We consider the mixed q-Gaussian algebras introduced by Speicher which are generated by the variables Xi = li + l ∗ i , i = 1, . . . , N, where l ∗ i lj − qij lj l ∗ i = δi,j and −1 < qij = qji < 1. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed q-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra L(FN ), provided that maxi,j |qij | is small enough. Similar results hold in the reduced C ∗ -algebra setting. The proof relies on some estimates which are generalizations of Dabrowski’s results for the special case qij ≡ q.

A modified theory of general relativity is proposed, where the gravitational constant is replaced by a dynamical variable in space-time. The dynamics of the gravitational coupling is described by a family of parametrized null geodesics, implying that the gravitational coupling at a space-time point is determined by solving transport equations along all null geodesics through this point. General relativity with dynamical gravitational coupling (DGC) is introduced. We motivate DGC from general considerations and explain how it arises in the context of causal fermion systems. The underlying physical idea is that the gravitational coupling is determined by microscopic structures on the Planck scale which propagate with the speed of light. In order to clarify the mathematical structure, we analyze the conformal behaviorn and prove local existence and uniqueness of the time evolution. The differences to Einstein’s theory are worked out in the examples of the Friedmann-RobertsonWalker model and the spherically symmetric collapse of a shell of matter. Potential implications for the problem of dark matter and for inflation are discussed. It is shown that the effects in the solar system are too small for being observable in present-day experiments.

Topological phases of matter are usually realized in deconfined phases of gauge theories. In this context, confined phases with strongly fluctuating gauge fields seem to be irrelevant to the physics of topological phases. For example, the low-energy theory of the two-dimensional (2D) toric code model (i.e. the deconfined phase of ℤ2 gauge theory) is a U(1)×U(1) Chern-Simons theory in which gauge charges (i.e., e and m particles) are deconfined and the gauge fields are gapped, while the confined phase is topologically trivial. In this paper, we point out a new route to constructing exotic 3D gapped fermionic phases in a confining phase of a gauge theory. Starting from a parton construction with strongly fluctuating compact U(1)×U(1) gauge fields, we construct gapped phases of interacting fermions by condensing two linearly independent bosonic composite particles consisting of partons and U(1)×U(1) magnetic monopoles. This can be regarded as a 3D generalization of the 2D Bais-Slingerland condensation mechanism. Charge fractionalization results from a Debye-H\"uckel-like screening cloud formed by the condensed composite particles. Within our general framework, we explore two aspects of symmetry-enriched 3D Abelian topological phases. First, we construct a new fermionic state of matter with time-reversal symmetry and Θ≠π, the fractional topological insulator. Second, we generalize the notion of anyonic symmetry of 2D Abelian topological phases to the charge-loop excitation symmetry (𝖢𝗁𝖺𝗋𝗅𝖾𝗌) of 3D Abelian topological phases. We show that line twist defects, which realize 𝖢𝗁𝖺𝗋𝗅𝖾𝗌 transformations, exhibit non-Abelian fusion properties.

A Calabi–Yau threefold is called of type K if it admits an ´etale Galois covering by the product of a K3 surface and an elliptic curve. In our previous paper [16], based on Oguiso–Sakurai’s fundamental work [24], we have provided the full classification of Calabi–Yau threefolds of type K and have studied some basic properties thereof. In the present paper, we continue the study, investigating them from the viewpoint of mirror symmetry. It is shown that mirror symmetry relies on duality of certain sublattices in the second cohomology of the K3 surface appearing in the minimal splitting covering. The duality may be thought of as a version of the lattice duality of the anti-symplectic involution on K3 surfaces discovered by Nikulin [23]. Based on the duality, we obtain several results parallel to what is known for Borcea–Voisin threefolds. Along the way, we also investigate the Brauer groups of Calabi–Yau threefolds of type K.

Recently, a number of new Ward identities for large gauge transformations and large diffeomorphisms have been discovered. Some of the identities are reinterpretations of previously known statements, while some appear to be genuinely new. We use Noether’s second theorem with the path integral as a powerful way of generating these kinds of Ward identities. We reintroduce Noether’s second theorem and discuss how to work with the physical remnant of gauge symmetry in gauge fixed systems. We illustrate our mechanism in Maxwell theory, Yang–Mills theory, p-form field theory, and Einstein–Hilbert gravity. We comment on multiple connections between Noether’s second theorem and known results in the recent literature. Our approach suggests a novel point of view with important physical consequences.