| T. Gannon | University of Alberta |
| A. Klemm | Humboldt University, Berlin |
| M. Mariño | CERN |
| E. Sharpe | University of Illinois |
| E. Aldrovandi | Florida State University |
| J. Bertin | Université Joseph Fourier |
| V. Chernov | Dartmouth College |
| P. Grange | CPHT, Ecole polytechnique |
| V. Marenitch | The University of Kalmar |
| A. Mikovic | Universidade Lusófona |
| A. Misra | Institut fuer Physik, Humboldt Universitaet |
| G. Mondello | Scuola Normale Superiore, Pisa |
| K.O. Ozansoy | Ankara University Faculty of Science |
| P. Pinto | Instituto Superior Técnico |
| E. Scheidegger | Institut fuer theoretische Physik der TU Wien |
| M. Tierz | Institut d'Estudis Espacials de Catalunya |
| S. Vacaru | CENTRA, Instituto
Superior Técnico,
Academy of Sciences (Republic of Moldova), ISS (Romania) |
| R. Vila Freyer | CIMAT |
Boundary Conformal Field Theory, Fusion Ring Representations and Strings
Boundary Conformal Field Theory, Fusion Ring Representations and Strings.
We'll begin with a gentle overview, discussing some of the background
context. This will include some words on Monstrous Moonshine, modular
functions, (boundary) conformal field theory, and string theory. Then
we'll
turn to fusion rings and the basic partition functions of the theory,
where
much of the data of the theory is conveniently encoded. We'll focus on
the partition function of the torus (relevant to closed string theory,
or the CFT of the bulk) and that of the cylinder (which is relevant to
open string theory and boundary CFT).
References:
Review and Lecture Notes:
Other References:
Although not strictly necessary, it would be convenient for the audience
to
have some familiarity with the elementary representation theory of affine
Kac-Moody algebras, and also the elementary theory of modular forms. These
are
reviewed for instance in the textbooks
More specific background for my lectures is provided for instance by:
Mirror Symmetry and the Topological String
We review the topological calculations in supersymmetric field theories in 0,1 and 2 dimensions, with special emphasis on the non-linear s model case. Then we define the A-model on general symplectic manifolds and the B-model on Calabi-Yau manifolds. We describe selected results in mirror symmetry on CY manifolds, such as the mirror symmetry predictions for the Gromow-Witten invariants and the definition of integer invariants for closed an open topological strings in (non-compact) Calabi-Yau backgrounds (with branes). Finally we describe aspects of the large N relation between topological gauge systems and the topological string and review what these theories calculate in the topological sector of 4d supersymmetric theories.
Chern-Simons Theory and Enumerative Geometry
The goal of these lectures is to give an introduction to the relations between enumerative geometry of non-compact, toric Calabi-Yau manifolds, and Chern-Simons theory. I will start with an introduction to the relevant aspects of Chern-Simons theory and its large N expansion. Then, I will explain the realization of Chern-Simons theory on the three-sphere in terms of closed and open topological strings, as well as the important idea of large N or geometric transition. Finally, I will extend this framework in order to compute topological string amplitudes of general toric manifolds using Chern-Simons ingredients.
References
Lecture Notes:
Other References:
D-branes and Sheaves
In these talks I will describe mathematical models of D-branes as sheaves and, more generally, derived categories. I will begin with a gentle introduction to some relevant mathematics (sheaves, Ext groups). After that, I will discuss mathematical models of D-branes on large-radius Calabi-Yau manifolds as sheaves, describing how sheaves can be used to calculate open string spectra, and how such sheaf-theoretic models can be extended in various directions to take into account B field backgrounds, orbifold structures, and nontrivial Higgs vevs. Finally, I will give a short introduction to derived categories, their physical realization as boundary states in the B model topological field theory, and stability issues.
Lecture Notes:
Basic references on sheaves & sheaf cohomology:
Basic references on Ext groups of sheaves:
Basic reference on the B model topological field theory:
Hermitian-holomorphic classes and tame symbols related to uniformization, the dilogarithm, and the Liouville Action
One possible approach to the uniformization of compact Riemann surfaces of genus greater than on is to look at metrics of constant negative curvature. Such metrics can be characterized as critical points of the classical Liouville action functional. We present an algebraic construction of this functional as the square of the metrized holomorphic tangent bundle in a suitably defined hermitian-holomorphic Deligne cohomology group. For a pair of line bundles on the Riemann surface, this construction generalizes Deligne's tame symbol and recovers the algebraic approach to the determinant of cohomology as pursued by Deligne, Brylinski, Gabber and others. We will also interpret the above results in terms of group cohomology for Kleinian groups and volume calculations in hyperbolic 3-space involving certain secondary classes, notably the Chern-Simons one. We will outline how this framework relates to recent results in Mathematics and Physics where a compact Riemann Surface is considered as the "hologram" of an associated hyperbolic 3-manifold.
References:
Coverings of algebraic curves, stacks, and Gromov Witten invariants
Study of Hurwitz stacks parametrizing branched covers of algebraic curves, and some aspects to Gromov Witten invariants. Study of Hurwitz stacks parametrizing branched covers of algebraic curves, and some aspects to Gromov Witten invariants.
Affine linking and winding numbers and the study of front propagation
Let M be an oriented n-dimensional manifold. We study the causal relations between the wave fronts W1 and W2 that originated at some points of M. We introduce a numerical topological invariant CR(W1 , W2) (the so-called causality relation invariant) that, in particular, gives the algebraic number of times the wave front W1 passed through the point that was the source of W2 before the front W2 originated. This invariant can be easily calculated from the current picture of wave fronts on M without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of M outside of a part bar-M of M such that W1 and W2 are null-homotopic in bar-M. We also construct the Affine winding number invariant win which is the generalization of the winding number to the case of nonzero-homologous shapes and manifolds other than R2. The win invariant gives the algebraic number of times the wave front has passed through a given point between two different time moments without the knowledge of the wave front propagation law. The invariants described above are particular cases of the general affine linking invariant al of nonzero homologous submanifolds N1 and N2 in M introduced by us. To construct al we introduce a new pairing on the bordism groups of space of mappings of N1 and N2 into M. For the case N1=N2=S1 this pairing can be regarded as an analog of the string-homology pairing constructed by Chas and Sullivan, and it is a generalization of the Goldman Lie bracket.
Star-products and open-string actions beyond the large-B limit
Duality between commutative and non-commutative gauge fields is encoded by the Seiberg-Witten equations [1], which have been solved in the U(1) case [2,3,4,5]. The solution leads to predictions for derivative corrections to the gauge sector of open-string effective actions, in presence of a large background magnetic field [6]. These predictions have been checked by explicit string-computations [7,8]. Going beyond the limit of a large background magnetic field involves taking the open-string metric into account, thus deforming the corrections [9,10]. These deformations can be translated into the language of non-commutative Yang-Mills theory.
References:
Note on geometry of twistor spaces
We consider a twistor space Z of an almost Kahler manifold M as a total space of a Riemannian submersion. Showing that it is itself an almost Kahler manifold we study branched minimal immersions of a closed surface into manifold M and their twistor lifts to Z. We establish some geometric properties of the restriction of a twistor bundle over a surface and obtain some inequalities for topological invariants of this bundle.
Spin Foam Models of String Theory
We propose a new approach to formulating string theory based on the connection between the representations of Kac-Moody algebras and the representations of quantum groups (for a good review see [BK]). The string scattering amplitudes are identified as certain spin network amplitudes in a two-dimensional spin foam state sum model associated to the isometry group of the background spacetime in which the string is propagating [M1,M2,M3]. The case of SU(2) group manifold spacetime is discussed in some detail.
References:
("Barely") G2 Manifolds, (Orientifolds of) a compact Calabi-Yau and nonperturbative N=1 Superpotentials
We discuss (a) membrane instanton superpotential in M-theory on G2 manifolds including non-rigid supersymmetric 3-cycles, (b) an N=1 triality involving Heterotic/M/F theories at the level of spectrum matching, (c) the Picard-Fuchs equation, its solution and monodromy properties associated with period integrals associated with the compact CY3(3,243) that becomes relevant when discussing the abovementioned triality, (d) null superpotential associated in particular with type IIA (that gets uplifted to a suitable M-theory background in the above triality), on a freely acting orientifold of CY3(243), away from the orbifold singularities associated with the (singular) Fermat hypersurface putting the aforementioned triality on a more firm footing, and finally (e) making a connection between the abovementioned topics and Witten's MQCD.
References:
Combinatorial classes on the moduli space of curves are tautological
Strebel's theorem of quadratic differentials gives an isomorphism between the moduli space of curves (over the complex numbers) and an orbi-simplicial complex A, whose simplices are indexed by some ribbon graphs. This description (due to Mumford) was used first by Harer to compute the virtual cohomological dimension of the mapping class group [1] and (with Zagier) the orbifold Euler characteristic of the moduli space of curves [2]; and then by Kontsevich [3] to prove Witten's conjecture (the generating series of intersection numbers of psi-classes on the moduli space of curves satisfy the KdV hierarchy). In this paper [3] Kontsevich proved that some interesting simplicial subcomplexes Wm of A (supported on simplices indexed by ribbon graphs with prescribed valencies m of their vertices) are in fact cycles and so define homology classes with noncompact support (and dually cohomology classes). Witten and Kontsevich conjectured that these W-classes are polynomials in the tautological classes (i.e. algebro-geometrically defined in terms of push-forward through the universal curve of powers of Chern classes of the relative cotangent bundle). Partial results were achived by Penner [4], Wolpert [5] and Arbarello-Cornalba [6] by different methods. In our preprint [7] we prove the conjecture and we exhibit a recursive formula to compute all the polynomials. To do so we use Kontsevich's explicit representatives of the psi-classes in the simplicial setting and the explicit description of the simplicial complex A and of a modification of A due to Looijenga [8].
References:
Geometric formulation of quantum mechanics
In this work, quantum mechanics has been reformulated in the geometric language of classical mechanics i.e. symplectic mechanics. In a quantum mechanical system, the states are represented by rays in a complex Hilbert space. The space constructed by this rays has a structure of a space. This provides a new geometric formula tion for quantum mechanics that is physically equivalent to standart algebraic framework but different as a mathematical appearance. In this formulation, the states of a quantum mechanical system are given by the points of a symplectic space as classical mechanics but this time, there is also a compatible Riemannian metric with the symplectic structure. The observables are represented by real-valued functions and the Scrödinger evolution is associated to symplectic flow generated by Hamiltonian function. This reformulation makes clear the main similarities and differences between quantum mechanics and classical mechanics.
References:
Fusion of sufferable modular invariant partition functions
We develop further fusion rule structure of modular invariants that can be realised by subfactors N \subset M started in [1]. This structure is a useful tool in the analysis of modular data from quantum doubles.
References:
D4-branes on toric Calabi-Yau hypersurfaces
We give an extensive description of rational D4-branes in toric Calabi-Yau hypersurfaces. We compute the dimension of their moduli space in geometry and in conformal field theory. The result yields compelling evidence for the decoupling conjecture stating that this dimension is independent of the Kaehler structure of the Calabi-Yau space. We thereby provide a large number examples of D-branes on algebraic 4-manifolds. To appear.
Soft Matrix Models and Chern-Simons partition functions
We present the properties of matrix models with very weak confining potentials [2]. These models are solved with q deformed orthogonal polynomials, and we show how they give rise to central quantities in Chern-Simons theory, such as the partition function of U(N) Chern-Simons theory on S3 [1,2]. We show general features of the density of states, correlation functions and loop averages of the models [2,4]. In particular, it is demonstrated how the partition function can be given by an infinite number of matrix models whose weight function includes a q-periodic function [2,4]. Finally, the special role played by the compactifications that give rise to these models [1,3] is explained in detail [4].
References:
Nonlinear Connection Geometry and Exact Solutions in String Gravity
We have developed a new method of integrating the Einstein equations in various models of gravity and string/brane theory for off-diagonal metric ansatz depending on three/four coordinates. The approach was derived from the geometry of anholonomic frames with associated nonlinear connection structures, in general, related anholonomic Riemannian-Finsler-Lagrange-Hamilton geometry and works also in general relativity. We analyze the spacetime geometries for different classes of exact solutions describing black holes with ellipsoid/toroidal symmetries, three/two dimensional spinor and solitonic configurations propagating and deforming black hole backgrounds in three, four and five dimensional spacetimes, various locally anisotoropic wormhole/flux tube and/or cosmological solutions with deformed symmetries, dilatons, monopoles, instantons and anisotropic Taub NUT spaces.
References
Asymptotic Holonomy
A flow of dimension one on a compact manifold, together with an invariant measure, gives rise to the Asymprotic cycle of Shwarztman, studied also by V. I. Arnold in a context in hydrodynamics in the 3 dimensional Euclidean space, to describe the average linking number. We will descibe that the computation of the asymptotic average holonomy through it, that gives rise to the Average "Wilson line" in the context of Witten's Chern Simons Theory. (See our paper in Comm. Math. Phys. 1994, No. 163, Pags. 73-88, where we constructed its trace). A similar construction for the orbit of Brownian motion on a Riemannian manifold will be described, it gives a random variable in the holonomy of the manifold.