Abstract
Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number—interpreted as area—which behaves additively under glueing. As opposed to topological theories, in area-dependent theories the state spaces can be infinite-dimensional. We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras. We also provide a state sum construction for area-dependent theories. Our main example is two-dimensional Yang–Mills theory with compact gauge group, which we treat in detail.
Highlights
Introduction and SummaryArea-dependent quantum field theory is a modification of 2-dimensional topological quantum field theory (TQFT): we consider the category of bordisms with area Bor d2area and symmetric monoidal functors from it to the category of Hilbert spaces Hilb which depend continuously on the area
Area-dependent quantum field theory is a modification of two-dimensional topological quantum field theory, where one equips each connected component of a bordism with a positive real number—interpreted as area—which behaves additively under glueing
We introduce the notion of regularised Frobenius algebras in Hilbert spaces and show that area-dependent theories are in one-to-one correspondence to commutative regularised Frobenius algebras
Summary
Area-dependent quantum field theory (aQFT1) is a modification of 2-dimensional topological quantum field theory (TQFT): we consider the category of bordisms with area Bor d2area and symmetric monoidal functors from it to the category of Hilbert spaces Hilb which depend continuously on the area. The commutative Frobenius algebra defining the TQFT obtained from this state sum construction is just the centre Z (A) The generalisation of these results to aQFTs is for the most part straightforward to the point of being mechanical: just add a positive real parameter to all maps in sight (“area parameters”) and impose the condition that everything just depends on the sum of these areas. Finite dimensional RFAs in turn are very simple: they are just usual Frobenius algebras A together with an element H in the centre Z (A) of A, and the area-dependence is obtained by exponentiating H (Corollary 2.15) This makes RFAs sound not very interesting, but note that, an infinite direct sum of finite-dimensional RFAs has to satisfy non-trivial bounds to again define an RFA (Proposition 2.16).
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