SYM on Quotients of Spheres and Complex Projective Spaces

We introduce a generic procedure to reduce a supersymmetric Yang-Mills (SYM) theory along the Hopf fiber of squashed \(S^{2r-1}\) with \(U(1)^r\) isometry, down to the \(\mathbb{CP}^{r-1}\) base. This amounts to fixing a Killing vector \(v\) generating a \(U(1)\subset U(1)^r\) rotation and dimensionally reducing either along \(v\) or along another direction contained in \(U(1)^r\). To perform such reduction we introduce a \(\mathbb{Z}_p\) quotient freely acting along one of the two fibers. For fixed \(p\) the resulting manifolds \(S^{2r-1}/\mathbb{Z}_p\equiv L^{2r-1}(p,\pm 1)\) are a higher dimensional generalization of lens spaces. In the large \(p\) limit the fiber shrinks and effectively we find theories living on the base manifold. Starting from \(\mathcal{N}=2\) SYM on \(S^3\) and \(\mathcal{N}=1\) SYM on \(S^5\) we compute the perturbative partition functions on \(L^{2r-1}(p,\pm 1)\) and, in the large \(p\) limit, on \(\mathbb{CP}^{r-1}\), respectively for \(r=2\) and \(r=3\). We show how the reductions along the two inequivalent fibers give rise to two distinct theories on the base. Reducing along \(v\) gives an equivariant version of Donaldson-Witten theory while the other choice leads to a supersymmetric theory closely related to Pestun's theory on \(S^4\). We use our technique to reproduce known results for \(r=2\) and we provide new results for \(r=3\). In particular we show how, at large \(p\), the sum over fluxes on \(\mathbb{CP}^2\) arises from a sum over flat connections on \(L^{5}(p,\pm 1)\). Finally, for \(r=3\), we also comment on the factorization of perturbative partition functions on non simply connected manifolds.