By O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz
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Additional info for Large N Field Theories, String Theory and Gravity
It was realized in the past 30 years that field theories generally exhibit a renormalization group flow from some scale-invariant (often free) UV fixed point to some scale-invariant (sometimes trivial) IR fixed point, and statistical mechanics systems also often have non-trivial IR scaleinvariant fixed points. Thus, studying scale-invariant theories is interesting for various physical applications. It is widely believed that unitary interacting scale-invariant theories are always invariant under the full conformal group, which is a simple group including scale invariance and Poincar´e invariance.
32). The choice of β for each scalar field depends on the theory we are considering. 38) is conserved only if the energy-momentum flux through the boundary at θ = π/2 vanishes, √ i dΩp gni T0|θ=π/2 = 0. 39) Sp This requirement reduces to the boundary condition (tan θ)p [(1 − 2β)∂θ + 2β tan θ] φ2 → 0 (θ → π/2). 34) is an integer. If we require the energy ω to be real, we find |ω|R = λ± + l + 2n, (n = 0, 1, 2, · · ·). 36) is real. 42) − (p + 1)2 ≤ m2 R2 . 4 This is known as the Breitenlohner-Freedman bound [101, 102].
The leading terms in the OPE of the energy-momentum tensor with primary fields are determined by the conformal algebra. For instance, for a scalar primary field φ of dimension ∆ in four dimensions, Tµν (x)φ(0) ∝ ∆φ(0)∂µ ∂ν ( 1 ) + ···. 8) One of the basic properties of conformal field theories is the one-to-one correspondence between local operators O and states |O in the radial quantization of the theory. In radial quantization the time coordinate is chosen to be the radial direction in Rd , with the origin corresponding to past infinity, so that the field theory lives on R ×S d−1 .
Large N Field Theories, String Theory and Gravity by O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz