Periodic Reporting for period 4 - GRAPHCPX (A graph complex valued field theory)
Berichtszeitraum: 2021-01-01 bis 2021-06-30
The objective of this project is to unify two important developments of mathematical physics and algebraic topology, namely perturbative topological quantum field theories and the Goodwillie-Weiss manifold calculus.
The fields of the quantum field theories we consider are essentially maps from a manifold (worldsheet) into another.
They are built so as to be symmetric under diffeomorphisms of the worldsheet and hence can be used to construct topological invariants. Such invariants have been studied extensively by mathematical physicists in the last decades.
On the topological side the Goodwillie-Weiss calculus also aims at defining topological invariants of (for example knot-)spaces by harnessing the natural algebraic structures on configuration spaces of points on manifolds.
The bridge between the physical and the topological side is provided by diagram complexes. From the physical viewpoint these complexes appear as the Feynman diagrams in the quantum field theories. From the topological viewpoints, the diagrams provide models (in a precise sense) of the spaces considered.
We work out this connection and hence unify the topological and physical viewpoint.
The fields of the quantum field theories we consider are essentially maps from a manifold (worldsheet) into another.
They are built so as to be symmetric under diffeomorphisms of the worldsheet and hence can be used to construct topological invariants. Such invariants have been studied extensively by mathematical physicists in the last decades.
On the topological side the Goodwillie-Weiss calculus also aims at defining topological invariants of (for example knot-)spaces by harnessing the natural algebraic structures on configuration spaces of points on manifolds.
The bridge between the physical and the topological side is provided by diagram complexes. From the physical viewpoint these complexes appear as the Feynman diagrams in the quantum field theories. From the topological viewpoints, the diagrams provide models (in a precise sense) of the spaces considered.
We work out this connection and hence unify the topological and physical viewpoint.
In the first half of the project we have used physical ideas (Feynman diagrams) to construct (rational or real) models for configuration spaces of points on manifolds.
Our models exhibit all the necessary algebraic structures to be used in the Goodwillie-Weiss calculus on the topological side.
They are hence the main building blocks of the bridges between the quantum field theory on the physical side and the topology.
We have already used these bridges, as far as built, to solve a few open conjectures and problems in topology. These include:
i) We showed that for a simply connected manifold the real homotopy type of the configuration space only depends on the real homotopy type of the manifold.
Generally, we provided models for configuration space of points on manifolds capturing their real or rational homotopy type.
ii) We computed the rational homotopy type of higher dimensional knot and embedding spaces (in codimension at least 3)
iii) We determined the rational homotopy type of the framed little n-disks operad, and the real homotopy types of configuration spaces of framed points.
iv) We computed the automorphisms of the (rationalized) little n-disks operad, expressing the answer through a graph complex of Feynman diagrams.
v) We defined a higher genus version of the Grothendieck-Teichmüller Lie algebras.
vi) We studied the cohomology of and algebraic structures on several graph complexes, as well their interrelation.
The results have presented to the scientific community through many conference and workshop talks be project members during the duration of the project.
Our models exhibit all the necessary algebraic structures to be used in the Goodwillie-Weiss calculus on the topological side.
They are hence the main building blocks of the bridges between the quantum field theory on the physical side and the topology.
We have already used these bridges, as far as built, to solve a few open conjectures and problems in topology. These include:
i) We showed that for a simply connected manifold the real homotopy type of the configuration space only depends on the real homotopy type of the manifold.
Generally, we provided models for configuration space of points on manifolds capturing their real or rational homotopy type.
ii) We computed the rational homotopy type of higher dimensional knot and embedding spaces (in codimension at least 3)
iii) We determined the rational homotopy type of the framed little n-disks operad, and the real homotopy types of configuration spaces of framed points.
iv) We computed the automorphisms of the (rationalized) little n-disks operad, expressing the answer through a graph complex of Feynman diagrams.
v) We defined a higher genus version of the Grothendieck-Teichmüller Lie algebras.
vi) We studied the cohomology of and algebraic structures on several graph complexes, as well their interrelation.
The results have presented to the scientific community through many conference and workshop talks be project members during the duration of the project.