Exact results for N $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants

203

466

We study symmetries of quantum field theories involving topologically distinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the qq-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger-type relations. These relations imply that the supersymmetric partition functions in the presence of Ω-deformation and defects obey the Ward identities of two dimensional conformal field theory and its q-deformations. The details will be discussed in the companion papers.

Section: Jhep03(2016)181mentioning

confidence: 99%

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

Nekrasov

2016

203

466

“…The expression (6.7) is precisely the critical value of the Seiberg-Witten prepotential of the dual field theory in the large N limit, see [76, (3.71)]. A natural conjecture inspired by [27,76,78,79] is that we need a gluing of the form…”

Section: Six Dimensionsmentioning

confidence: 99%

Gluing gravitational blocks for AdS black holes

Hosseini

Hristov

Zaffaroni

2019

124

We provide a unifying entropy functional and an extremization principle for black holes and black strings in AdS 4 × S 7 and AdS 5 × S 5 with arbitrary rotation and generic electric and magnetic charges. This is done by gluing gravitational blocks, basic building blocks that are directly inspired by the holomorphic blocks appearing in the factorization of supersymmetric partition functions in three and four dimensions. We also provide an explicit realization of the attractor mechanism by identifying the values of the scalar fields at the horizon with the critical points of the entropy functional. We give examples based on dyonic rotating black holes with a twist in AdS 4 × S 7 , rotating black strings in AdS 5 × S 5 , dyonic Kerr-Newman black holes in AdS 4 × S 7 and Kerr-Newman black holes in AdS 5 × S 5 . In particular, our entropy functional extends existing results by adding rotation to the twisted black holes in AdS 4 and by adding flavor magnetic charges for the Kerr-Newman black holes in AdS 4 . We also discuss generalizations to higher-dimensional black objects. arXiv:1909.10550v1 [hep-th] 23 Sep 20191 The BPS conditions impose a linear constraint on the magnetic charges and some non-linear con-5 This has been checked at large N in full generality for static mAdS 4 black holes [1,41], rotating black strings in AdS 5 [19], KN-AdS 5 black holes with equal angular momenta [4] and at large N but in the Cardy limit for general purely electric KN-AdS 4 and KN-AdS 5 [3,42]. It is still not known if the large N limit of the superconformal index of N = 4 SYM reproduces the entropy functional, and therefore the entropy, in the case of KN-AdS 5 black holes with unequal angular momenta.6 See also [43] for a different approach.10 In particular, in the regime considered in (4.15) the minus sign in the chemical potential is equivalent to a complex conjugate. This might explain the complex conjugate we detect in gravity, see (4.9). 11 We correct a misprint in the formula for the entropy in [21]. Note also that we have redefined the angular momentum as J there = − 1 2 J here (see footnote 21).An important symmetry of the equations of motion of supergravity is the electromagnetic duality. As the name suggests, the n V + 1 electric and magnetic gauge field strengths F Λ and G Λ (Λ = 0, . . . , n V ) can be transformed among each other under the symplectic group Sp(2(n V + 1), Z), resulting in a rotation of the electromagnetic charges,

“…Otherwise, (1.5) should be replaced by log Z Σg 1 ×T d ≈ (1 − g 1 )D (1) d W d (∆ I ), where D where a (l) , (l) 1 and (l) 2 are determined by the toric data near each fixed point and encode the dependence on the fluxes m. For brevity, we suppressed the dependence on the flavor fluxes and fugacities s I and y I that can be easily reinstated by considering them as components of a background vector multiplet. This formula is the five-dimensional analogue of a similar four-dimensional expression that has been successfully used to evaluate equivariant Donaldson invariants [53][54][55][56].…”

Section: The Five-dimensional Indexmentioning

confidence: 99%

Topologically twisted indices in five dimensions and holography

Hosseini

Yaakov

Zaffaroni

2018

168

We provide a formula for the partition function of five-dimensional N = 1 gauge theories on M 4 × S 1 , topologically twisted along M 4 in the presence of general background magnetic fluxes, where M 4 is a toric Kähler manifold. The result can be expressed as a contour integral of the product of copies of the K-theoretic Nekrasov's partition function, summed over gauge magnetic fluxes. The formula generalizes to five dimensions the topologically twisted index of three-and four-dimensional field theories. We analyze the large N limit of the partition function and some related quantities for two theories: N = 2 SYM and the USp(2N ) theory with N f flavors and an antisymmetric matter field. For P 1 × P 1 × S 1 , which can be easily generalized to Σ g 2 × Σ g 1 × S 1 , we conjecture the form of the relevant saddle point at large N . The resulting partition function for N = 2 SYM scales as N 3 and is in perfect agreement with the holographic results for domain walls in AdS 7 × S 4 . The large N partition function for the USp(2N ) theory scales as N 5/2 and gives a prediction for the entropy of a class of magnetically charged black holes in massive type IIA supergravity. arXiv:1808.06626v3 [hep-th] 27 Nov 2018 14 See [87, sect. 4.2] for a more complete explanation..(3.96) 26 The free energy on Σ g2 × S 3 as a function of ∆ was explicitly computed in field theory in [67] after the completion of this work. The result in [67] perfectly agrees with (3.95).