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The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded$d$-dimensional cobordism category for all$d$. For$d$= 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].

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We explore various 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π (SU(2)θ=π YM). Its higher 't Hooft anomalies of generalized global symmetries indicate that the 4d SU(2)θ=π YM, in order to realize all global symmetries locally, necessarily couples to a 5d higher symmetry-protected topological state (SPTs, as an invertible TQFT, or as a 5d 1-form-center-symmetry-protected interacting "topological superconductor" in condensed matter). We revisit the 4d SU(2)θ=π YM-5d SRE-higher-SPTs coupled systems in [arXiv:1812.11968] and find their "Fantastic Four Siblings" with four sets of new higher anomalies associated with the Kramers singlet/doublet and bosonic/fermionic properties of Wilson lines. Following Weyl's gauge principle, by dynamically gauging the 1-form center symmetry, we transform a 5d bulk SRE SPTs into an LRE symmetry-enriched topologically ordered state (SETs); thus we obtain the 4d SO(3)θ=π YM-5d LRE-higher-SETs coupled system with dynamical higher-form gauge fields. Apply the tool introduced in [arXiv:1612.09298], we derive new exotic anyonic statistics of extended objects such as 2-worldsheet of strings and 3-worldvolume of branes, which physically characterize the 5d SETs. We discover new triple and quadruple link invariants potentially associated with the underlying 5d higher-gauge TQFTs, hinting a new intrinsic relation between non-supersymmetric 4d pure YM and topological links in 5d. We provide lattice simplicial complex regularizations and "condensed matter" realizations.

We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the bracket are weightedhomogeneous polynomials in the derivatives of the dependent variables with respect to the space variable. In the particular case of a conformal (homogeneous) Frobenius structure, our hierarchy coincides with the Dubrovin–Zhang hierarchy that is canonically associated to the underlying Frobenius structure. Therefore, our approach allows to prove the polynomiality of the equations, Hamiltonians, and one of the Poisson brackets of these hierarchies, as conjectured by Dubrovin and Zhang.