Abstract
A Cardy formula for 6d superconformal field theories (SCFTs) conjectured by Di Pietro and Komargodski in [1] governs the universal behavior of the supersymmetric partition function on S^1_\beta \times S^5Sβ1×S5 in the limit of small \betaβ and fixed squashing of the S^5S5. For a general 6d SCFT, we study its 5d effective action, which is dominated by the supersymmetric completions of perturbatively gauge-invariant Chern-Simons terms in the small \betaβ limit. Explicitly evaluating these supersymmetric completions gives the precise squashing dependence in the Cardy formula. For SCFTs with a pure Higgs branch (also known as very Higgsable SCFTs), we determine the Chern-Simons levels by explicitly going onto the Higgs branch and integrating out the Kaluza-Klein modes of the 6d fields on S^1_\betaSβ1. We then discuss tensor branch flows, where an apparent mismatch between the formula in [1] and the free field answer requires an additional contribution from BPS strings. This ``missing contribution’’ is further sharpened by the relation between the fractional part of the Chern-Simons levels and the (mixed) global gravitational anomalies of the 6d SCFT. We also comment on the Cardy formula for 4d \mathcal{N}=2𝒩=2 SCFTs in relation to Higgs branch and Coulomb branch flows.
Highlights
Operators, to the Weyl anomaly coefficient c in the high temperature limit1 log ZSβ1 ×S1
In [1], Di Pietro and Komargodski introduced analogs of the Cardy formula for 4d and 6d superconformal field theories (SCFTs), establishing universal relations between perturbative anomalies and the Cardy limit of the superconformal index.2. These higher-dimensional Cardy formulae involve a sum of terms that depend on the background geometry and gauge fields, whose overall coefficients are determined by the perturbative anomaly coefficients
Supersymmetry plays a key role in their higher dimensional Cardy formulae, because supersymmetric partition functions are geometric invariants – roughly speaking, quantities that depend only on a subset of the bosonic background data [9,10,11,12,13]
Summary
For any N ≥ 1 SCFT in 4d, we define the supersymmetric Sβ1 × S3 partition function (or superconformal index) ZSβ1 ×S3 by [33,34,35,36],5. Under a large background U(1)KK gauge transformation, the partition function will pick up a phase This is nothing but a manifestation of the global (mixed) gravitational anomaly of the 4d CFT [48,49,50]. The coefficient κ receives contributions from the massive KK-modes via 1-loop diagrams Summing over such contributions under the appropriate regularization, one finds the relation (1.5) between the coefficient κ and the mixed gravitational-R anomaly coefficient k, computed at the free point. Since both κ and k are invariant under RG flows triggered by deformations in the action, the relation (1.5) holds at the superconformal fixed point as well
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