arXiv subjects

Igor R. Klebanov

Publications and source records attributed to Igor R. Klebanov.

At least 55 records · Page 3Linked to original sources

arXiv2016

On $C_J$ and $C_T$ in the Gross-Neveu and $O(N)$ Models

We apply large $N$ diagrammatic techniques for theories with double-trace interactions to the leading corrections to $C_J$, the coefficient of a conserved current two-point function, and $C_T$, the coefficient of the stress-energy tensor two-point function. We study in detail two famous conformal field theories in continuous dimensions, the scalar $O(N)$ model and the Gross-Neveu model. For the $O(N)$ model, where the answers for the leading large $N$ corrections to $C_J$ and $C_T$ were derived long ago using analytic bootstrap, we show that the diagrammatic approach reproduces them correctly. We also carry out a new perturbative test of these results using the $O(N)$ symmetric cubic scalar theory in $6-ε$ dimensions. We go on to apply the diagrammatic method to the Gross-Neveu model, finding explicit formulae for the leading corrections to $C_J$ and $C_T$ as a function of dimension. We check these large $N$ results using regular perturbation theory for the Gross-Neveu model in $2+ε$ dimensions and the Gross-Neveu-Yukawa model in $4-ε$ dimensions. For small values of $N$, we use Pade approximants based on the $4-ε$ and $2+ε$ expansions to estimate the values of $C_J$ and $C_T$ in $d=3$. For the $O(N)$ model our estimates are close to those found using the conformal bootstrap. For the GN model, our estimates suggest that, even when $N$ is small, $C_T$ differs by no more than $2\%$ from that in the theory of free fermions. We find that the inequality $C_T^{\textrm{UV}} > C_T^{\textrm{IR}}$ applies both to the GN and the scalar $O(N)$ models in $d=3$.

hep-th↗

arXiv2015

Conformal QED$_d$, $F$-Theorem and the $ε$ Expansion

We calculate the free energies $F$ for $U(1)$ gauge theories on the $d$ dimensional sphere of radius $R$. For the theory with free Maxwell action we find the exact result as a function of $d$; it contains the term $\frac{d-4}{2} \log R$ consistent with the lack of conformal invariance in dimensions other than 4. When the $U(1)$ gauge theory is coupled to a sufficient number $N_f$ of massless 4 component fermions, it acquires an interacting conformal phase, which in $d<4$ describes the long distance behavior of the model. The conformal phase can be studied using large $N_f$ methods. Generalizing the $d=3$ calculation in arXiv:1112.5342, we compute its sphere free energy as a function of $d$, ignoring the terms of order $1/N_f$ and higher. For finite $N_f$, following arXiv:1409.1937 and arXiv:1507.01960, we develop the $4-ε$ expansion for the sphere free energy of conformal QED$_d$. Its extrapolation to $d=3$ shows very good agreement with the large $N_f$ approximation for $N_f>3$. For $N_f$ at or below some critical value $N_{\rm crit}$, the $SU(2N_f)$ symmetric conformal phase of QED$_3$ is expected to disappear or become unstable. By using the $F$-theorem and comparing the sphere free energies in the conformal and broken symmetry phases, we show that $N_{\rm crit}\leq 4$. As another application of our results, we calculate the one loop beta function in conformal QED$_6$, where the gauge field has a 4-derivative kinetic term. We show that this theory coupled to $N_f$ massless fermions is asymptotically free.

hep-th↗

arXiv2015

Generalized $F$-Theorem and the $ε$ Expansion

Some known constraints on Renormalization Group flow take the form of inequalities: in even dimensions they refer to the coefficient $a$ of the Weyl anomaly, while in odd dimensions to the sphere free energy $F$. In recent work arXiv:1409.1937 it was suggested that the $a$- and $F$-theorems may be viewed as special cases of a Generalized $F$-Theorem valid in continuous dimension. This conjecture states that, for any RG flow from one conformal fixed point to another, $\tilde F_{\rm UV} > \tilde F_{\rm IR}$, where $\tilde F=\sin (πd/2)\log Z_{S^d}$. Here we provide additional evidence in favor of the Generalized $F$-Theorem. We show that it holds in conformal perturbation theory, i.e. for RG flows produced by weakly relevant operators. We also study a specific example of the Wilson-Fisher $O(N)$ model and define this CFT on the sphere $S^{4-ε}$, paying careful attention to the beta functions for the coefficients of curvature terms. This allows us to develop the $ε$ expansion of $\tilde F$ up to order $ε^5$. Pade extrapolation of this series to $d=3$ gives results that are around $2-3\%$ below the free field values for small $N$. We also study RG flows which include an anisotropic perturbation breaking the $O(N)$ symmetry; we again find that the results are consistent with $\tilde F_{\rm UV} > \tilde F_{\rm IR}$.

hep-th↗

arXiv2015

Accidental Symmetries and the Conformal Bootstrap

We study an ${\cal N} = 2$ supersymmetric generalization of the three-dimensional critical $O(N)$ vector model that is described by $N+1$ chiral superfields with superpotential $W = g_1 X \sum_i Z_i^2 + g_2 X^3$. By combining the tools of the conformal bootstrap with results obtained through supersymmetric localization, we argue that this model exhibits a symmetry enhancement at the infrared superconformal fixed point due to $g_2$ flowing to zero. This example is special in that the existence of an infrared fixed point with $g_1,g_2\neq 0$, which does not exhibit symmetry enhancement, does not generally lead to any obvious unitarity violations or other inconsistencies. We do show, however, that the $F$-theorem excludes the models with $g_1,g_2\neq 0$ for $N>5$. The conformal bootstrap provides a stronger constraint and excludes such models for $N>2$. We provide evidence that the $g_2=0$ models, which have the enhanced $O(N)\times U(1)$ symmetry, come close to saturating the bootstrap bounds. We extend our analysis to fractional dimensions where we can motivate the nonexistence of the $g_1,g_2\neq 0$ models by studying them perturbatively in the $4-ε$ expansion.

hep-th↗

arXiv2015

Critical $Sp(N)$ Models in $6-ε$ Dimensions and Higher Spin dS/CFT

Theories of anti-commuting scalar fields are non-unitary, but they are of interest both in statistical mechanics and in studies of the higher spin de Sitter/Conformal Field Theory correspondence. We consider an $Sp(N)$ invariant theory of $N$ anti-commuting scalars and one commuting scalar, which has cubic interactions and is renormalizable in 6 dimensions. For any even $N$ we find an IR stable fixed point in $6-ε$ dimensions at imaginary values of coupling constants. Using calculations up to three loop order, we develop $ε$ expansions for several operator dimensions and for the sphere free energy $F$. The conjectured $F$-theorem is obeyed in spite of the non-unitarity of the theory. The $1/N$ expansion in the $Sp(N)$ theory is related to that in the corresponding $O(N)$ symmetric theory by the change of sign of $N$. Our results point to the existence of interacting non-unitary 5-dimensional CFTs with $Sp(N)$ symmetry, where operator dimensions are real. We conjecture that these CFTs are dual to the minimal higher spin theory in 6-dimensional de Sitter space with Neumann future boundary conditions on the scalar field. For $N=2$ we show that the IR fixed point possesses an enhanced global symmetry given by the supergroup $OSp(1|2)$. This suggests the existence of $OSp(1|2)$ symmetric CFTs in dimensions smaller than 6. We show that the $6-ε$ expansions of the scaling dimensions and sphere free energy in our $OSp(1|2)$ model are the same as in the $q \rightarrow 0$ limit of the $q$-state Potts model.

hep-th↗

arXiv2015

Interpolating between $a$ and $F$

We study the dimensional continuation of the sphere free energy in conformal field theories. In continuous dimension $d$ we define the quantity $\tilde F=\sin (πd/2)\log Z$, where $Z$ is the path integral of the Euclidean CFT on the $d$-dimensional round sphere. $\tilde F$ smoothly interpolates between $(-1)^{d/2}π/2$ times the $a$-anomaly coefficient in even $d$, and $(-1)^{(d+1)/2}$ times the sphere free energy $F$ in odd $d$. We calculate $\tilde F$ in various examples of unitary CFT that can be continued to non-integer dimensions, including free theories, double-trace deformations at large $N$, and perturbative fixed points in the $ε$ expansion. For all these examples $\tilde F$ is positive, and it decreases under RG flow. Using perturbation theory in the coupling, we calculate $\tilde F$ in the Wilson-Fisher fixed point of the $O(N)$ vector model in $d=4-ε$ to order $ε^4$. We use this result to estimate the value of $F$ in the 3-dimensional Ising model, and find that it is only a few percent below $F$ of the free conformally coupled scalar field. We use similar methods to estimate the $F$ values for the $U(N)$ Gross-Neveu model in $d=3$ and the $O(N)$ model in $d=5$. Finally, we carry out the dimensional continuation of interacting theories with 4 supercharges, for which we suggest that $\tilde F$ may be calculated exactly using an appropriate version of localization on $S^d$. Our approach provides an interpolation between the $a$-maximization in $d=4$ and the $F$-maximization in $d=3$.

hep-th↗

arXiv2014

Three Loop Analysis of the Critical $O(N)$ Models in $6-ε$ Dimensions

We continue the study, initiated in arXiv:1404.1094, of the $O(N)$ symmetric theory of $N+1$ massless scalar fields in $6-ε$ dimensions. This theory has cubic interaction terms $\frac{1}{2}g_1 σ(ϕ^i)^2 + \frac{1}{6}g_2 σ^3$. We calculate the 3-loop beta functions for the two couplings and use them to determine certain operator scaling dimensions at the IR stable fixed point up to order $ε^3$. We also use the beta functions to determine the corrections to the critical value of $N$ below which there is no fixed point at real couplings. The result suggests a very significant reduction in the critical value as the dimension is decreased to $5$. We also study the theory with $N=1$, which has a $Z_2$ symmetry under $ϕ\rightarrow -ϕ$. We show that it possesses an IR stable fixed point at imaginary couplings which can be reached by flow from a nearby fixed point describing a pair of $N=0$ theories. We calculate certain operator scaling dimensions at the IR fixed point of the $N=1$ theory and suggest that, upon continuation to two dimensions, it describes a non-unitary conformal minimal model.

hep-th↗

arXiv2014

Partition Functions and Casimir Energies in Higher Spin AdS_{d+1}/CFT_d

Recently, the one-loop free energy of higher spin (HS) theories in Euclidean AdS_{d+1} was calculated and matched with the order N^0 term in the free energy of the large N "vectorial" scalar CFT on the S^d boundary. Here we extend this matching to the boundary theory defined on S^1 x S^{d-1}, where the length of S^1 may be interpreted as the inverse temperature. It has been shown that the large N limit of the partition function on S^1 x S^2 in the U(N) singlet sector of the CFT of N free complex scalars matches the one-loop thermal partition function of the Vasiliev theory in AdS_4, while in the O(N) singlet sector of the CFT of N real scalars it matches the minimal theory containing even spins only. We extend this matching to all dimensions d. We also calculate partition functions for the singlet sectors of free fermion CFT's in various dimensions and match them with appropriately defined higher spin theories, which for d>3 contain massless gauge fields with mixed symmetry. In the zero-temperature case R x S^{d-1} we calculate the Casimir energy in the scalar or fermionic CFT and match it with the one-loop correction in the global AdS_{d+1}. For any odd-dimensional CFT the Casimir energy must vanish on general grounds, and we show that the HS duals obey this. In the U(N) symmetric case, we exhibit the vanishing of the regularized 1-loop Casimir energy of the dual HS theory in AdS_{d+1}. In the minimal HS theory the vacuum energy vanishes for odd d while for even d it is equal to the Casimir energy of a single conformal scalar in R x S^{d-1} which is again consistent with AdS/CFT, provided the minimal HS coupling constant is ~ 1/(N-1). We demonstrate analogous results for singlet sectors of theories of N Dirac or Majorana fermions. We also discuss extensions to CFT's containing N_f flavors in the fundamental representation of U(N) or O(N).

hep-th↗

arXiv2014

Critical $O(N)$ Models in $6-ε$ Dimensions

We revisit the classic $O(N)$ symmetric scalar field theories in $d$ dimensions with interaction $(ϕ^i ϕ^i)^2$. For $2 1038$. We show that the $1/N$ expansions of various operator scaling dimensions match the known results for the critical $O(N)$ theory continued to $d=6-ε$. These results suggest that, for sufficiently large $N$, there are 5-dimensional unitary $O(N)$ symmetric interacting CFT's; they should be dual to the Vasiliev higher-spin theory in AdS$_6$ with alternate boundary conditions for the bulk scalar. Using these CFT's we provide a new test of the 5-dimensional $F$-theorem, and also find a new counterexample for the $C_T$ theorem.

hep-th↗

arXiv2014

Higher Spin AdS$_{d+1}$/CFT$_d$ at One Loop

Following arXiv:1308.2337, we carry out one loop tests of higher spin AdS$_{d+1}$/CFT$_d$ correspondences for $d\geq 2$. The Vasiliev theories in AdS$_{d+1}$, which contain each integer spin once, are related to the $U(N)$ singlet sector of the $d$-dimensional CFT of $N$ free complex scalar fields; the minimal theories containing each even spin once -- to the $O(N)$ singlet sector of the CFT of $N$ free real scalar fields. Using analytic continuation of higher spin zeta functions, which naturally regulate the spin sums, we calculate one loop vacuum energies in Euclidean AdS$_{d+1}$. In even $d$ we compare the result with the $O(N^0)$ correction to the $a$-coefficient of the Weyl anomaly; in odd $d$ -- with the $O(N^0)$ correction to the free energy $F$ on the $d$-dimensional sphere. For the theories of integer spins, the correction vanishes in agreement with the CFT of $N$ free complex scalars. For the minimal theories, the correction always equals the contribution of one real conformal scalar field in $d$ dimensions. As explained in arXiv:1308.2337, this result may agree with the $O(N)$ singlet sector of the theory of $N$ real scalar fields, provided the coupling constant in the higher spin theory is identified as $G_N\sim 1/(N-1)$. Our calculations in even $d$ are closely related to finding the regularized $a$-anomalies of conformal higher spin theories. In each even $d$ we identify two such theories with vanishing $a$-anomaly: a theory of all integer spins, and a theory of all even spins coupled to a complex conformal scalar. We also discuss an interacting UV fixed point in $d=5$ obtained from the free scalar theory via an irrelevant double-trace quartic interaction. This interacting large $N$ theory is dual to the Vasiliev theory in AdS$_6$ where the bulk scalar is quantized with the alternate boundary condition.

hep-th↗

arXiv2013

One Loop Tests of Higher Spin AdS/CFT

Vasiliev's type A higher spin theories in AdS4 have been conjectured to be dual to the U(N) or O(N) singlet sectors in 3-d conformal field theories with N-component scalar fields. We compare the O(N^0) correction to the 3-sphere free energy F in the CFTs with corresponding calculations in the higher spin theories. This requires evaluating a regularized sum over one loop vacuum energies of an infinite set of massless higher spin gauge fields in Euclidean AdS4. For the Vasiliev theory including fields of all integer spin and a scalar with Delta=1 boundary condition, we show that the regularized sum vanishes. This is in perfect agreement with the vanishing of subleading corrections to F in the U(N) singlet sector of the theory of N free complex scalar fields. For the minimal Vasiliev theory including fields of only even spin, the regularized sum remarkably equals the value of F for one free real scalar field. This result may agree with the O(N) singlet sector of the theory of N real scalar fields, provided the coupling constant in the Vasiliev theory is identified as G_N ~ 1/(N-1). Similarly, consideration of the USp(N) singlet sector for N complex scalar fields, which we conjecture to be dual to the husp(2;0|4) Vasiliev theory, requires G_N ~ 1/(N+1). We also test the higher spin AdS3/CFT2 conjectures by calculating the regularized sum over one loop vacuum energies of higher spin fields in AdS3. We match the result with the O(N^0) term in the central charge of the W_N minimal models; this requires a certain truncation of the CFT operator spectrum so that the bulk theory contains two real scalar fields with the same boundary conditions.

hep-th↗

arXiv2013

AdS Description of Induced Higher-Spin Gauge Theory

We study deformations of three-dimensional large N CFTs by double-trace operators constructed from spin s single-trace operators of dimension Δ. These theories possess UV fixed points, and we calculate the change of the 3-sphere free energy δF= F_{UV}- F_{IR}. To describe the UV fixed point using the dual AdS_4 space we modify the boundary conditions on the spin s field in the bulk; this approach produces δF in agreement with the field theory calculations. If the spin s operator is a conserved current, then the fixed point is described by an induced parity invariant conformal spin s gauge theory. The low spin examples are QED_3 (s=1) and the 3-d induced conformal gravity (s=2). When the original CFT is that of N conformal complex scalar or fermion fields, the U(N) singlet sector of the induced 3-d gauge theory is dual to Vasiliev's theory in AdS_4 with alternate boundary conditions on the spin s massless gauge field. We test this correspondence by calculating the leading term in δF for large N. We show that the coefficient of (1/2)\log N in δF is equal to the number of spin s-1 gauge parameters that act trivially on the spin s gauge field. We discuss generalizations of these results to 3-d gauge theories including Chern-Simons terms and to theories where s is half-integer. We also argue that the Weyl anomaly a-coefficients of conformal spin s theories in even dimensions d, such as that of the Weyl-squared gravity in d=4, can be efficiently calculated using massless spin s fields in AdS_{d+1} with alternate boundary conditions. Using this method we derive a simple formula for the Weyl anomaly a-coefficients of the d=4 Fradkin-Tseytlin conformal higher-spin gauge fields. Similarly, using alternate boundary conditions in AdS_3 we reproduce the well-known central charge c=-26 of the bc ghosts in 2-d gravity, as well as its higher-spin generalizations.

hep-th↗

arXiv2013

A Crack in the Conformal Window

In {\cal N} = 2 superconformal three-dimensional field theory the R-symmetry is determined by locally maximizing the free energy F on the three-sphere. Using F-maximization, we study the {\cal N} = 2 supersymmetric U(N_c) gauge theory coupled to N_f pairs of fundamental and anti-fundamental superfields in the Veneziano large N_c limit, where x = N_f / N_c is kept fixed. This theory has a superconformal window 1 \leq x \leq \infty, while for x < 1 supersymmetry is broken. As we reduce x we find "a crack in the superconformal window" - a critical value x_c \approx 1.45 where the monopole operators reach the unitarity bound. To continue the theory to x < x_c we assume that the monopoles become free fields, leading to an accidental global symmetry. Using the Aharony dual description of the theory for x < x_c allows us to determine the R-charges and F for 1 \leq x < x_c. Adding a Chern-Simons term removes the transition at x_c. In these more general theories we study the scaling dimensions of meson operators as functions of x and κ= |k| / N_c. We find that there is an interesting transition in behavior at κ= 1.

hep-th↗

arXiv2012

Is Renormalized Entanglement Entropy Stationary at RG Fixed Points?

The renormalized entanglement entropy (REE) across a circle of radius R has been proposed as a c-function in Poincaré invariant (2+1)-dimensional field theory. A proof has been presented of its monotonic behavior as a function of R, based on the strong subadditivity of entanglement entropy. However, this proof does not directly establish stationarity of REE at conformal fixed points of the renormalization group. In this note we study the REE for the free massive scalar field theory near the UV fixed point described by a massless scalar. Our numerical calculation indicates that the REE is not stationary at the UV fixed point.

hep-th↗

arXiv2012

On Shape Dependence and RG Flow of Entanglement Entropy

We use a mix of field theoretic and holographic techniques to elucidate various properties of quantum entanglement entropy. In (3+1)-dimensional conformal field theory we study the divergent terms in the entropy when the entangling surface has a conical or a wedge singularity. In (2+1)-dimensional field theory with a mass gap we calculate, for an arbitrary smooth entanglement contour, the expansion of the entropy in inverse odd powers of the mass. We show that the shape-dependent coefficients that arise are even powers of the extrinsic curvature and its derivatives. A useful dual construction of a (2+1)-dimensional theory, which allows us to exhibit these properties, is provided by the CGLP background. This smooth warped throat solution of 11-dimensional supergravity describes renormalization group flow from a conformal field theory in the UV to a gapped one in the IR. For this flow we calculate the recently introduced renormalized entanglement entropy and confirm that it is a monotonic function.

hep-th↗

arXiv2012

Strange Metals in One Spatial Dimension

We consider 1+1 dimensional SU(N) gauge theory coupled to a multiplet of massive Dirac fermions transforming in the adjoint representation of the gauge group. The only global symmetry of this theory is a U(1) associated with the conserved Dirac fermion number, and we study the theory at variable, non-zero densities. The high density limit is characterized by a deconfined Fermi surface state with Fermi wavevector equal to that of free gauge-charged fermions. Its low energy fluctuations are described by a coset conformal field theory with central charge c=(N^2-1)/3 and an emergent N=(2,2) supersymmetry: the U(1) fermion number symmetry becomes an R-symmetry. We determine the exact scaling dimensions of the operators associated with Friedel oscillations and pairing correlations. For N>2, we find that the symmetries allow relevant perturbations to this state. We discuss aspects of the N->infty limit, and its possible dual description in AdS3 involving string theory or higher-spin gauge theory. We also discuss the low density limit of the theory by computing the low lying bound state spectrum of the large N gauge theory numerically at zero density, using discretized light cone quantization.

hep-th↗

arXiv2012

Renyi Entropies for Free Field Theories

Renyi entropies S_q are useful measures of quantum entanglement; they can be calculated from traces of the reduced density matrix raised to power q, with q>=0. For (d+1)-dimensional conformal field theories, the Renyi entropies across S^{d-1} may be extracted from the thermal partition functions of these theories on either (d+1)-dimensional de Sitter space or R x H^d, where H^d is the d-dimensional hyperbolic space. These thermal partition functions can in turn be expressed as path integrals on branched coverings of the (d+1)-dimensional sphere and S^1 x H^d, respectively. We calculate the Renyi entropies of free massless scalars and fermions in d=2, and show how using zeta-function regularization one finds agreement between the calculations on the branched coverings of S^3 and on S^1 x H^2. Analogous calculations for massive free fields provide monotonic interpolating functions between the Renyi entropies at the Gaussian and the trivial fixed points. Finally, we discuss similar Renyi entropy calculations in d>2.

hep-th↗

arXiv2012

Entanglement Entropy of 3-d Conformal Gauge Theories with Many Flavors

Three-dimensional conformal field theories (CFTs) of deconfined gauge fields coupled to gapless flavors of fermionic and bosonic matter describe quantum critical points of condensed matter systems in two spatial dimensions. An important characteristic of these CFTs is the finite part of the entanglement entropy across a circle. The negative of this quantity is equal to the finite part of the free energy of the Euclidean CFT on the three-sphere, and it has been proposed to satisfy the so called F-theorem, which states that it decreases under RG flow and is stationary at RG fixed points. We calculate the three-sphere free energy of non-supersymmetric gauge theory with a large number N_F of bosonic and/or fermionic flavors to the first subleading order in 1/N_F. We also calculate the exact free energies of the analogous chiral and non-chiral {\cal N} = 2 supersymmetric theories using localization, and find agreement with the 1/N_F expansion. We analyze some RG flows of supersymmetric theories, providing further evidence for the F-theorem.

hep-th↗