We classify three fold isolated quotient Gorenstein singularity C^ 3/G . These singularities are rigid, ie there is no non-trivial deformation, and we conjecture that they define 4d C^ 3/G SCFTs which do not have a Coulomb branch.

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We study a variational problem whose critical point determines the Reeb vector field for a SasakiEinstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that the manifolds are toric. We show that the EinsteinHilbert action, restricted to a space of Sasakian metrics on a link <i>L</i> in a CalabiYau cone <i>X</i>, is the volume functional, which in fact is a function on the space of Reeb vector fields. We relate this function both to the DuistermaatHeckman formula and also to a limit of a certain equivariant index on <i>X</i> that counts holomorphic functions. Both formulae may be evaluated by localisation. This leads to a general formula for the volume function in terms of topological fixed point data. As a result we prove that the volume of a SasakiEinstein manifold, relative to that of the round sphere, is always an algebraic number. In complex dimension <i>n</i>=3 these results provide, via AdS

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to compute the Poisson moduli space of the sphere in the dual of a compact semisimple Lie algebra [17].

Consider an N-Boson system interacting via a two-body repulsive short-range potential $V$ in a three dimensional box  of side length $L$. We take the $limit N,L \to \infty $ while keeping the density $\rho = N/L^3 fixed and small. We prove a new lower bound for its ground state energy per particle $$ \frac{E(N,\Lambda)}{N} \ge 4 \pi a \rho [1- O(\rho^{1/3} | log \rho |^3) ] $$ as $\rho \to 0$, where $a$ is the scattering length of $V$.