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Register a new userFlat Connections in Open String Mirror Symmetry
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We study a flat connection defined on the open-closed deformation space of open string mirror symmetry for type II compactifications on Calabi-Yau threefolds with D-branes. We use flatness and integrability conditions to define distinguished flat coordinates and the superpotential function at an arbitrary point in the open-closed deformation space. Integrability conditions are given for concrete deformation spaces with several closed and open string deformations. We study explicit examples for expansions around different limit points, including orbifold Gromov-Witten invariants, and brane configurations with several brane moduli. In particular, the latter case covers stacks of parallel branes with non-Abelian symmetry.
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We describe supersymmetric A-branes and B-branes in open N=(2,2) dynamically gauged nonlinear sigma models (GNLSM), placing emphasis on toric manifold target spaces. For a subset of toric manifolds, these equivariant branes have a mirror description as branes in gauged Landau-Ginzburg models with neutral matter. We then study correlation functions in the topological A-twisted version of the GNLSM, and identify their values with open Hamiltonian Gromov-Witten invariants. Supersymmetry breaking can occur in the A-twisted GNLSM due to nonperturbative open symplectic vortices, and we canonically BRST quantize the mirror theory to analyze this phenomenon.
We analyze the locus, together with multiplicities, of bad conformal field theories in the compactified moduli space of N=(2,2) superconformal field theories in the context of the generalization of the Batyrev mirror construction using the gauged linear sigma-model. We find this discriminant of singular theories is described beautifully by the GKZ A-determinant but only if we use a noncompact toric Calabi-Yau variety on the A-model side and logarithmic coordinates on the B-model side. The two are related by local mirror symmetry. The corresponding statement for the compact case requires changing multiplicities in the GKZ determinant. We then describe a natural structure for monodromies around components of this discriminant in terms of spherical functors. This can be considered a categorification of the GKZ A-determinant. Each component of the discriminant is naturally associated with a category of massless D-branes.
Using string scattering amplitudes of open bosonic string on a single $D$-brane, we construct a local field theoretical action for tachyon fields. Cubic local interactions between various particles, belonging to the particle spectrum of string may be directly followed from three-string scattering amplitude. These cubic local interactions may generate perturbative non-local four-particle interactions, which may contribute to four-string scattering amplitude. It was observed that tachyon field in open bosonic string theory must be represented by a complex field in order to reproduce the Veneziano amplitude, describing four-tachyon scattering. The Veneziano amplitude, expanded in terms of $s$-channel poles was compared with the four-tachyon scattering amplitudes in $s$-channel generated perturbatively and it was found that a quartic potential term is needed in the local field theoretical action, which describes open string theory effectively in the low energy regime. With this quartic term, the tachyon potential has a stable minimum point and the tachyon field may condensate. As a result, both tachyon and gauge fields become massive at Planck scale and completely disappear from the low energy particle spectrum.
Localization methods have produced explicit expressions for the sphere partition functions of (2,2) superconformal field theories. The mirror symmetry conjecture predicts an IR duality between pairs of Abelian gauged linear sigma models, a class of which describe families of Calabi-Yau manifolds realizable as complete intersections in toric varieties. We investigate this prediction for the sphere partition functions and find agreement between that of a model and its mirror up to the scheme-dependent ambiguities inherent in the definitions of these quantities.
The moduli space of flat SU(2) connections on a punctured surface, having prescribed holonomy around the punctures, is a compact smooth manifold if the prescription is generic. This paper gives a direct, elementary proof that the trace of the holonomy around a certain loop determines a Bott-Morse function on the moduli space which is perfect, meaning that the Morse inequalities are equalities. This leads to an attractive recursion for the Betti numbers of the moduli space, which agrees with the Harder-Narasimhan formula in the case of one puncture with holonomy -1.
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