We compute the topologically twisted index for general N = 2 supersymmetric field theories on H 2 × S 1 . We also discuss asymptotically AdS 4 magnetically charged black holes with hyperbolic horizon, in four-dimensional N = 2 gauged supergravity. With certain assumptions, put forward by Benini, Hristov and Zaffaroni, we find precise agreement between the black hole entropy and the topologically twisted index, for ABJM theories.

Using localization techniques, we compute the path integral of N = 2 SUSY gauge theory coupled to matter on the hemisphere HS 4 , with either Dirichlet or Neumann supersymmetric boundary conditions. The resulting quantities are wave-functions of the theory depending on the boundary data. The one-loop determinant are computed using SO(4) harmonics basis. We solve kernel and co-kernel equations for the relevant differential operators arising from gauge and matter localizing actions. The second method utilizes full SO(5) harmonics to reduce the computation to evaluating Q 2 SU SY eigenvalues and its multiplicities. In the Dirichlet case, we show how to glue two wavefunctions to get back the partition function of round S 4 . We will also describe how to obtain the same results using SO(5) harmonics basis.Appendix H Analysis for |q L | < j L 43 3 SU (2) L generators J L only, implies in particular that SU (2) R commutes with the equations, and therefore the solutions will organize in SU (2) R multiplets of dimensions 2j R + 1, where the possible values of j R are j L ± 1, j L , depending on the spherical harmonics involved, as it is detailed in the appendix B. In more detail, if one introduces the expressions:with c a 1 (θ, r) an auxiliary variable. Now we show that all these equations can be expressed in terms of SU (2) L generators. θ, r)), J + c a 1 = e −i(q L +1)ψ e −iq R φ l +µ ∂ µ (e i(q L )ψ e iq R φ c a 1 (θ, r)) J + J − c = e −i(q L )ψ e −iq R φ l +µ D ν (l −ν D µ (e i(q L )ψ e i(q R )φ c(θ, r))), J − J + c = e −i(q L )ψ e −iq R φ l −µ D ν (l +ν D µ (e i(q L )ψ e i(q R )φ c(θ, r))), J 3 J 3 c = q 2 L c(θ, r).(3.17)

We study mixing boundary conditions in AdS 2 motivated by a family of 1/6 BPS Wilson loops in ABJM theory which interpolates between the bosonic 1/6 and the 1/2 BPS loops. The deformation that takes the 1/6 loop to the 1/2 loop is Q-exact and can be thought as an exact marginal deformation in the defect CFT 1 defined by the loop. Insertions along the loop lead to conformal