5d partition functions with a twist

Abstract

We derive the partition function of 5d N = 1 gauge theories on the manifold S 3 b × Σ g with a partial topological twist along the Riemann surface, Σ g . This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large N limit, where it is related to holographic RG flows between asymptotically locally AdS 6 and AdS 4 spacetimes, reproducing known holographic relations between the corresponding free energies on S 5 and S 3 and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric N = 2 Yang-Mills theory, in which case the partition function computes the 4d index of general class S theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to M 3 × Σ g with more general three-manifolds M 3 and focus in particular on M 3 = Σ g × S 1 , in which case the partition function relates to the entropy of black holes in AdS 6 . β 67 A.1 5d maximal theory and class S 68 A.2 Class S k theories 70 A.3 E-string theory 72 B Instanton partition function for N = 2 U(N ) and SU(N ) SYM 74This preserves an SU(2) × U(1) isometry, and the Killing vector appearing in the superalgebra is K = ∂ ψ . Then, in this background we have 44], and so the central charge in the N = (2, 2) algebra (2.15) iswhich defines a variable, σ, valued in the complexification of the Cartan subalgebra of the flavor symmetry, which is a holomorphic combination of the background real scalar, z, and component, A ψ , of the background gauge field along the direction K, as discussed in more detail in [11]. The 2d A-twist and Bethe Ansatz equationsAs observed above, the supersymmetric background on S 3 b × Σ g , when considered along the Σ g directions, takes the form of a 2d topological twist. Such setups, where a d-dimensional 5 We will return to consider more general lens spaces in section 5.

Universal 3d Cardy block and black hole entropy

“…which we derive for 3d SCFTs with Lagrangian descriptions. Also the same relation is expected by the supergravity analysis [48] for 3d theories obtained as the twisted compactification of 5d SCFTs [85]. From those results, we may expect that our Cardy analysis for Lagrangian theories would hold for a broader class of 3d SCFTs, which will be interesting to clarify.…”

Universal spinning black holes and theories of class $$ \mathcal{R} $$

“…An interesting class of 3d N = 2 theories, which is much less explored, is obtained by twisted compactification of 5d SCFTs on a Riemann surface Σ g , and whose various partition functions are accessible via supersymmetric localization [8,9]. In particular, the S 3 partition function was computed in [8], and one can show that for g > 1 and large N there is a universal relation, F S 3 = − 8 9 (g − 1)F S 5 , with F S 5 the free energy on the round five-sphere, as predicted by supergravity [26]. Combining this with (2.29) leads to the following prediction for the partition function of the 5d SCFT on S 1 × S 2 ω × Σ g>1 in large N limit:…”

“…where ds 2 4 is the black hole metric and A is the gauge field in (2.3). 8 Here we have split the constrained coordinates µ i=1,...,5 on S 4 according to SO(3) × SO(2) ⊂ SO (5), To match the conventions of [47] to ours we need to set g 2 = √ 2 and rescale the gauge field as gA there…”

“…Alternatively one can utilize (p, q)-fivebrane webs in type IIB string theory [62][63][64], which was studied holographically in [65,66]. Upon a twisted compactification on a Riemann surface Σ g of genus g one obtains a 3d N = 2 theory whose S 3 partition function can be computed via the localization results of [8] where a universal relation to the free energy on S 5 was derived; we comment on this further in Section 5. These 3d N = 2 SCFTs admit a bulk description in terms of warped AdS 4 vacua of massive IIA or IIB supergravity, see for example [26] and [67].…”

“…This was first achieved for five-dimensional black holes in asymptotically flat spacetime in the seminal work [1]. For black holes in asymptotically locally AdS spacetimes this has only been recently achieved, starting with certain magnetic black holes in 4d in [2], extended in various ways in [3][4][5][6][7], for magnetic black holes in 6d in [8][9][10][11][12][13], and for spinning black holes in 5d in [14][15][16][17][18][19]. These calculations rely on holography, exploiting the properties of the asymptotic SCFT dual and the powerful supersymmetric localization techniques developed in the past decade.…”