$c$-Functions in Flows Across Dimensions

Abstract

We explore the notion of c-functions in renormalization group flows between theories in different spacetime dimensions. We discuss functions connecting central charges of the UV and IR fixed point theories on the one hand, and functions which are monotonic along the flow on the other. First, using the geometric properties of the holographic dual RG flows across dimensions and the constraints from the null energy condition, we construct a monotonic holographic c-function and thereby establish a holographic c-theorem across dimensions. Second, we use entanglement entropies for two different types of entangling regions in a field theory along the RG flow across dimensions to construct candidate c-functions which satisfy one of the two criteria but not both. In due process we also discuss an interesting connection between corner contributions to the entanglement entropy and the topology of the compact internal space. As concrete examples for both approaches, we holographically study twisted compactifications of 4d N = 4 SYM and compactifications of 6d N = (2, 0) theories.

Compactification of 6d ${\cal N}=(1,0)$ quivers, 4d SCFTs and their holographic dual Massive IIA backgrounds

“…The plot in Figure 6 shows this. This plot is very similar to the analogous quantity calculated from the Entanglement Entropy in reference [27]. The discussion about the factors in N is similar to that in the previous subsection.…”

“…In that paper the authors define a quantity c monotonic using a calculation of Entanglement Entropy. We believe that c monotonic in [27] and our eq. (3.7) should be related by a proportionality factor when studied in our family of backgrounds.…”

“…(3.4)-(3.8) for a four dimensional QFT (we have set d = 3) is not able to recognise that the system is approximately approaching a six dimensional SCFT. This same arguments were expressed by the authors of [27]. In that paper the authors define a quantity c monotonic using a calculation of Entanglement Entropy.…”

“…This calls for a quantity that is actually able to detect both fixed points (this was also one of the objectives in [27]). A quantity called c f low in [26], [28] can play this role.…”

“…r) Γ r . (A 27). Using the projection(A.19) leads to 2Γ r θ δψ θ = h (r) + i k a(r) e −h(r) Γ r θ σ 1 − i 2 k a (r)e −h(r) Γ θ σ 1 − 1 2k 1 − k a 2 (r) e −2h(r) Γ r − m2e −2Φ(r) Γ r .…”