Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

Geometry of Matrix Product States: metric, parallel transport and curvature

Abstract: We study the geometric properties of the manifold of states described as (uniform) matrix product states. Due to the parameter redundancy in the matrix product state representation, matrix product states have the mathematical structure of a (principal) fiber bundle. The total space or bundle space corresponds to the parameter space, i.e. the space of tensors associated to every physical site. The base manifold is embedded in Hilbert space and can be given the structure of a K\"ahler manifold by inducing the Hilbert space metric. Our main interest is in the states living in the tangent space to the base manifold, which have recently been shown to be interesting in relation to time dependence and elementary excitations. By lifting these tangent vectors to the (tangent space) of the bundle space using a well-chosen prescription (a principal bundle connection), we can define and efficiently compute an inverse metric, and introduce differential geometric concepts such as parallel transport (related to the Levi-Civita connection) and the Riemann curvature tensor.

The restricted tangent bundle of a rational curve in P2

Abstract: (1988). The restricted tangent bundle of a rational curve in P2. Communications in Algebra: Vol. 16, No. 11, pp. 2193-2208.

Riemann-Roch variations

Abstract: We construct a mirror-type correspondence that assigns variations (that is, local systems, -modules or -adic sheaves) to pairs , where is a variety and is a complex of densely filtered vector bundles over . We consider Calabi-Yau complete intersections in projective spaces. In the particular case when the complex is quasi-isomorphic to the tangent bundle on a generic Calabi-Yau complete intersection, this construction yields the variation that arises in the relative cohomology of the mirror-dual pencil. We call it the Riemann-Roch variation. The Riemann-Roch data of the divisorial sublattice in the -group can be read off the Riemann-Roch local system since it encodes the information about the Euler characteristics of all sheaves (in an essentially non-commutative way).

Gromov-Witten invariants of jumping curves

Abstract: Given a vector bundle E on a smooth projective variety X, we can define subschemes of the Kontsevich moduli space of genus-zero stable maps M 0,0 (X,β) parameterizing maps f: P 1 → X such that the Grothendieck decomposition of f*E has a specified splitting type. In this paper, using a "compactification" of this locus, we define Gromov-Witten invariants of jumping curves associated to the bundle E. We compute these invariants for the tautological bundle of Grassmannians and the Horrocks-Mumford bundle on P 4 . Our construction is a generalization of jumping lines for vector bundles on P". Since for the tautological bundle of the Grassmannians the invariants are enumerative, we resolve the classical problem of computing the characteristic numbers of unbalanced scrolls.

Stiefel—Whitney currents

Abstract: A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v1,…,vm, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ2-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle.