Basic results in Vacuum Statistics
A review of recent work on constructing and finding statistics of string theory vacua, done in collaboration with Frederik Denef, Bogdan Florea, Bernard Shiffman and Steve Zelditch.
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Michael R. Douglas
Publications and source records attributed to Michael R. Douglas.
Building a Better Racetrack
We find IIb compactifications on Calabi-Yau orientifolds in which all Kahler moduli are stabilized, along lines suggested by Kachru, Kallosh, Linde and Trivedi.
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Statistical analysis of the supersymmetry breaking scale
We discuss the question of what type and scale of supersymmetry breaking might be statistically favored among vacua of string/M theory, building on comments in Denef and Douglas, hep-th/0404116.
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Critical points and supersymmetric vacua
Supersymmetric vacua (`universes') of string/M theory may be identified with certain critical points of a holomorphic section (the `superpotential') of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed. The present paper initiates the study of the statistics of critical points $\nabla s = 0$ of Gaussian random holomorphic sections with respect to a connection $\nabla$. Even the expected number of critical points depends on the curvature of $\nabla$. The principal results give formulas for the expected density and number of critical points of Gaussian random sections relative to $\nabla$ in a variety of settings. The results are particularly concrete for Riemann surfaces. Analogous results on the density of critical points of fixed Morse index are given.
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Distributions of flux vacua
We give results for the distribution and number of flux vacua of various types, supersymmetric and nonsupersymmetric, in IIb string theory compactified on Calabi-Yau manifolds. We compare this with related problems such as counting attractor points.
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Counting Flux Vacua
We develop a technique for computing expected numbers of vacua in Gaussian ensembles of supergravity theories, and apply it to derive an asymptotic formula for the index counting all flux supersymmetric vacua with signs in Calabi-Yau compactification of type IIb string theory, which becomes exact in the limit of a large number of fluxes. This should give a reasonable estimate for actual numbers of vacua in string theory, for CY's with small b_3.
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Statistics of String vacua
We give an introduction to the statistical approach to studying vacua of string/M theory, and discuss recent results of Ashok and Douglas on counting supersymmetric flux vacua in type IIb Calabi-Yau compactification. To appear in the proceedings of the 2003 String Phenomenology workshop in Durham, UK.
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The statistics of string/M theory vacua
We discuss systematic approaches to the classification of string/M theory vacua, and physical questions this might help us resolve. To this end, we initiate the study of ensembles of effective Lagrangians, which can be used to precisely study the predictive power of string theory, and in simple examples can lead to universality results. Using these ideas, we outline an approach to estimating the number of vacua of string/M theory which can realize the Standard Model.
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Chiral Rings and Anomalies in Supersymmetric Gauge Theory
Motivated by recent work of Dijkgraaf and Vafa, we study anomalies and the chiral ring structure in a supersymmetric U(N) gauge theory with an adjoint chiral superfield and an arbitrary superpotential. A certain generalization of the Konishi anomaly leads to an equation which is identical to the loop equation of a bosonic matrix model. This allows us to solve for the expectation values of the chiral operators as functions of a finite number of ``integration constants.'' From this, we can derive the Dijkgraaf-Vafa relation of the effective superpotential to a matrix model. Some of our results are applicable to more general theories. For example, we determine the classical relations and quantum deformations of the chiral ring of $\N=1$ super Yang-Mills theory with SU(N) gauge group, showing, as one consequence, that all supersymmetric vacua of this theory have a nonzero chiral condensate.
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Dirichlet branes, homological mirror symmetry, and stability
We discuss some mathematical conjectures which have come out of the Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in Calabi-Yau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich's homological mirror symmetry conjecture, and "physics proofs" for many of the subsequent conjectures based on it, such as the representation of Calabi-Yau monodromy by autoequivalences of the derived category.
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Seiberg Duality for Quiver Gauge Theories
A popular way to study N=1 supersymmetric gauge theories is to realize them geometrically in string theory, as suspended brane constructions, D-branes wrapping cycles in Calabi-Yau manifolds, orbifolds, and otherwise. Among the applications of this idea are simple derivations and generalizations of Seiberg duality for the theories which can be so realized. We abstract from these arguments the idea that Seiberg duality arises because a configuration of gauge theory can be realized as a bound state of a collection of branes in more than one way, and we show that different brane world-volume theories obtained this way have matching moduli spaces, the primary test of Seiberg duality. Furthermore, we do this by defining ``brane'' and all the other ingredients of such arguments purely algebraically, for a very large class of N=1 quiver supersymmetric gauge theories, making physical intuitions about brane-antibrane systems and tachyon condensation precise using the tools of homological algebra. These techniques allow us to compute the spectrum and superpotential of the dual theory from first principles, and to make contact with geometry and topological string theory when this is appropriate, but in general provide a more abstract notion of ``noncommutative geometry'' which is better suited to these problems. This makes contact with mathematical results in the representation theory of algebras; in this language, Seiberg duality is a tilting equivalence between the derived categories of the quiver algebras of the dual theories.
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Open String Star as a Continuous Moyal Product
We establish that the open string star product in the zero momentum sector can be described as a continuous tensor product of mutually commuting two dimensional Moyal star products. Let the continuous variable $κ\in [~0,\infty)$ parametrize the eigenvalues of the Neumann matrices; then the noncommutativity parameter is given by $θ(κ) =2\tanh(πκ/4)$. For each $κ$, the Moyal coordinates are a linear combination of even position modes, and the Fourier transform of a linear combination of odd position modes. The commuting coordinate at $κ=0$ is identified as the momentum carried by half the string. We discuss the relation to Bars' work, and attempt to write the string field action as a noncommutative field theory.
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D-branes on Calabi-Yau Manifolds and Superpotentials
We show how to compute terms in an expansion of the world-volume superpotential for fairly general D-branes on the quintic Calabi-Yau using linear sigma model techniques, and show in examples that this superpotential captures the geometry and obstruction theory of bundles and sheaves on this Calabi-Yau.
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D-Brane Stability and Monodromy
We review the idea of Pi-stability for B-type D-branes on a Calabi-Yau manifold. It is shown that the octahedral axiom from the theory of derived categories is an essential ingredient in the study of stability. Various examples in the context of the quintic Calabi-Yau threefold are studied and we plot the lines of marginal stability in several cases. We derive the conjecture of Kontsevich, Horja and Morrison for the derived category version of monodromy around a "conifold" point. Finally, we propose an application of these ideas to the study of supersymmetry breaking.
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Noncommutative Field Theory
We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. To appear in Reviews of Modern Physics.
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D-Branes and N=1 Supersymmetry
We discuss the recent proposal that BPS D-branes in Calabi-Yau compactification of type II string theory are Pi-stable objects in the derived category of coherent sheaves.
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D-branes, Categories and N=1 Supersymmetry
We show that boundary conditions in topological open string theory on Calabi-Yau manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the BPS branes at any point in CY moduli space, completing the proposal of Pi-stability.
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D-Branes on Calabi-Yau Manifolds
We give an overview of recent work on Dirichlet branes on Calabi-Yau threefolds which makes contact with Kontsevich's homological mirror symmetry proposal, proposes a new definition of stability which is appropriate in string theory, and provides concrete quiver categories equivalent to certain categories of branes on CY. To appear in the proceedings of the 3rd European Congress of Mathematics.
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