Past String Theory Seminar

Abstract: I will describe how computations are done using "Groebner bases" in quotient rings of polynomial rings, and I will describe explicitly the form of a particular Groebner basis for the ideal defining the ring parametrizing all factorizations of a monic polynomial of degree a+b+...+e into monic factors of degree a,b,...,e. That can be and is used in practice to compute intersection numbers involving of algebraic cycles arising as Chern classes on flag bundles of vector bundles. Simplest example: how many lines in 3-space meet four fixed lines?

Abstract: Different phenomenological scenarios emerging from the exponentially large volume string constructions will be discussed with and without low energy supersymmetry.

Abstract: I will talk about how the reformulation of perturbative Yang-Mills theory in terms of MHV rules accounts for one-loop amplitudes for gluons of the same helicity, and some of the effects of introducing a regulator.

Abstract: Compact $G_2$ manifolds with isolated conical singularities arise naturally in M-theory. I will discuss such manifolds, and explain a method to ``desingularize'' them by glueing in pieces of asymptotically conical $G_2$ manifolds. There are topological obstructions to such desingularizations that depend on the rate of convergence to the cone at the singularities, and on the geometry of the links of the cones. If time permits, I will also briefly discuss a new related project with Dominic Joyce which could provide the first examples of such manifolds, as well as a possible new construction of smooth compact $G_2$ manifolds.

Abstract: We consider deformations of torsion-free $ G_2 $ structures, defined by the $ G_2 $-invariant 3-form $ \phi $ and compute the expansion of the Hodge star of $ \phi $ to fourth order in the deformations of $ \phi $. By considering M-theory compactified on a $ G_2 $ manifold, the $ G_2 $ moduli space is naturally complexified, and we get a Kahler metric on it. Using the expansion of the Hodge star of $ \phi $ we work out the full curvature of this metric and relate it to the Yukawa coupling.

Abstract: We will discuss string $AdS_4$ domain wall solutions with stabilized moduli. These solutions are interesting, since they potentially induce decay processes between different vacua within the string landscape. Moreover, we discuss how black hole physics provide another tool of seeing through the vacuum landscape.

Abstract: We discuss the string-inspired approach to gauge theory amplitudes prompted by the work of Alday and Maldacena, in particular its application to weak coupling.

Abstract: Quantum Hall systems are characterized by a spectacular set of observations (universal low-temperature conductivity, critical behaviour and semi-circle laws for transitions between Quantum Hall states) that are more robust than would be expected from the detailed theory of underlying electron dynamics. The talk starts with a summary of these observations, and their derivation from the assumption that the important charge carriers at the low energies relevant to conductivity measurements are weakly interacting particles or vortices. This implies a large emergent duality symmetry (a level two subgroup of SL(2,Z)), whose presence underlies the robustness of the observations in question. The newly-discovered and unusual Quantum Hall properties of graphene are discussed as providing a new test of this picture.

Abstract: The alternative action for Yang-Mills theory, which Lionel Mason formulated in twistor space, explains some of the simplicities of gluon scattering amplitudes. We will review the derivation of the familiar CSW rules concerning tree-level scattering, show that the `missing' three-point amplitude can be correctly recovered and elucidate the connection with the canonical Lagrangian approach of Mansfied, Morris, et. al.

Abstract: For a large class of compactifications of interest in string phenomenology, the task of finding vacua of the four dimensional effective theories can be rewritten as a simple problem in algebraic geometry. Using recent developments in computer algebra, the task can then be rapidly dealt with in a completely algorithmic fashion. I shall review the main points of hep-th/0606122 and hep-th/0703249 in which this approach to finding vacua was set out, before moving on to a description of the Mathematica package STRINGVACUA (as described in arXiv:0801.1508 [hep-th]). This package uses the power of the computer algebra system Singular and provides a user-friendly implementation of our methods, intended for use by physicists, within the comfortable working environment of Mathematica.

Abstract: I show that the effective action of string compactifications has astructure that can naturally solve the supersymmetric flavour and CP problems. At leading order in the $g_s$ and $\alpha'$ expansions, the hidden sector factorises. The moduli space splits into two mirror parts that depend on K\"ahler and complex structure moduli. Holomorphy implies the flavour structure of the Yukawa couplings arises in only one part. In type IIA string theory flavour arises through the K\"ahler moduli sector and in type IIB flavour arises through the complex structure moduli sector. This factorisation gives a simple solution to the supersymmetric flavour and CP problems: flavour physics is generated in one sector while supersymmetry is broken in the mirror sector. This mechanism does not require the presence of gauge, gaugino or anomaly mediation and is explicitly realised by phenomenological models of IIB flux compactifications.

Abstract: The moduli space of Calabi-Yau manifolds have a natural geometrical structure that has come to be known as special geometry. This geometry will be reviewed in the complex context and it will be shown that much of the structure persists for p-adic Calabi-Yau manifolds.

Abstract: In this talk, I will describe recent work in string phenomenology from the perspective of computational algebraic geometry. I will begin by reviewing some of the long-standing issues in heterotic model building and describe the difficult task of producing realistic particle physics from heterotic string theory. This goal can be approached by creating a large class of heterotic models which can be algorithmically scanned for physical suitability. I will outline a well-defined set of heterotic compactifications over complete intersection Calabi-Yau manifolds using the monad construction of vector bundles. Further, I will describe how a combination of analytic methods and computer algebra can provide efficient techniques for proving stability and calculating particle spectra.

Abstract: The deconfinement transition in gauge theories, in which the Polyakov loop acquires a non-zero expectation value, is described in AdS/CFT as the formation of a black hole in the dual graviational theory. We will explain how to compute the free energy diagram for the Polyakov loop by a constrained gravitational path integral, leading to a new class of black hole solutions.

Abstract: I will give an introduction to, and overview of, the AdS/CFT correspondence from a geometric perspective. As I hope to explain, the correspondence leads to some remarkable relationships between string theory, conformal field theory, algebraic geometry, differential geometry and combinatorics.

Abstract: Twistor-string theory is reformulated as a `half-twisted heterotic' theory with target $CP^3$. This in effect gives a Dolbeault formulation of a theory of holomorphic curves in twistor space and gives a clearer picture of the mathematical structures underlying the theory and how they arise from the original Witten and Berkovits models. It is also explained how space-time physics arises from the model. It intended that the lecture be, to a certain extent, pedagogical.

Abstract: Self-dual connections on ALF spaces can be encoded in terms of bow diagrams, which are natural generalizations of quivers. This provides a convenient description of the moduli spaces of these self-dual connections. We make some comments about the associated twistor data. Via the Nahm transform we construct two explicit examples: a single instanton and a single monopole on a Taub-NUT space.

Abstract: Toric geometry provides powerful and efficient combinatorial tools for the construction and analysis of Calabi-Yau manifolds. After recollections of the hypersurface case I present recent results on new Calabi-Yau 3-folds and their mirrors via conifold transitions, ideas for generalizations to higher codimensions and applications to string theory.

Abstract: Following Donaldson's approach we compute the Calabi-Yau metric on quintics, a four-generation quotient, Schoen threefolds and quotients thereof. Using the explicit Calabi-Yau metric, we then compute eigenvalues and eigenmodes of the Laplace operator.

Abstract: Supersymmetric gauge theories have a spectrum of chiral operators which are preserved under at least 2 supercharges. These operators are sometimes called BPS operators in the chiral ring. The problem of counting operators in the chiral ring is reasonably simple and reveals information about the moduli space of vacua for the supersymmetric gauge theory. In this talk I will review the counting problem and present exact results on the moduli space of both mesonic and baryonic operators for a large class of gauge theories