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Synthetic Quantum Field Theory
talks at
(talk slides, details on example 1, based on arXiv:1310.7930, Higher toposes of laws of motion)
on the modern version of Hilbert's 6th problem: the formulation of fundamental physics – hence of local boundary/defect prequantum field theory and its motivic quantization to quantum gauge field theory – internal to the foundations given by homotopy type theory/(∞,1)-topos theory via differential cohesion.
Among the examples that we derive and study from the axioms are
classical mechanics in terms of prequantized Lagrangian correspondences,
see at Classical field theory via Cohesive homotopy types for more on this;
the refinement of that to a holographic motivic quantization of Poisson manifolds;
see at Motivic quantization of prequantum field theory for more on this;
the higher local prequantum field theory with defects of ∞-Chern-Simons-type;
the brane bouquet of string theory/M-theory.
More details at
Attempts to axiomatize classical continuum mechanics inside toposes famously had led W. Lawvere to his “synthetic” axioms for differential geometry formulated internal to toposes, and more recently to his formulation of “axiomatic cohesion”. In this talk I report on interesting results that one finds when interpreting these axioms not in ordinary toposes, but in their homotopy theoretic incarnation as infinity-toposes; in other words, when implementing them in homotopy type theory. The central claim is that this way one obtains beyond a synthetic theory of differential equations/D-geometry and hence of classical physics a natural internal formulation of modern fundamental physics, namely of local gauge quantum field theory. I will try to explain this in general and go through some examples.
differential cohomology in a cohesive topos
Synthetic Quantum Field Theory