SYM on quotients of spheres and complex projective spaces

Abstract

Abstract 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 ℂℙ r−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 ℤ p quotient freely acting along one of the two fibers. For fixed p the resulting manifolds S 2r−1 /ℤ p ≡ L 2r−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 $$ \mathcal{N} $$ = 2 SYM on S 3 and N $$ \mathcal{N} $$ = 1 SYM on S 5 we compute the perturbative partition functions on L 2r−1(p, ±1) and, in the large p limit, on 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 ℂℙ2 arises from a sum over flat connections on L 5(p, ±1). Finally, for r = 3, we also comment on the factorization of perturbative partition functions on non simply connected manifolds.

Keywords

• Supersymmetric Gauge Theory
• Extended Supersymmetry
• Field Theories in Higher Dimensions
• Topological Field Theories