SYM on Quotients of Spheres and Complex Projective Spaces
Authors: Jim Lundin and Lorenzo Ruggeri
Preprint number: UUITP-52/21
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 N=2 SYM on S3 and N=1 SYM on S5 we compute the 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.