Pseudoholomorphic quilt

In mathematics , the theory of pseudoholomorphen quilts the basis for a general approach to the construction of symplectic versions of out of gauge theory derived invariants .

definition

A quilted surface consists of a family of Riemannian surfaces with diffeomorphisms between some of its edge components . ${\displaystyle (S_{k})_{k}}$ ${\displaystyle \phi _{\sigma }\colon I_{k,b}\to I_{k^{\prime },b^{\prime }}}$ ${\displaystyle I_{k,b},b\in \pi _{0}\partial S_{k}}$

For a quilted surface one has a family of symplectic manifolds and for each one a Lagrangian submanifold . ${\displaystyle (M_{k})_{k}}$ ${\displaystyle \phi _{\sigma }}$ ${\displaystyle L_{\sigma }\subset M_{k}\times M_{k^{\prime }}}$

A pseudoholomorphic quilt is a family of pseudoholomorphic images with for everyone . ${\displaystyle u_{k}\colon S_{k}\to M_{k}}$ ${\displaystyle (u_{k}(x),u_{k^{\prime }}(\phi _{\sigma }(x)))\in L_{\sigma }}$ ${\displaystyle x\in I_{k,b}}$

literature

• K. Wehrheim and C. Woodward: Quilted Floer cohomology. Geometry & Topology, 14: 833-902, 2010.
• K. Wehrheim and C. Woodward: Floer field theory for tangles. https://arxiv.org/abs/1503.07615
• K. Wehrheim and C. Woodward: Floer field theory for coprime rank and degree. https://arxiv.org/abs/1601.04924

Web links

• Wehrheim website with lectures and lectures on quilts.