We study S duality of four dimensional N= 2 Argyres-Douglas (AD) theory engineered from 6d A_ {N-1}(2, 0) theory. We find a (p, q) sequence of SCFTs, here (p, q) is co-prime and class S theory defined on sphere corresponds to class (0, 1) theory. We represent these theories by a sphere with marked points, and S duality is interpreted as different pants decompositions of the same punctured sphere. The weakly coupled gauge theory description involves gauging AD matter which is represented by three punctured sphere.

We examine gravitational wave memory in the case where sources and detector are in an expanding cosmology. For simplicity, we treat the case where the cosmology is de Sitter spacetime, and discuss the possibility of generalizing our results to the case of a more realistic cosmology. We find results very similar to those of gravitational wave memory in an asymptotically flat spacetime, but with the magnitude of the effect multiplied by a redshift factor.

A bstract We conjecture that every three dimensional canonical singularity defines a five dimensional N N= 1 SCFT. Flavor symmetry can be found from singularity structure: non-abelian flavor symmetry is read from the singularity type over one dimensional singular locus. The dimension of Coulomb branch is given by the number of compact crepant divisors from a crepant resolution of singularity. The detailed structure of Coulomb branch is described as follows: a) a chamber of Coulomb branch is described by a crepant resolution, and this chamber is given by its Nef cone and the prepotential is computed from triple intersection numbers; b) Crepant resolution is not unique and different resolutions are related by flops; Nef cones from crepant resolutions form a fan which is claimed to be the full Coulomb branch.

We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I _2 + I _2 collision of singularities. The resulting gauge theory has a semi-simple Lie algebra _2 or _2 . We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representations by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is _2 but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplifted. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.

IN 1970 Lawson [22] proved that two embedded closed diffeomorphic minimal surfaces in the unit three-dimensional sphere S3 in lRJ are ambiently isotopic in S3. Lawson proved this theorem by first proving that an embedded orientable closed minimal surface of genus y in a closed orientable Riemannian three-manifold Mwith positive Ricci curvature disconnects M-into two genus-8 handlebodics. A result of Frankel [7] was used to prove this. Lawson then applied a deep result of Waldhauscn [33] that states that decompositions of SJ into two genus-cl handlebodies arc unique up to ambient isotopy. More prcciscly. Waldhauscns uniqucncss thcorcm states that whenever a closed surface of genus y in S separates SJ into handlcbodics, then the embedding of the surface is as simple as possible; in other words, the surface is obtained from a two-sphere Sz c Sby adding handles in an unknotted manner.