Partition functions and double-trace deformations in AdS/CFT

Abstract

Abstract:We study the effect of a relevant double-trace deformation on the partition function (and conformal anomaly) of a CF T d at large N and its dual picture in AdS d+1 . Three complementary previous results are brought into full agreement with each other: bulk [1] and boundary [2] computations, as well as their formal identity [3]. We show the exact equality between the dimensionally regularized partition functions or, equivalently, fluctuational determinants involved. A series of results then follows: (i) equality between the renormalized partition functions for all d; (ii) for all even d, correction to the conformal anomaly; (iii) for even d, the mapping entails a mixing of UV and IR effects on the same side (bulk) of the duality, with no precedent in the leading order computations; and finally, (iv) a subtle relation between overall coefficients, volume renormalization and IR-UV connection. All in all, we get a clean test of the AdS/CFT correspondence beyond the classical SUGRA approximation in the bulk and at subleading O(1) order in the large-N expansion on the boundary.

The heat kernel on AdS

“…Now since logZ (S) = − This expression matches exactly with that which is obtained with the method of [22] (see also, earlier work by [23,24]). Note: After version 1 of this paper appeared on the arXiv, we learned of [25] where expressions for the one-loop partition function for general spin on thermal AdS were obtained by means of a Hamiltonian computation.…”

Non-supersymmetric Wilson loop in $$ \mathcal{N} $$ = 4 SYM and defect 1d CFT

“…To compute this correction we may use the general result for the difference of effective actions with standard (D or +) and alternate (N or -) boundary conditions for a scalar with mass m in AdS d+1 [43,44] …”

“…One may also give an alternative derivation of (4.8) using the relation between the AdS d+1 bulk field and S d boundary conformal field partition functions: Z − /Z + = Z conf (see [44][45][46]). For a massive scalar in AdS d+1 associated to an operator with dimension The induced boundary CFT has thus the kinetic operator ∂ ≡ (−∂ 2 ) 1/2 defined on S 1 and thus we find again (4.8)…”

“…Explicitly, in the case of 5 massless scalars in AdS d+1 with spherical boundary and mixed boundary conditions (4.17) the analog of (4.6) gives [43,44] (see eqs. (3.2), (5.2) in [44]) …”

“…Note that this expression comes out of the general expression in [44] or (4.23) if we interchange the roles of ∆ + and ∆ − (i.e. set ν = − .8)).…”