SYM on quotients of spheres and complex projective spaces

Abstract

We introduce a generic procedure to reduce a supersymmetric Yang-Mills (SYM) theory along the Hopf fiber of squashed S2r−1 with U(1)r isometry, down to the ℂℙr−1 base. This amounts to fixing a Killing vector v generating a U(1) ⊂ 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 ℤp quotient freely acting along one of the two fibers. For fixed p the resulting manifolds S2r−1/ℤp ≡ L2r−1(p, ±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 S3 and mathcal{N} = 1 SYM on S5 we compute the perturbative partition functions on L2r−1(p, ±1) and, in the large p limit, on ℂℙr−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 S4. 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 ℂℙ2 arises from a sum over flat connections on L5(p, ±1). Finally, for r = 3, we also comment on the factorization of perturbative partition functions on non simply connected manifolds.