Ron Donagi

Supersymmetric Yang-Mills theory 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,Z) S-duality group (with the central element −1 of SL(2,Z) acting as charge conjugation); SL(2,Z) permutes the Higgs, confining, and oblique confining phases in the expected fashion. We also study more exotic phases.

An SU(5) heterotic standard model

Abstract We introduce a new heterotic Standard Model which has precisely the spectrum of the Minimal Supersymmetric Standard Model (MSSM), with no exotic matter. The observable sector has gauge group SU ( 3 ) C × SU ( 2 ) L × U ( 1 ) Y . Our model is obtained from a compactification of heterotic strings on a Calabi–Yau threefold with Z 2 fundamental group, coupled with an invariant SU ( 5 ) bundle. Depending on the region of moduli space in which the model lies, we obtain a spectrum consisting of the three generations of the Standard Model, augmented by 0, 1 or 2 Higgs doublet conjugate pairs. In particular, we get the first compactification involving a heterotic string vacuum (i.e., a stable bundle) yielding precisely the MSSM with a single pair of Higgs.

The Particle spectrum of heterotic compactifications

Techniques are presented for computing the cohomology of stable, holomorphic vector bundles over elliptically fibered Calabi-Yau threefolds. These cohomology groups explicitly determine the spectrum of the low energy, four-dimensional theory. Generic points in vector bundle moduli space manifest an identical spectrum. However, it is shown that on subsets of moduli space of co-dimension one or higher, the spectrum can abruptly jump to many different values. Both analytic and numerical data illustrating this phenomenon are presented. This result opens the possibility of tunneling or phase transitions between different particle spectra in the same heterotic compactification. In the course of this discussion, a classification of SU(5) GUT theories within a specific context is presented.

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.

Moduli in N = 1 heterotic/F-theory duality

Abstract The moduli in a 4D N = 1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into “base” parameters which are even (under the natural involution of the elliptic curves), and “fiber” or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5-brane/F-theoretic 3-brane impurities.

Standard-model bundles on non-simply connected Calabi-Yau threefolds

We give a proof of the existence of G = SU(5), stable holomorphic vector bundles on elliptically bered Calabi-Yau threefolds with fundamental group Z2. ThebundlesweconstructhaveEulercharacteristic3andananomalythatcanbe absorbed by M-theory ve-branes. Such bundles provide the basis for constructing the standard model in heterotic M-theory. They are also applicable to vacua of the weakly coupled heterotic string. We explicitly present a class of three family models with gauge group SU(3)CSU(2)LU(1)Y.

The spectra of heterotic standard model vacua

A formalism for determining the massless spectrum of a class of realistic heterotic string vacua is presented. These vacua, which consist of SU(5) holomorphic bundles on torus-fibered Calabi-Yau threefolds with fundamental group Z_2, lead to low energy theories with standard model gauge group (SU(3)_C x SU(2)_L x U(1)_Y)/Z_6 and three families of quarks and leptons. A methodology for determining the sheaf cohomology of these bundles and the representation of Z_2 on each cohomology group is given. Combining these results with the action of a Z_2 Wilson line, we compute, tabulate and discuss the massless spectrum.

Superpotentials for vector bundle moduli

We present a method for explicitly computing the non-perturbative superpotentials associated with the vector bundle moduli in heterotic superstrings and M-theory. This method is applicable to any stable, holomorphic vector bundle over an elliptically fibered Calabi–Yau threefold. For specificity, the vector bundle moduli superpotential, for a vector bundle with structure group G=SU(3), generated by a heterotic superstring wrapped once over an isolated curve in a Calabi–Yau threefold with base B=F1, is explicitly calculated. Its locus of critical points is discussed. Superpotentials of vector bundle moduli potentially have important implications for small instanton phase transitions and the vacuum stability and cosmology of superstrings and M-theory.

The structure of the prym map

I. THE DEGREE OF THE PRYM MAP . . . . . . . . . . . . . . . . . . . . . . . . 29 1. T h e p a r t i a l c o m p a c t i f i c a t i o n . . . . . . . . . . . . . . . . . . . . . . . . 29 2. T h e m a i n r e s u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3. C o m p u t a t i o n of loca l deg rees . . . . . . . . . . . . . . . . . . . . . . . . 35 4. T h e cod i f f e ren t i a l of P r y m . . . . . . . . . . . . . . . . . . . . . . . . . 38

Vector bundle moduli superpotentials in heterotic superstrings and M theory

The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.