We show that the generating function of electrically charged 1 2 -BPS states in N = 4 supersymmetric CHL 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 Z N -orbifolds when N is non-prime. We study the Z 4 -CHL orbifold in detail and show that the associated Siegel modular forms, Φ 3 (Z) and Φ 3 (Z), 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 1 4 -BPS states, i.e., Φ 3 (Z) has a parabolic root system with a light-like Weyl vector and the walls of its fundamental Weyl chamber are mapped to the walls of marginal stability of the 1 4 -BPS states. * suresh@physics.iitm.ac.in † gkrishna@imsc.res.in

We consider D-branes wrapped around supersymmetric cycles of Calabi-Yau manifolds from the viewpoint of N=2 Landau-Ginzburg models with boundary as well as by consideration of boundary states in the corresponding Gepner models. The Landau-Ginzburg approach enables us to provide a target space interpretation for the boundary states. The boundary states are obtained by applying Cardy's procedure to combinations of characters in the Gepner models which are invariant under spectral flow. We are able to relate the two descriptions using the common discrete symmetries of the two descriptions. We are thus able to provide an extension to the boundary of the bulk correspondence between Landau-Ginzburg orbifolds and the corresponding Gepner models.Comment: 28 pages, LaTeX with revtex; (v2) Condition involving superpotential in the boundary LG model imposed, references included ; (v3) final version to appear in journa

We use the framework of matrix factorizations to study topological B-type Dbranes on the cubic curve. Specifically, we elucidate how the brane RR charges are encoded in the matrix factors, by analyzing their structure in terms of sections of vector bundles in conjunction with equivariant R-symmetry. One particular advantage of matrix factorizations is that explicit moduli dependence is built in, thus giving us full control over the open-string moduli space. It allows one to study phenomena like discontinuous jumps of the cohomology over the moduli space, as well as formation of bound states at threshold. One interesting aspect is that certain gauge symmetries inherent to the matrix formulation lead to a non-trivial global structure of the moduli space. We also investigate topological tachyon condensation, which enables us to construct, in a systematic fashion, higher-dimensional matrix factorizations out of smaller ones; this amounts to obtaining branes with higher RR charges as composites of ones with minimal charges. As an application, we explicitly construct all rank two matrix factorizations.December, 2005 † On leave from the

Following Sen, we study the counting of ('twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with 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.

We study both A-type and B-type D-branes in the gauged linear sigma model by considering worldsheets with boundary. The boundary conditions on the matter and vector multiplet fields are first considered in the largevolume phase/non-linear sigma model limit of the corresponding CalabiYau manifold, where we also find that we need to add a contact term on the boundary. These considerations enable to us to derive the boundary conditions in the full gauged linear sigma model, including the addition of the appropriate boundary contact terms, such that these boundary conditions have the correct non-linear sigma model limit. Most of our results are derived for the quintic Calabi-Yau manifold, though we comment on possible generalisations.

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.

We present a method based on mutations of helices which leads to the construction (in the large volume limit) of exceptional coherent sheaves associated with the ( a l a = 0) orbits in Gepner models. This is explicitly verified for a few examples including some cases where the ambient weighted projective space has singularities not inherited by the CalabiYau hypersurface. The method is based on two conjectures which lead to the analog,in the general case, of the Beilinson quiver for P n . We discuss how one recovers the McKay quiver using the gauged linear sigma model (GLSM) near the orbifold or Gepner point in Kähler moduli space.

We provide evidence for the existence of a family of generalized Kac-Moody(GKM) superalgebras, G N , whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N , ∆ k/2 (Z), for (N, k) = (1, 10), (2, 6), (3, 4), and possibly (5, 2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric Z N -orbifolds of the heterotic string compactified on T 6 . The new generalized Kac-Moody superalgebras all arise as different 'automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, ∆ k/2 (Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, G 1 leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the 'algebra of BPS states' in CHL compactifications. * suresh@physics.iitm.ac.in † gkrishna@imsc.res.in 12 2 also happens to be the size of a 1/4 BPS multiplet of states.