We propose a new basis in Witten’s open string field theory, in which the star product simplifies considerably. For a convenient choice of gauge, the classical string field equation of motion yields straightforwardly an exact analytic solution that represents the nonperturbative tachyon vacuum. The solution is given in terms of Bernoulli numbers and the equation of motion can be viewed as novel Euler–Ramanujan-type identity. It turns out that the solution is the Euler–Maclaurin asymptotic expansion of a sum over wedge states with certain insertions. This new form is fully regular from the point of view of level truncation. By computing the energy difference between the perturbative and nonperturbative vacua, we prove analytically Sen’s first conjecture.

Stable, holomorphic vector bundles are constructed on a torus fibered, non-simply connected Calabi–Yau three-fold using the method of bundle extensions. Since the manifold is multiply connected, we work with equivariant bundles on the elliptically fibered covering space. The cohomology groups of the vector bundle, which yield the low energy spectrum, are computed using the Leray spectral sequence and fit the requirements of particle phenomenology. The physical properties of these vacua were discussed previously. In this paper, we systematically compute all relevant cohomology groups and explicitly prove the existence of the necessary vector bundle extensions. All mathematical details are explained in a pedagogical way, providing the technical framework for constructing heterotic standard model vacua.

We compute Seidel’s mirror map for abelian varieties by constructing the homogeneous coordinate rings from the Fukaya category of the symplectic mirrors. The computations are feasible, as only linear holomorphic disks contribute to the Fukaya composition in the case of the planar Lagrangians used. The map depends on a symplectomorphism ρ representing the large complex structure monodromy. For the example of the two-torus, different families of elliptic curves are obtained by using different ρ’s which are linear in the universal cover. In the case where ρ is merely affine linear in the universal cover, the commutative elliptic curve mirror is embedded in noncommutative projective space. The case of Kummer surfaces is also considered.