguished Prym component as parameter space for higgs bundles.

alg-geom↗

arXiv1995

A Non-Linear Deformation of the Hitchin Dynamical System

Mukai's space, parametrizing simple sheaves on a K3 surface S whose numerical invariants are those of a line bundle on a curve C in S, is interpreted as a deformation of Hitchin's system on C. This is used to show that the nilpotent cone in Mukai space is Lagrangian. In rank 2, components of this nilpotent cone are described as affine bundles over symmetric products of the curve. The underlying vector bundles give the corresponding components of the Hitchin nilpotent cone.

alg-geom↗

arXiv1994

Cubics, Integrable Systems, and Calabi-Yau Threefolds

In this work we construct an analytically completely integrable Hamiltonian system which is canonically associated to any family of Calabi-Yau threefolds. The base of this system is a moduli space of gauged Calabi-Yaus in the family, and the fibers are Deligne cohomology groups (or intermediate Jacobians) of the threefolds. This system has several interesting properties: the multivalued sections obtained as Abel-Jacobi images, or ``normal functions'', of a family of curves on the generic variety of the family, are always Lagrangian; the natural affine coordinates on the base, which are used in the mirror correspondence, arise as action variables for the integrable system; and the Yukawa cubic, expressing the infinitesimal variation of Hodge structure in the family, is essentially equivalent to the symplectic structure on the total space.

alg-geom↗

arXiv1993

Theta Functions for $\SL(n)$ versus $\GL(n)$

Over a smooth complex projective curve $C$ of genus $g$ let $\M (n,d)$ be the moduli space of semistable bundles of rank $n$ and degree $d$ on $C$, and $\SM (n,L)$, the moduli space of those bundles whose determinant is isomorphic to a fixed line bundle $L$ over $C$. Let $θ_F$ and $θ$ be theta bundles over these two moduli spaces. We prove a simple formula relating their spaces of sections: if $h=\gcd (n,d)$ is the greatest common divisor of $n$ and $d$, and $L\in \Pic ^d(C)$, then $$\dim H^0(\SM (n,L), θ^k) \cdot k^g=\dim H^0(\M(n,d),θ_F^k)\cdot h^g.$$ We also formulate a conjectural duality between these two types of spaces of sections.

alg-geom↗

arXiv1992

The fibers of the Prym map

In this work we use the bigonal, trigonal and tetragonal constructions to describe the fibers of the Prym map P : R_{g} ---->A_{g-1} inthe cases when it is dominant, i.e. for g < 7. The most interesting cases are g = 5, where the fiber is a double cover of the Fano surface of lines on a cubic threefold, and g=6, where the map is generically finite (of degree 27) with Galois group WE_{6}, so that the general fiber has the structure of the 27 lines on a cubic surface. For g > 6, the map is known to be generically injective. The tetragonal construction gives many counterexamples to injectivity, and we conjecture that all noninjectivity is due to the tetragonal construction.

alg-geom↗