We study the natural scheme-independent quantity obtained from the three-sphere partition function of a ( 2 + 1 )-dimensional quantum field theory by removing all local counterterm ambiguities. At conformal fixed points this quantity equals the standard F -theorem invariant. Conformal perturbation theory shows that it locally decreases at O ( g 2 ) under any relevant scalar deformation of a three-dimensional Conformal Field Theory. However, an exact analysis of the free massive scalar on S 3 shows that this sphere-free-energy interpolant is not monotone along the full renormalization-group flow: it dips below its infrared value and then returns to it. Thus the natural counterterm-subtracted quantity built from sphere thermodynamics is not, by itself, a monotone F -function. We trace the obstruction to the second-order differential operator required to eliminate the local ambiguities.
We study the natural scheme-independent quantity obtained from the three-sphere partition function of a ( 2 + 1 )-dimensional quantum field theory by removing all local counterterm ambiguities. At conformal fixed points this quantity equals the standard F -theorem invariant. Conformal perturbation theory shows that it locally decreases at O ( g 2 ) under any relevant scalar deformation of a three-dimensional Conformal Field Theory. However, an exact analysis of the free massive scalar on S 3 shows that this sphere-free-energy interpolant is not monotone along the full renormalization-group flow: it dips below its infrared value and then returns to it. Thus the natural counterterm-subtracted quantity built from sphere thermodynamics is not, by itself, a monotone F -function. We trace the obstruction to the second-order differential operator required to eliminate the local ambiguities.
The F function, the three-dimensional counterpart of the central charge in the two-dimensional (2D) conformal field theory, measures the effective number of degrees of freedom in three-dimensional (3D) quantum field theory, and it is monotonically decreasing under the renormalization group flow. However, unlike the 2D central charge, the F function is a nonlocal quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first nonperturbative computation of the F function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our computation yields FIsing=0.0612(5), slightly smaller than the F function of a real free scalar, ${F}_{\text{free}}=\frac{log2}{8}\ensuremath{-}\frac{3\ensuremath{\zeta}(3)}{16{\ensuremath{\pi}}^{2}}\ensuremath{\approx}0.0638$, consistent with the F theorem, and close to the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion estimates of ${F}_{\text{Ising}}\ensuremath{\approx}0.061\ensuremath{\sim}0.062$.
Entropic F-function of 3D Ising conformal field theory via the fuzzy sphere regularization
The F-function, the three-dimensional counterpart of the central charge in the 2D conformal field theory, measures the effective number of degrees of freedom in 3D quantum field theory, and it is monotonically decreasing under the renormalization group flow. However, unlike the 2D central charge, the F-function is a non-local quantity and cannot be computed using correlators of local operators. Utilizing the recently proposed fuzzy sphere regularization, we have performed the first non-perturbative computation of the F-function for the paradigmatic 3D Ising conformal field theory through entanglement entropy. Our estimate yields FIsing≈0.0612(5), slightly smaller than the F-function of a real free scalar, Ffree=8log2−16π23ζ(3)≈0.0638, consistent with the F-theorem, and close to the 4−ε expansion estimates of FIsing≈0.0610∼0.0623.
The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.
Abstract We explore a C-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate C-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases but the entanglement entropy increases along a certain renormalization group (RG) flow triggered by a defect localized perturbation which is assumed to have a trivial IR fixed point without defects. We hence propose a C-theorem in DCFTs stating that the increment of the regularized sphere free energy due to the defect does not increase under any defect RG flow. We also provide a proof of our proposal in several holographic models of defect RG flows.
We analyze the renormalization group (RG) flow of the temperature independent term of the entropy in the high temperature limit β/a ≪ 1 of a massive field theory in 3-dimensional spherical spaces, M 3, with constant curvature 6/a 2. For masses lower than β2π , this term can be identified with the free energy of the same theory on M 3 considered as a 3-dimensional Euclidean space-time. The non-extensive part of this free energy, S hol, is generated by the holonomy of the spatial metric connection. We show that for homogeneous spherical spaces the holonomy entropy S hol decreases monotonically when the RG scale flows to the infrared. At the conformal fixed points the values of the holonomy entropy do coincide with the genuine topological entropies recently introduced. The monotonic behavior of the RG flow leads to an inequality between the topological entropies of the conformal field theories connected by such flow, i.e. S top > S top . From a 3-dimensional viewpoint the same term arises in the 3-dimensional Euclidean effective action and has the same monotonic behavior under the RG group flow. We conjecture that such monotonic behavior is generic, which would give rise to a 3-dimensional generalization of the c-theorem, along the lines of the 2-dimensional c-theorem and the 4-dimensional a-theorem. The conjecture is related to recent formulations of the F -theorem. In particular, the holonomy entropy on lens spaces is directly related to the topological Rényi entanglement entropy on disks of 2-dimensional flat spaces.
Extensive systems have a simple thermodynamic signature; the dimensionless canonical free energy scales homogeneously with the size of the system. We show that the failure of this scaling, measured by the replica energy E , provides a useful bridge between statistical mechanics and quantum field theory. The associated differential operator ( 1 − 1 d L ∂ L ) removes the leading bulk contribution to F = β F can = − log Z and isolates the part that is sensitive to boundaries, topology, defects, long-range forces, or other sources of nonadditivity. In quantum field theory this thermodynamic idea has two closely related uses. For ordinary finite-volume or spherical partition functions, suitable higher-order versions of the same filter remove local counterterms and extract universal fixed-point data such as the central charge, the sphere free energy F , and the Euler anomaly coefficient a . For replica geometries with entangling defects, the same filtering principle gives the renormalized defect free energy. In 2 + 1 dimensions, its n → 1 limit gives the entropic F -function, with the sign fixed below by the standard free-energy convention. We use this perspective to distinguish ordinary finite-size corrections, topology-dependent constants in gapped phases, subextensive fracton degeneracies, and genuinely nonextensive systems with long-range interactions such as self-gravitating thermal matter. Replica energy therefore offers a common thermodynamic language for additivity, defect free energies, and renormalization-group irreversibility.