Bundle tanning

A bundle tannin is an object from the algebraic topology that was defined in 1994 by Michael K. Murray. It is a special type of tanner in the general sense, the particular advantage of which is that it allows additional geometric structures - for example a connection . These in turn make bundle tanning with context an object of interest for parts of physics .

The physical interest is based on the correspondence of categorization on the mathematical side and stringification on the physical side (both are not well-defined terms): A gauge theory for point-like particles is described by a Hermitian bundle of lines with a connection. If one goes over to a string theory , then particles are replaced by strings, and Hermitian straight bundles with connection by Hermitian U (1) bundle tanning with connection. In this case, Abelian bundle tanning becomes particularly interesting. But non-Abelian bundle tanning also seem to have applications in the M-theory to find.

Definition: A Hermitian U (1) -bundle tannin over a smooth manifold is a surjective submersion together with a Hermitian line bundle and an isomorphism of Hermitian line bundles over . ${\displaystyle X}$ ${\displaystyle \pi \colon Y\to X}$ ${\displaystyle L\to Y\times _{X}Y}$ ${\displaystyle \mu \colon \pi _{12}^{*}L\otimes \pi _{23}^{*}L\to \pi _{13}^{*}L}$ ${\displaystyle Y\times _{X}Y\times _{X}Y}$