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.

A familiar anomaly affects SU(2) gauge theory in four dimensions: a theory with an odd number of fermion multiplets in the spin 1/2 representation of the gauge group, and more generally in representations of spin 2r+1/2, is inconsistent. We describe here a more subtle anomaly that can affect SU(2) gauge theory in four dimensions under the condition that fermions transform with half-integer spin under SU(2) and bosons with integer spin. Such a theory, formulated in a way that requires no choice of spin structure, and with an odd number of fermion multiplets in representations of spin 4r+3/2, is inconsistent. The theory is consistent if one picks a spin or spin_c structure. Under Higgsing to U(1), the new SU(2) anomaly reduces to a known anomaly of "all-fermion electrodynamics." Like that theory, an SU(2) theory with an odd number of fermion multiplets in representations of spin 4r+3/2 can provide a boundary state for a five-dimensional gapped theory whose partition function on a closed five-manifold Y is (−1)^{∫_Y w2w3}. All statements have analogs with SU(2) replaced by Sp(2N). There is also an analog in five dimensions.

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We apply quantum gravitational results to spatially unbounded Friedmann universes and try to answer some questions related to dark energy, dark matter, inflation and the missing antimatter.

We prove trace class estimates for self-adjoint elliptic operators defined in $\R[n]$ or $\R[]_+$. These estimates are also applicable when a physical system is governed by a wave equation by employing separation of variables to obtain corresponding temporal and spatial Hamiltonians. It is shown, in one important example, that the resulting Hamiltonians are of trace class such that quantum statistics can be applied to the system.