Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.

We study the space of 3d ${\cal N} = 6$ SCFTs by combining numerical bootstrap techniques with exact results derived using supersymmetric localization. First we derive superconformal block decomposition four-point function stress tensor multiplet primary. then use localization for $U(N)_k\times U(N+M)_{-k}$ Chern-Simons-matter theories to determine two protected OPE coefficients many values $N,...

The M(atrix) theory contains membranes and longitudinal 5-branes (L5-branes) as extended objects. The transverse components of these brane solutions can be described by fuzzy CP (k = 1, 2), where k = 1 corresponds to spherical membranes and k = 2 to L5-branes of CP×S1 world-volume geometry. In addition to these, we show the existence of L7-branes of CP×S1 geometry, introducing extra potentials ...

Quantization, at least in some formulations, involves replacing some algebra of observables by a (more non-commutative) deformed algebra. In view of the fundamental role played by K-theory in non-commutative geometry and topology, it is of interest to ask to what extent K-theory remains \rigid" under this process. We show that some positive results can be obtained using ideas of Gabber, Gillet-...

We study Thurston’s norm on the second homology of a 3-manifold. The novelty of our approach consists in the use of methods of the C∗algebra theory. Namely, for Stallings’ fibration M with pseudo-Anosov monodromy, we associate a C∗-algebra whose K-theory gives rise to an algebraic number field K. We show that the trace function on the ring of integers of K induces a norm on the second homology ...

An M ∨-matrix has the form A = sI − B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨-matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.