@article{14756,
  abstract     = {We prove the r-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer r: the 2-groupoid of 2-dimensional fully extended r-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced Spin 2r -action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the rth power of their Serre automorphisms. For r=1, we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to r=2.
To construct examples, we explicitly describe Spin 2r​-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau–Ginzburg models gives rise to fully extended spin TQFTs and that half of these do not factor through the oriented bordism 2-category.},
  author       = {Carqueville, Nils and Szegedy, Lorant},
  issn         = {1663-487X},
  journal      = {Quantum Topology},
  keywords     = {Geometry and Topology, Mathematical Physics},
  number       = {3},
  pages        = {467--532},
  publisher    = {European Mathematical Society},
  title        = {{Fully extended r-spin TQFTs}},
  doi          = {10.4171/qt/193},
  volume       = {14},
  year         = {2023},
}

@article{8816,
  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.},
  author       = {Runkel, Ingo and Szegedy, Lorant},
  issn         = {14320916},
  journal      = {Communications in Mathematical Physics},
  number       = {1},
  pages        = {83–117},
  publisher    = {Springer Nature},
  title        = {{Area-dependent quantum field theory}},
  doi          = {10.1007/s00220-020-03902-1},
  volume       = {381},
  year         = {2021},
}

@article{10176,
  abstract     = {We give a combinatorial model for r-spin surfaces with parameterized boundary based on Novak (“Lattice topological field theories in two dimensions,” Ph.D. thesis, Universität Hamburg, 2015). The r-spin structure is encoded in terms of ℤ𝑟-valued indices assigned to the edges of a polygonal decomposition. This combinatorial model is designed for our state-sum construction of two-dimensional topological field theories on r-spin surfaces. We show that an example of such a topological field theory computes the Arf-invariant of an r-spin surface as introduced by Randal-Williams [J. Topol. 7, 155 (2014)] and Geiges et al. [Osaka J. Math. 49, 449 (2012)]. This implies, in particular, that the r-spin Arf-invariant is constant on orbits of the mapping class group, providing an alternative proof of that fact.},
  author       = {Runkel, Ingo and Szegedy, Lorant},
  issn         = {00222488},
  journal      = {Journal of Mathematical Physics},
  number       = {10},
  publisher    = {AIP Publishing},
  title        = {{Topological field theory on r-spin surfaces and the Arf-invariant}},
  doi          = {10.1063/5.0037826},
  volume       = {62},
  year         = {2021},
}