In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension Δ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in [1] for conformal blocks of external scalar operators. We then generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator. Finally we specialize to the case in which the vector operator is a conserved current.

We compute the 4d superconformal index for $\mathcal{N}=1$, 2 gauge theories on $S^1\times L(p,1)$, where $L(p,1)$ is a lens space. We find that the 4d $\mathcal{N}=1$, 2 index on $S^1\times L(p,1)$ reduces to a 3d $\mathcal{N}=2$, 4 index on $S^1\times S^2$ in the large $p$ limit, and to a 3d partition function on a squashed $L(p,1)$ when the size of the temporal $S^1$ shrinks to zero. As an application of our index, we study 4d $\mathcal{N}=2$ superconformal field theories arising from the 6d $\mathcal{N}=(2,0)$ $A_1$ theory on a punctured Riemann surface $\Sigma$, and conjecture the existence of a 2d topological quantum field theory on $\Sigma$ whose correlation function coincides with the 4d $\mathcal{N}=2$ index on $S^1\times L(p,1)$.

A low-energy effective theory is said to be in the swampland if it does not have any consistent UV completion inside a theory of quantum gravity. The natural question is if the standard model of particle physics, possibly with some minimal extensions, are in the swampland — we are in trouble if the answer to this question is yes. We discuss this question in view of the recent swampland conjectures. We prove a no-go theorem concerning the modification of the Higgs sector. Moreover, we find that QCD axion is incompatible with the recent swampland conjectures, unless some sophisticated possibilities are considered. We discuss the implications of this result for spontaneous breaking of CP symmetry. We comment on dynamical supersymmetry breaking as well as the issue of multi-valuedness of the potential. We also propose a refinement of the de Sitter swampland conjecture.