The flow of homogeneous fluids through porous media: analogies with other physical problems

http://cds.cern.ch/record/994225/files/0610327.pdf

Four-dimensional string compactifications with D-branes, orientifolds and fluxes

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

/pdf/the-topological-vertex-3lrad1kwkt.pdf

The Topological Vertex

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

/pdf/moduli-spaces-of-local-systems-and-higher-teichmuller-theory-36h32nsutb.pdf

Moduli spaces of local systems and higher Teichmüller theory

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.

https://iopscience.iop.org/article/10.1088/1126-6708/2003/05/046/pdf

The statistics of string/M theory vacua

We consider Landau-Ginzburg (LG) models with boundary conditions pre- serving A-type N = 2 supersymmetry. We show the equivalence of a linear class of boundary conditions in the LG model to a particular class of boundary states in the cor- responding CFT by an explicit computation of the open-string Witten index in the LG model. We extend the linear class of boundary conditions to general non-linear bound- ary conditions and determine their consistency with A-type N = 2 supersymmetry. This enables us to provide a microscopic description of special Lagrangian submani- folds in C n due to Harvey and Lawson. We generalise this construction to the case of hypersurfaces in P n . We nd that the boundary conditions must necessarily have vanishing Poisson bracket with the combination (W () W ()), where W ( )i s the appropriate superpotential for the hypersurface. An interesting application considered is the T 3 supersymmetric cycle of the quintic in the large complex structure limit.

http://cds.cern.ch/record/432444/files/0003242.pdf

On the Landau-Ginzburg description of boundary CFTs and special Lagrangian submanifolds

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.

http://cds.cern.ch/record/543703/files/0203173.pdf

D-branes on Calabi{Yau Manifolds and Superpotentials

We show that the generating function of electrically charged $$ \frac{1}{2} $$
 -BPS states in $$ \mathcal{N} = 4 $$
 supersymmetric CHL $$ {\mathbb{Z}_N} $$
 -orbifolds of the heterotic string on T
 6 are given by multiplicative η-products. The η-products are determined by the cycle shape of the corresponding symplectic involution in the dual type II picture. This enables us to complete the construction of the genus-two Siegel modular forms due to David, Jatkar and Sen [arXiv:hep-th/0609109] for $$ {\mathbb{Z}_N} $$
 -orbifolds when N is non-prime. We study the $$ {\mathbb{Z}_4} $$
 CHL orbifold in detail and show that the associated Siegel modular forms, $$ {\Phi_3}\left( \mathbb{Z} \right) $$
 and $$ {\widetilde\Phi_3}\left( \mathbb{Z} \right) $$
 , are given by the square of the product of three even genus-two theta constants. Extending work by us as well as Cheng and Dabholkar, we show that the ‘square roots’ of the two Siegel modular forms appear as the denominator formulae of two distinct Borcherds-Kac-Moody (BKM) Lie superalgebras. The BKM Lie superalgebra associated with the generating function of $$ \frac{1}{4} $$
 -BPS states, i.e., $$ {\widetilde\Phi_3}\left( \mathbb{Z} \right) $$
 has a parabolic root system with a lightlike Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the $$ \frac{1}{4} $$
 -BPS states.

BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N

Boundary Fermions, Coherent Sheaves and D-branes on Calabi-Yau manifolds

Following Sen, we study the counting of (‘twisted’) BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with $ \mathcal{N} = 4 $
 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M
 24, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to a periodic table of BKM Lie superalgebras with properties that are consistent with physical expectations.

Suresh Govindarajan

Papers

On the Landau-Ginzburg description of boundary CFTs and special Lagrangian submanifolds

D-branes on Calabi{Yau Manifolds and Superpotentials

BKM Lie superalgebras from dyon spectra in Z N CHL orbifolds for composite N

Boundary Fermions, Coherent Sheaves and D-branes on Calabi-Yau manifolds

BKM Lie superalgebras from counting twisted CHL dyons