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AdS7 black-hole entropy and 5D N$$ \mathcal{N} $$ = 2 Yang-Mills

G. Kántor, C. Papageorgakis, P. Richmond

2020
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Journal of High Energy Physics
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We generalise the work of 1810.11442 for the case of AdS 7 /CFT 6 . Starting from the 2-equivalent charge, 3-equivalent rotation non-extremal black-hole solution in 7D gauged supergravity, we consider the supersymmetric and then the extremal limit and evaluate the associated thermodynamic quantities. Away from extremality, the black-hole solution becomes complex. The entropy is then given by the Legendre transform of the on-shell action with respect to two complex chemical potentials subject to
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... a constraint. At the conformal boundary we derive the dual background and evaluate the corresponding partition function for the A N −1 6D (2,0) theory at large N in a Cardy-like limit. This is carried out via a 5D N = 2 super Yang-Mills calculation on S 5 . The gravitational on-shell action is found to be exactly reproduced by the boundary partition function at large N . We argue that this agreement puts strong constraints on the form of possible higher-derivative corrections to the 5D gauge theory that is used in the S 5 evaluation. where I is the superconformal index, β the radius of the thermal circle and F a quantity related to the vacuum Casimir energy [6] [7] [8] [9] . Unlike the index, this "supersymmetric Casimir energy" was shown to exhibit the expected scaling of degrees of freedom at large N . However matching the precise coefficient predicted by AdS/CFT, and corresponding entropy of BPS black holes in the gravity dual, turns out to be subtler, see for example [10] . More recently, the black-hole entropy for even D has been recovered from field theory through the extremisation of a quantity closely resembling the supersymmetric Casimir energy [11, 12] . Over the last few months, there has been a significant acceleration of activity in this direction. In [13], a complete gravitational derivation of the field-theoretic entropy function was performed for AdS 5 /CFT 4 , 1 while in [16, 17] the entropy function was reproduced from field theory in the Cardy limit of large charges, for a variety of bulk dimensions. In yet another line of attack, the superconformal index itself (and not the supersymmetric Casimir energy) was shown to be capturing larger than previously thought degeneracies for a particular complexification of chemical potentials, through a formulation that involves solutions for auxiliary Thermodynamic Bethe Ansatz (TBA) equations [18, 19] . Finally, the general behaviour seen in [18, 19] was reproduced for 4D N = 1 SCFTs in the Cardy limit without resorting to the TBA equations in [20, 21] . In this note, we will focus on the relationship between the AdS 7 black-hole entropy and the superconformal index of the six-dimensional (2,0) theory. 2 We will first generalise the AdS 5 analysis of [13] for the case of 2-equivalent charge and 3-equivalent rotation non-extremal black-hole solutions. A study of the regularity conditions for the metric and Killing spinors in the bulk leads to a specific background at the boundary of AdS 7 . The AdS/CFT correspondence then dictates that the black-hole entropy should be related to an R-twisted, supersymmetric partition function for the six-dimensional A N −1 (2,0) theory on this particular background in the large-N limit. As the interacting (2,0) theory does not admit a Lagrangian description one cannot directly employ the method of supersymmetric localisation to evaluate the boundary partition function. We therefore turn to the existing literature, where it has instead been conjectured to arise from a 5D supersymmetric partition function on S 5 for the maximallysupersymmetric (N = 2) Yang-Mills theory (MSYM) with SU(N ) gauge group [7, 22-28]the circle reduction of the (2,0) theory on S 5 × S 1 β . A modification of these results at large N and in a Cardy-like limit reproduces a (generalised) supersymmetric Casimir energy that exactly matches the gravitational prediction. 3 Lastly, we argue that the 6D Casimir energy is sensitive to the choice of 5D theory for which one evaluates the partition function on S 5 . Since 5D MSYM is conventionally thought of as a Wilsonian effective field theory, it is expected to differ from the microscopic (2,0) theory in the UV by an infinite tower of higher-derivative corrections. The latter 1 For generalisations see [14, 15] . 2 Note that the AdS7 entropy function was first written down in [12], while it was reproduced in the Cardy limit in [16] . However, our scope here will be to provide a microscopic derivation of this quantity from the six-dimensional CFT dual theory. 3 It would be very interesting to revisit the original work of [3] and investigate the scaling of degrees of freedom directly in the 6D superconformal index along the lines of [18, 19] . JHEP01(2020)017 can be organised into D-terms and F-terms, out of which only the F-terms affect the S 5 partition-function computation [29] [30] [31] [32] . The precision agreement between AdS 7 and CFT 6 hence constrains F-term-correction coefficients. Setting all these coefficients to zero, as done in [7, [22] [23] [24] [25] [26] [27] [28] , provides an obvious solution. However, since the inclusion of higherdimension operators lead to different matrix models, we cannot prove that there is no infinite sequence of F-term corrections with non-zero coefficients that also reproduces the result for the Casimir energy. At the very least, the AdS 7 black-hole entropy severely restricts the form of the completion of 5D MSYM towards the 6D (2,0) theory in the UV. We also comment on the connection between our results and the conjecture that 5D MSYM, without the need for higher-derivative corrections, captures all the information about the (2,0) theory on S 1 [33] [34] [35] . We should emphasise that the bulk approach of [13] determining the black-hole entropy that we follow in this paper is an interesting recent alternative to Sen's entropy-function formalism [36, 37] . The latter employs the attractor mechanism to identify the leading degrees of freedom of the full spacetime with the Bekenstein-Hawking entropy at the black-hole horizon. Therefore, the associated extremisation principle derived in [36, 37] is expected to agree to leading order with the one used here. However, due to considering the near-horizon geometry, Sen's approach does not explicitly determine the dual supersymmetric partition function at the boundary, which is where we would like to focus our attention in this work. Note also that the statistical entropy of various asymptotically-AdS black holes (not only for D even) can be reproduced microscopically from a different, "topologicallytwisted index". 4 The latter can be evaluated through supersymmetric localisation for a topologically-twisted gauge theory along the lines of [38] ; see e.g. [39, 40] for applications. The rest of this note is organised as follows: in section 2, we analyse the 2-equivalent charge, 3-equivalent rotation black-hole solution in AdS 7 and determine how it fixes the form of the boundary partition function that is AdS/CFT dual to the corresponding onshell action at large N . We then discuss the evaluation of this boundary partition function from 5D MSYM using supersymmetric localisation in section 3. Finally, in section 4 we discuss the implications of this precision matching for the UV completion of 5D MSYM by examining the contributions of higher-derivative corrections. Note added. Shortly before v1 of this manuscript appeared on the arXiv, we received [14] , which overlaps with the results of sections 2.1-2.6. * 2 . (A.6)

doi:10.1007/jhep01(2020)017
fatcat:fput6i2sebd2lddgafbn2kkauq