Betti multiplets, flows across dimensions and c-extremization

The gravitational dual of c-extremization for a class of (0, 2) two-dimensional theories obtained by twisted compactifications of D3-brane gauge theories living at a toric Calabi-Yau three-fold has been recently proposed. The equivalence of this construction with c-extremization has been checked in various examples and holds also off-shell. In this note we prove that such equivalence holds for an arbitrary toric Calabi-Yau. We do it by generalizing the proof of the equivalence between a-maximization and volume minimization for four-dimensional toric quivers. By an explicit parameterization of the R-charges we map the trial right-moving central charge c r into the off-shell functional to be extremized in gravity. We also observe that the similar construction for M2-branes on C 4 is equivalent to the I-extremization principle that leads to the microscopic counting for the entropy of magnetically charged black holes in AdS 4 × S 7 . Also this equivalence holds off-shell.

“…That c-extremization correctly reproduces the central charge predicted by the gravitational dual has been tested in many examples [2,3,40,[42][43][44].…”

Section: Twisted Compactification To Two Dimensionsmentioning

confidence: 92%

Proving the equivalence of c-extremization and its gravitational dual for all toric quivers

Hosseini

Zaffaroni

2019

“…Therefore, our strategy is to first reduce the seven-dimensional gauged supergravity on Σ g 2 down to five dimensions and then do a further reduction on the circle inside AdS 3 to four-dimensional N = 2 gauged supergravity, where the attractor mechanism for static BPS black holes is well-understood. An analogous argument has been used in [140] to explain the extremization of the R-symmetry in the two-dimensional CFT.…”

Section: )mentioning

confidence: 95%

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).

“…Extremal magnetic black strings interpolate between (so-called 'magnetic') AdS 5 at infinity and AdS 3 × Σ (with Σ a two-dimensional space of constant curvature) at the horizon. Holographically, this corresponds to an RG flow across dimensions from a 4d field theory to a two-dimensional CFT in the infrared [19][20][21][22][23]. By plugging the near-horizon data into the flow equations, one gets the attractor equations.…”

Section: Jhep01(2017)106mentioning

confidence: 99%

“…It may be of some interest to use the flow equations obtained in section 5 to study more complicated type IIB configurations, as was initiated in [23].…”

Section: Final Remarksmentioning

confidence: 99%

Black string first order flow in N = 2, d = 5 abelian gauged supergravity

Klemm

Petri

Rabbiosi

2017

We derive both BPS and non-BPS first-order flow equations for magnetically charged black strings in five-dimensional N = 2 abelian gauged supergravity, using the Hamilton-Jacobi formalism. This is first done for the coupling to vector multiplets only and U(1) Fayet-Iliopoulos (FI) gauging, and then generalized to the case where also hypermultiplets are present, and abelian symmetries of the quaternionic hyperscalar target space are gauged. We then use these results to derive the attractor equations for near-horizon geometries of extremal black strings, and solve them explicitely for the case where the constants appearing in the Chern-Simons term of the supergravity action satisfy an adjoint identity. This allows to compute in generality the central charge of the two-dimensional conformal field theory that describes the black strings in the infrared, in terms of the magnetic charges, the CY intersection numbers and the FI constants. Finally, we extend the r-map to gauged supergravity and use it to relate our flow equations to those in four dimensions.