The order polytope O(P ) and the chain polytope C(P ) associated to a partially ordered set P are studied. In this paper, we introduce the convex polytope Γ(O(P ),−C(Q)) which is the convex hull of O(P ) ∪ (−C(Q)), where both P and Q are partially ordered sets with |P | = |Q| = d. It will be shown that Γ(O(P ),−C(Q)) is a normal and Gorenstein Fano polytope by using the theory of reverse lexico...

Non-critical string field theory for 2d quantum gravity coupled to (p,q)–conformal fields Abstract We propose a non-critical string field theory for 2d quantum gravity coupled to (p,q) conformal fields. The Hamiltonian is described by the generators of the W p algebra, and the Schwinger-Dyson equation is equivalent to a vacuum condition imposed on the generators of W p algebra.

Recently an infinite family of explicit Sasaki–Einstein metrics Y p,q on S2 × S3 has been discovered, where p and q are two coprime positive integers, with q < p. These give rise to a corresponding family of Calabi–Yau cones, which moreover are toric. Aided by several recent results in toric geometry, we show that these are Kähler quotients C4//U(1), namely the vacua of gauged linear sigma mode...

We study the simplest examples of minimal string theory whose worldsheet description is the unitary (p, q) minimal model coupled to twodimensional gravity (Liouville field theory). In the Liouville sector, we show that four-point correlation functions of ‘tachyons’ exhibit logarithmic singularities, and that the theory turns out to be logarithmic. The relation with Zamolodchikov’s logarithmic d...

We study the simplest examples of minimal string theory, which are the unitary (p, q) minimal models coupled to two-dimensional gravity (Liouville field theory). In the Liouville sector, we show that four-point correlation functions of ‘tachyons’ exhibit logarithmic singularities, and that the theory turns out to be logarithmic. The relation with Zamolodchikov’s logarithmic degenerate fields is...

Let $(M,g)$ be a pseudo-Riemannian manifold of signature $(p,q)$. We compute the obstruction for vector bundle $S$ over to admit Dirac operator whose principal symbol induces on structure irreducible real Clifford modules complex type, that is, spinor type. In order do this, we use theory Lipschitz structures in $p-q\equiv_8 3,7$ reformulate problem as $\mathrm{Spin}^{o}_{\alpha}$ with $\alpha ...