The interdisciplinary research at Penn arose from the convergence of two very different historical developments.
In mathematics, great progress had been made in the understanding of algebraically integrable systems, especially those related to Higgs bundles. This theory turned out to have important applications to both string theory and field theory. Together with his student Markman, Donagi exhibited an integrable structure on the moduli space of Calabi-Yau manifolds, a basic ingredient of the conjectural mirror symmetry. In subsequent work, Donagi and Witten used integrable systems to extend the theory of Seiberg and Witten to include adjoint matter. Together with Grassi, Donagi and Witten were also able to calculate a non-perturbative superpotential in an F-theory compactification to four dimensions. Finally, Donagi realized that his theory of integrable systems also applied to describe the moduli space of G-bundles on elliptically fibered spaces, including Calabi-Yau spaces which appear in string dualities. This result, together with the work of Friedman, Morgan and Witten on related questions, allowed Donagi and his collaborators to establish the geometric part of the heterotic/F-theory duality. Closely related results concerning this duality in four dimensions were also obtained by Pantev and his collaborators. Together with Bershadsky, Johansen, Sadov and Vafa they were able to produce a concrete model for the Seiberg duality in N=1 theories, to give a geometric interpretation of the gauge symmetries of the heterotic string in terms of the F-theory dual and to analyze the vacua for the compactifications without vector structure of the Chaudhuri-Hockney-Lykken string.
Meanwhile, there have been significant developments in physics. Superstrings and M-theory naturally occur in higher dimensional spacetime, ten-dimensions for superstrings and eleven-dimensions for M-theory. In order to describe our physical world, one must explain how a four-dimensional spacetime (three space and one time) emerges from these higher dimensions. It was shown by Ovrut and his collaborators that this could occur if our physical world "lived" on the surface of an extended object, a brane, embedded in the higher dimensional bulk space. This work, which was carried out within the context of M-theory but also extends to superstrings, was a fundamental paradigm for so-called "brane world" theories of our Universe, which have attracted much attention recently. A key requirement of these brane world theories is that the standard model or a grand unified theory should arise naturally on the physical brane.
It is at this point that the interdisciplinary collaboration at Penn began. It was realized that the geometric techniques for analyzing moduli spaces of bundles were precisely the needed tools for understanding the brane-world scenarios. Donagi, Ovrut, Pantev and collaborators were able to show that grand unified theories of physical interest could be constructed if the vacuum state of the theory was an N=1 supersymmetric gauge "instanton" on an appropriate geometrical background. They showed that these instantons could be explicitly constructed as holomorphic vector bundles over elliptically fibered Calabi-Yau three-folds. Within this context, they were able to produce a large number of physically acceptable grand unified brane world theories. To directly obtain the standard model, it was necessary to extend the available construction techniques for bundles to manifolds of non-trivial homotopy. This was accomplished in several papers and shown to lead to standard model brane world scenarios. These were the first consistent grand unified and standard model brane world models within the context of M-theory. They are now called heterotic M-theory. An important new aspect of these theories is the prediction that there must be branes, in addition to our physical world, located in the higher dimensional bulk space. The local properties of these bulk branes involved studying the mathematical aspects of their moduli spaces. Furthermore, the question of what happens if a bulk brane collides with our physical brane was asked, and answered, in a paper co-authored by Ovrut, Pantev and Park. In this paper, it was shown that this corresponds to a so-called "small instanton" phase transition where the bulk brane is absorbed into our physical world, changing its properties dramatically. Again, the solution of this problem was a quintessential interdisciplinary result, requiring the mathematics of torsion free sheaves and holomorphic bundle theory to solve a new, physical problem. The bulk branes can have a very interesting spectrum of particles in their own right. These arise from a careful study of the degeneracy of two-cycles in the Calabi-Yau three-fold in certain regions of its moduli space. This work, again at the boundary of mathematics and physics, was carried out by Grassi, Ovrut and Guralnik.
This brane world approach to particle physics, either heterotic M-theory or its superstring analogs, suggests a radically new solution to the problems of early universe cosmology. Working in collaboration with Paul Steinhardt and Justin Khoury at Princeton University and Neil Turok at Cambridge University, Ovrut constructed the Ekpyrotic Universe, a possible alternative to inflation scenarios. This work relies heavily on the results obtained by the Math/Physics collaboration at Penn. For example, in Ekpyrotic theory, the Big Bang occurs as the catastrophic collision of a bulk brane with our physical brane. The analysis and physics of this collision depend entirely on the theory of holomorphic vector bundles and their small instanton phase transitions worked out with Donagi and Pantev. The Ekpyrotic collision, in addition to producing the expansion and heat of the Big Bang, simultaneously, via a small instanton transition, can produce a standard-like model with the correct gauge group and three families of quarks and leptons. Detailed models of Ekpyrosis require certain technical data, such as the tensions of the associated branes and the potentials between them. Again, this data must be obtained from an interplay of modern mathematics with physics. Brane tensions were found to depend on a detailed knowledge of the Chern characters on physically relevant vector bundles and were recently computed. The theory of superpotentials for the transition moduli of vector bundles is of importance in Ekpyrotic theory and is currently under study. Ekpyrotic theory is the direct outgrowth of truly interdisciplinary collaboration between mathematicians, physicists and cosmologists.
Some of the research projects currently being carried out by the Penn Math/Physics research group, or anticipated for future study, are listed below. Of course, part of the focus of the M/PRG will change as new research directions manifest themselves.
Metrics with G2 Holonomy and Calabi-Yau Geometry
Constructing metrics with holonomy G2 has always been a difficult task. A very interesting new insight into the structure of G2-spaces and their singularities was recently provided by the geometry of string backgrounds in M-theory. This motivates two closely related projects: the systematic construction of new and interesting singular G2 spaces, and the analysis of their degenerations. In addition to the intrinsic mathematical interest in such results, it is expected that they will also provide a broad class of G2 compactifications of M-theory with the properties required for realistic models of particle physics. There are also remarkable relationships between G2 and Calabi-Yau manifolds.
The study of S1-gerbes and S1-bound gerbes on smooth manifolds has attracted considerable attention in the last couple of years. Our interest in the subject stems from two main sources. The first is differential geometric and topological - it turns out that the theory of self-dual connections on abelian gerbes and higher abelian gerbes is a very powerful framework for understanding the higher abelian gauge theories that naturally arise in stringy physics. The second is algebro-geometric: abelian gerbes bounded by a subgroup emerge as the natural habitat of Fourier-Mukai transforms of sheaves on genus one fibered varieties and thus allow for major improvements in the explicit construction of vector bundles on higher dimensional spaces. In view of this, we are working on two complementary projects concerning gerbes: non-abelian Fourier-Mukai duality, and K-theory of gerbes.
Many moduli problems in geometry and physics give rise to setups in which
the objects whose variation one wants to study live naturally in a triangulated
category rather than in an abelian one. Typical examples include moduli
problems for coherent sheaves on stacks and moduli problems for D-branes
in string theory. We are developing a notion of stability for objects
in a triangulated category D which is intrinsic to D and does not require
the choice of a heart or a t-structure on D. The problem of finding such
a stability notion plays a central role in understanding the braid action
on the derived category of sheaves of a Calabi-Yau manifold at the large
complex structure limit and in determining which D-branes in a tqft are
physical, that is, exist in string theory.
Instantons, Vector Bundles and Small Instanton Transitions
Within these topics we are studying the following.
a) The theory of stable, holomorphic G-bundles on torus-fibered Calabi-Yau three-folds thus far was only presented for G=SU(5). It is of importance in particle physics phenomenology to extend these results to other structure groups, notably SO(10) and E6. This is a very non-trivial extension which will require new technology in bundle theory and, of necessity, illuminate the structure of such bundles. Construction of these bundles will produce new standard model vacua and, in particlular, allow for a study of important physical processes, such as proton decay.
b) The mathematical theory of small instanton transitions has, thus far, been limited to the collision of a bulk five-brane with a visible brane with a grand unified gauge group; that is, to bundles over elliptic three-folds. However, it is clearly important to extend this theory to transitions involving standard model bundles; that is, bundles over torus fibered Calabi-Yau spaces. This is a significant extension of the transition theory mathematics and introduces non-perturbative phase transitions into the standard model.
c) An important aspect of brane universes is the force of attraction between various branes. In previous work, part of the Penn Math/Physics collaboration computed the non-perturbative superpotentials for the separation moduli between branes. These superpotentials arise from the exchange of membrane instantons. However, the holomorphic vector bundles on the visible (and hidden) branes also have moduli, and it is of physical relevence to compute their superpotentials as well. We propose to compute these superpotentials at least for the bundle moduli associated with small instanton transitions and, hopefully, for all bundle moduli.
The Math/Physics of Ekpyrotic Cosmology
a) We are studying, in detail, small instanton transitions resulting in standard model vector bundles. Of particular importance will be the elucidation, exactly at the point of transition, of the singular torsion free sheaf and the massless spectrum of this sheaf. This sheaf and spectrum control the transfer of data, such as scale-invariant fluctuations, from the bulk brane to the visible brane and, hence, a detailed understanding of their mathematics and physics is essential. In addition to doing this in the physically realistic case of the strongly coupled heterotic string, we will study the collision of D-branes in weakly coupled superstring theories. These systems are mathematically more tractable and should give considerable insight into, and explicit examples of, small instanton phase transitions.
b) As the two branes approach each other prior to collision, the metric of the low energy effective theory is contracting, albeit very slowly. After collision, however, the effective metric describes a subluminally expanding Friedman-Robinson-Walker universe. The reversal from contraction to expansion takes place precisely at the moment of collision. Although previous work has demonstrated the plausibility of this reversal, it is essential to prove its existence, both in the case of a torsion free sheaf transition or in the case of a "bounce" of branes described by a weakly coupled heterotic superstring. We propose to study the general relativity and string theory of this reversal in detail. We conjecture that this is somewhat like a "flop" or conical transition, whose divergent geometric aspects are "softened" by M-theory or string theory.
In addition to these fundamental aspects of the Ekpyrotic scenario, another important feature involves the transfer of energy-momentum from the bulk brane to the G-bundle moduli.
c) Given the bundle moduli superpotential discussed previously, we are studying the associated bundle Kahler potential and potential energy and to use these results, as well as the potential for the separation moduli of the branes, to discuss the dynamics of the brane collision in detail. In fact, these results will have a wider applicability than Ekpyrotic cosmology. Knowledge of both the superpotential and Kahler potential of the vector bundle moduli will give us a dynamical theory of G-instantons, allowing study of their dynamical stability and phase transitions.
We are pursuing the following research in these topics.
a) The structure of conical singularities on G2 manifolds has recently been studied from a purely mathematical point of view. In addition, these singularities have been shown to possess Abelian anomalies which are cancelled in M-theory by the appearance of localized chiral fermions. Further work, in the neighborhood of A-D-E spaces fibered over a real three-dimensional base Q, has revealed that gauge anomalies can occur at isolated points where the fibers become more singular than A-D-E. The consistency of M-theory then requires that chiral fermions transforming non-trivially under the A-D-E groups must exist at each such singularity. There are two obvious issues here that we want to study. First, we will consider the mathematical conditions under which both conical and A-D-E singularities can coalesce. We will then study the anomaly structure of these overlapping singularities, thus constructing new theories of chiral fermions in four-dimensions. Secondly, if there are several separated A-D-E singularities, then there will generically exist lower dimensional spaces where the individual singular planes overlap. These spaces will be fibered by overlapping A-D-E fibers. We would like to study the gauge groups of these overlapping fibrations, their anomalies and the associated four-dimensional theories of chiral fermions. Within these new classes of chiral vacua, we will search for those describing realistic grand unified and standard model theories of particle physics.
b) It is of considerable interest to understand how the strongly coupled heterotic string, associated with M-theory compactification on a Calabi-Yau three-fold times an interval, can arise as a special point in the moduli space of G2 manifolds with conical and fiber singularities. This might arise from the existence of two conical singularies in the base Q or from other base or fiber singularities. However it occurs, this relationship will shed much light on the relationship of generic M-theory G2 vacua and the heterotic string as well as on the associated four-dimensional chiral theories.
c) Another question is whether it is possible to wrap five-branes on certain six-cycles in G2 manifolds, preserving N=1 supersymmetry. These six-cycles have to be singular and would lead to a non-trivial superpotential. This superpotential has been shown to be crucial to understanding the relation among G2 seven-manifolds and Calabi-Yau three-folds and four-folds. We are currently studying, in detail, the examples of G2 holonomy manifolds constructed by Joyce which have a fibration over the interval. The fibers over the boundary points of the interval are six-cycles obtained by "partially resolving" the quotients of the Calabi-Yau three-fold by an anti-holomorphic involution. Another question is whether these models are related by a "birational transformation" to seven-dimensional varieties with smaller holonomy groups, which are a suitable quotient of the product of a Calabi-Yau manifold times a circle.
d) Nonperturbative properties of QFT often show up classically or semiclassically in string and M theory. Especially important is supersymmetry-breaking physics, which is not well understood in string theory, and which is an essential part of any realistic string model. This can only be studied in N=1 models with chiral fermions, such as arise in G2 manifolds. The mathematical techniques for this study have not yet been developed. We hope to address these questions.
Five-Branes, Gerbes and K-Theory
There are a number of very fundamental issues in this context that we are studying in detail.
a) When N D-branes coalesce, the U(1) gauge groups on each brane combine to form a non-Abelian U(N) group on the worldvolume. By analogy, one would expect that when N M-five-branes coincide, the Abelian two-forms on each brane would combine to form a "non-Abelian two-form" on the worldvolume. However, neither the local definition nor the dynamics of such a worldvolume field is understood. We will study this mathematical structure with the aim of defining this important worldvolume field locally, presenting its consistent dynamics and studying its global topological structure. Success here would have important impact on our understanding of M-theory.
b) The mathematical and physical structure of K-brane, anti-D-brane pairs
has recently been a topic of intense study. The open strings quantizing
such solitons have a "tachyon" or instability, which is understood
via the K-theory associated with the relative vector bundles of the two
branes. There is an obvious physical analog in M-theory; that is, the
structure of M-five-brane, anti-M-five-brane pairs. Progress here depends
on understanding the relative gerbe "bundles" for the five-brane
pair and the associated extended K-theory. Understanding the mathematics
of this type of K-theory would allow one to study the stability of M-brane
pairs and, after compactification, stability of NS 5-brane pairs of IIA
string theory. The latter system involves a closed string tachyon, an
instability of a kind that also afflicts the bosonic string theory in
26 dimensions. It has been a longstanding goal of string theory to understand
the dynamics of such instabilities. In addition, the study of M5-brane
pair stability has applications to supersymmetry breaking in semi-realistic
standard model vacua.