arXiv subjects

Simone Giombi

Publications and source records attributed to Simone Giombi.

At least 37 records · Page 2Linked to original sources

arXiv2023

Wilson loops at large $N$ and the quantum M2 brane

The Wilson loop operator in the $U(N)_k \times U(N)_{-k}$ ABJM theory at large $N$ and fixed level $k$ has a dual description in terms of a wrapped M2 brane in the M-theory background AdS$_4 \times S^7/\mathbb Z_k$. We consider the localization result for the $1\over 2$-BPS circular Wilson loop expectation value $W$ in this regime, and compare it to the prediction of the M2 brane theory. The leading large $N$ exponential factor is matched as expected by the classical action of the M2 brane solution with AdS$_2\times S^1$ geometry. We show that the subleading $k$-dependent prefactor in $W$ is also exactly reproduced by the one-loop term in the partition function of the wrapped M2 brane (with all Kaluza-Klein modes included). This appears to be the first case of an exact matching of the overall numerical prefactor in the Wilson loop expectation value against the dual holographic result. It provides an example of a consistent quantum M2 brane computation, suggesting various generalizations.

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arXiv2022

Fermions in AdS and Gross-Neveu BCFT

We study the boundary critical behavior of conformal field theories of interacting fermions in the Gross-Neveu universality class. By a Weyl transformation, the problem can be studied by placing the CFT in an anti de Sitter space background. After reviewing some aspects of free fermion theories in AdS, we use both large $N$ methods and the epsilon expansion near 2 and 4 dimensions to study the conformal boundary conditions in the Gross-Neveu CFT. At large $N$ and general dimension $d$, we find three distinct boundary conformal phases. Near four dimensions, where the CFT is described by the Wilson-Fisher fixed point of the Gross-Neveu-Yukawa model, two of these phases correspond respectively to the choice of Neumann or Dirichlet boundary condition on the scalar field, while the third one corresponds to the case where the bulk scalar field acquires a classical expectation value. One may flow between these boundary critical points by suitable relevant boundary deformations. We compute the AdS free energy on each of them, and verify that its value is consistent with the boundary version of the F-theorem. We also compute some of the BCFT observables in these theories, including bulk two-point functions of scalar and fermions, and four-point functions of boundary fermions.

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arXiv2022

Long Range, Large Charge, Large $N$

We study operators with large charge $j$ in the $d$-dimensional $O(N)$ model with long range interactions that decrease with the distance as $1/r^{d+s}$, where $s$ is a continuous parameter. We consider the double scaling limit of large $N$, large $j$ with $j/N=\hat{j}$ fixed, and identify the semiclassical saddle point that captures the two-point function of the large charge operators in this limit. The solution is given in terms of certain ladder conformal integrals that have recently appeared in the literature on fishnet models. We find that the scaling dimensions for general $s$ interpolate between $Δ_j \sim \frac{(d-s)}{2}j$ at small $\hat{j}$ and $Δ_j \sim \frac{(d+s)}{2}j$ at large $\hat{j}$, which is a qualitatively different behavior from the one found in the short range version of the $O(N)$ model. We also derive results for the structure constants and 4-point functions with two large charge and one or two finite charge operators. Using a description of the long range models as defects in a higher dimensional local free field theory, we also obtain the scaling dimensions in a complementary way, by mapping the problem to a cylinder in the presence of a chemical potential for the conserved charge.

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arXiv2022

Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: Matrix Model and Classical String

We study the large charge sector of the defect CFT defined by the half-BPS Wilson loop in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. Specifically, we consider correlation functions of two large charge insertions and several light insertions in the double-scaling limit where the 't Hooft coupling $λ$ and the large charge $J$ are sent to infinity, with the ratio $J/\sqrtλ$ held fixed. They are holographically dual to the expectation values of light vertex operators on a classical string solution with large angular momentum, which we evaluate in the leading large $J$ limit. We also compute the two-point function of large charge insertions by evaluating the on-shell string action, supplemented by the boundary terms that generalize the one introduced by Drukker, Gross and Ooguri for the Wilson loop without insertions. For a special class of correlation functions, we reproduce the string results from field theory by using supersymmetric localization. The results are given by correlation functions in an "emergent" matrix model whose matrix size is proportional to $J$ and whose spectral curve coincides with that of the classical string. Similar matrix models appeared in the study of extremal correlators in rank-1 $\mathcal{N}=2$ superconformal field theories, but our results hold also for non-extremal cases.

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arXiv2022

Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: II. Quantum Fluctuations, OPE, and Spectral Curve

We continue our study of large charge limits of the defect CFT defined by the half-BPS Wilson loop in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. In this paper, we compute $1/J$ corrections to the correlation function of two heavy insertions of charge $J$ and two light insertions, in the double scaling limit where the charge $J$ and the 't Hooft coupling $λ$ are sent to infinity with the ratio $J/\sqrtλ$ fixed. Holographically, they correspond to quantum fluctuations around a classical string solution with large angular momentum, and can be computed by evaluating Green's functions on the worldsheet. We derive a representation of the Green's functions in terms of a sum over residues in the complexified Fourier space, and show that it gives rise to the conformal block expansion in the heavy-light channel. This allows us to extract the scaling dimensions and structure constants for an infinite tower of non-protected dCFT operators. We also find a close connection between our results and the semi-classical integrability of the string sigma model. The series of poles of the Green's functions in Fourier space corresponds to points on the spectral curve where the so-called quasi-momentum satisfies a quantization condition, and both the scaling dimensions and the structure constants in the heavy-light channel take simple forms when written in terms of the spectral curve. These observations suggest extensions of the results by Gromov, Schafer-Nameki and Vieira on the semiclassical energy of closed strings, and in particular hint at the possibility of determining the structure constants directly from the spectral curve.

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arXiv2022

Wilson loop in general representation and RG flow in 1d defect QFT

The generalized Wilson loop operator interpolating between the supersymmetric and the ordinary Wilson loop in ${\cal N}=4$ SYM theory provides an interesting example of renormalization group flow on a line defect: the scalar coupling parameter $ζ$ has a non-trivial beta function and may be viewed as a running coupling constant in a 1d defect QFT. In this paper we continue the study of this operator, generalizing previous results for the beta function and Wilson loop expectation value to the case of an arbitrary representation of the gauge group and beyond the planar limit. Focusing on the scalar ladder limit where the generalized Wilson loop reduces to a purely scalar line operator in a free adjoint theory, and specializing to the case of the rank $k$ symmetric representation of $SU(N)$, we also consider a certain semiclassical limit where $k$ is taken to infinity with the product $k\, ζ^2$ fixed. This limit can be conveniently studied using a 1d defect QFT representation in terms of $N$ commuting bosons. Using this representation, we compute the beta function and the circular loop expectation value in the large $k$ limit, and use it to derive constraints on the structure of the beta function for general representation. We discuss the corresponding 1d RG flow and comment on the consistency of the results with the 1d defect version of the F-theorem.

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arXiv2021

Defect CFT in the 6d (2,0) theory from M2 brane dynamics in AdS$_7 \times$S$^4$

Surface operators in the 6d (2,0) theory at large $N$ have a holographic description in terms of M2 branes probing the AdS$_7 \times S^4$ M-theory background. The most symmetric, 1/2-BPS, operator is defined over a planar or spherical surface, and it preserves a 2d superconformal group. This includes, in particular, an $SO(2,2)$ subgroup of 2d conformal transformations, so that the surface operator may be viewed as a conformal defect in the 6d theory. The dual M2 brane has an AdS$_3$ induced geometry, reflecting the 2d conformal symmetry. Here we use the holographic description to extract the defect CFT data associated to the surface operator. The spectrum of transverse fluctuations of the M2 brane is found to be in one-to-one correspondence with a protected multiplet of operator insertions on the surface, which includes the displacement operator. We compute the one-loop determinants of fluctuations of the M2 brane, and extract the conformal anomaly coefficient of the spherical surface to order $N^0$. We also briefly discuss the RG flow from the non-supersymmetric to the 1/2-BPS defect operator, and its consistency with a "$b$-theorem" for the defect CFT. Starting with the M2 brane action, we then use AdS$_3$ Witten diagrams to compute the 4-point functions of the elementary bosonic insertions on the surface operator, and extract some of the defect CFT data from the OPE. The 4-point function is shown to satisfy superconformal Ward identities, and we discuss a related subsector of "twisted" scalar insertions, whose correlation functions are constrained by the residual superconformal symmetry.

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arXiv2021

Higher order RG flow on the Wilson line in $\mathcal{N}=4$ SYM

Extending earlier work, we find the two-loop term in the beta-function for the scalar coupling $ζ$ in a generalized Wilson loop operator of the $\mathcal{N}=4$ SYM theory, working in the planar weak-coupling expansion. The beta-function for $ζ$ has fixed points at $ζ=\pm 1$ and $ζ=0$, corresponding respectively to the supersymmetric Wilson-Maldacena loop and to the standard Wilson loop without scalar coupling. As a consequence of our result for the beta-function, we obtain a prediction for the two-loop term in the anomalous dimension of the scalar field inserted on the standard Wilson loop. We also find a subset of higher-loop contributions (with highest powers of $ζ$ at each order in `t Hooft coupling) coming from the scalar ladder graphs determining the corresponding terms in the five-loop beta-function. We discuss the related structure of the circular Wilson loop expectation value commenting, in particular, on consistency with a 1d defect version of the F-theorem. We also compute (to two loops in the planar ladder approximation) the two-point correlators of scalars inserted on the Wilson line.

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arXiv2021

Monodromy Defects from Hyperbolic Space

We study monodromy defects in $O(N)$ symmetric scalar field theories in $d$ dimensions. After a Weyl transformation, a monodromy defect may be described by placing the theory on $S^1\times H^{d-1}$, where $H^{d-1}$ is the hyperbolic space, and imposing on the fundamental fields a twisted periodicity condition along $S^1$. In this description, the codimension two defect lies at the boundary of $H^{d-1}$. We first study the general monodromy defect in the free field theory, and then develop the large $N$ expansion of the defect in the interacting theory, focusing for simplicity on the case of $N$ complex fields with a one-parameter monodromy condition. We also use the $ε$-expansion in $d=4-ε$, providing a check on the large $N$ approach. When the defect has spherical geometry, its expectation value is a meaningful quantity, and it may be obtained by computing the free energy of the twisted theory on $S^1\times H^{d-1}$. It was conjectured that the logarithm of the defect expectation value, suitably multiplied by a dimension dependent sine factor, should decrease under a defect RG flow. We check this conjecture in our examples, both in the free and interacting case, by considering a defect RG flow that corresponds to imposing alternate boundary conditions on one of the low-lying Kaluza-Klein modes on $H^{d-1}$. We also show that, adapting standard techniques from the AdS/CFT literature, the $S^1\times H^{d-1}$ setup is well suited to the calculation of the defect CFT data, and we discuss various examples, including one-point functions of bulk operators, scaling dimensions of defect operators, and four-point functions of operator insertions on the defect.

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arXiv2021

Non-supersymmetric Wilson loop in N=4 SYM and defect 1d CFT

Following Polchinski and Sully (arXiv:1104.5077), we consider a generalized Wilson loop operator containing a constant parameter $ζ$ in front of the scalar coupling term, so that $ζ=0$ corresponds to the standard Wilson loop, while $ζ=1$ to the locally supersymmetric one. We compute the expectation value of this operator for circular loop as a function of $ζ$ to second order in the planar weak coupling expansion in N=4 SYM theory. We then explain the relation of the expansion near the two conformal points $ζ=0$ and $ζ=1$ to the correlators of scalar operators inserted on the loop. We also discuss the $AdS_5\times S^5$ string 1-loop correction to the strong-coupling expansion of the standard circular Wilson loop, as well as its generalization to the case of mixed boundary conditions on the five-sphere coordinates, corresponding to general $ζ$. From the point of view of the defect 1d CFT defined on the Wilson line, the $ζ$-dependent term can be seen as a perturbation driving a RG flow from the standard Wilson loop in the UV to the supersymmetric Wilson loop in the IR. Both at weak and strong coupling we find that the logarithm of the expectation value of the standard Wilson loop for the circular contour is larger than that of the supersymmetric one, which appears to be in agreement with the 1d analog of the F-theorem.

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arXiv2021

Aspects of CFTs on Real Projective Space

We present an analytic study of conformal field theories on the real projective space $\mathbb{RP}^d$, focusing on the two-point functions of scalar operators. Due to the partially broken conformal symmetry, these are non-trivial functions of a conformal cross ratio and are constrained to obey a crossing equation. After reviewing basic facts about the structure of correlators on $\mathbb{RP}^d$, we study a simple holographic setup which captures the essential features of boundary correlators on $\mathbb{RP}^d$. The analysis is based on calculations of Witten diagrams on the quotient space $AdS_{d+1}/\mathbb{Z}_2$, and leads to an analytic approach to two-point functions. In particular, we argue that the structure of the conformal block decomposition of the exchange Witten diagrams suggests a natural basis of analytic functionals, whose action on the conformal blocks turns the crossing equation into certain sum rules. We test this approach in the canonical example of $ϕ^4$ theory in dimension $d=4-ε$, extracting the CFT data to order $ε^2$. We also check our results by standard field theory methods, both in the large $N$ and $ε$ expansions. Finally, we briefly discuss the relation of our analysis to the problem of construction of local bulk operators in AdS/CFT.

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arXiv2020

On the Large Charge Sector in the Critical $O(N)$ Model at Large $N$

We study operators in the rank-$j$ totally symmetric representation of $O(N)$ in the critical $O(N)$ model in arbitrary dimension $d$, in the limit of large $N$ and large charge $j$ with $j/N\equiv \hat{j}$ fixed. The scaling dimensions of the operators in this limit may be obtained by a semiclassical saddle point calculation. Using the standard Hubbard-Stratonovich description of the critical $O(N)$ model at large $N$, we solve the relevant saddle point equation and determine the scaling dimensions as a function of $d$ and $\hat{j}$, finding agreement with all existing results in various limits. In $4<d<6$, we observe that the scaling dimension of the large charge operators becomes complex above a critical value of the ratio $j/N$, signaling an instability of the theory in that range of $d$. Finally, we also derive results for the correlation functions involving two "heavy" and one or two "light" operators. In particular, we determine the form of the "heavy-heavy-light" OPE coefficients as a function of the charges and $d$.

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arXiv2020

$O(N)$ Models with Boundary Interactions and their Long Range Generalizations

We study the critical properties of scalar field theories in $d+1$ dimensions with $O(N)$ invariant interactions localized on a $d$-dimensional boundary. By a combination of large $N$ and epsilon expansions, we provide evidence for the existence of non-trivial $O(N)$ BCFTs in $1<d<4$. Due to having free fields in the bulk, these models possess bulk higher-spin currents which are conserved up to terms localized on the boundary. We suggest that this should lead to a set of protected spinning operators on the boundary, and give evidence that their anomalous dimensions vanish. We also discuss the closely related long-range $O(N)$ models in $d$ dimensions, and in particular study a weakly coupled description of the $d=1$ long range $O(N)$ model near the upper critical value of the long range parameter, which is given in terms of a non-local non-linear sigma model. By combining the known perturbative descriptions, we provide some estimates of critical exponents in $d=1$.

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arXiv2020

Giant Wilson Loops and AdS$_2$/dCFT$_1$

The 1/2-BPS Wilson loop in $\mathcal{N}=4$ supersymmetric Yang-Mills theory is an important and well-studied example of conformal defect. In particular, much work has been done for the correlation functions of operator insertions on the Wilson loop in the fundamental representation. In this paper, we extend such analyses to Wilson loops in the large-rank symmetric and antisymmetric representations, which correspond to probe D3 and D5 branes with $AdS_2 \times S^2$ and $AdS_2 \times S^4$ worldvolume geometries, ending at the $AdS_5$ boundary along a one-dimensional contour. We first compute the correlation functions of protected scalar insertions from supersymmetric localization, and obtain a representation in terms of multiple integrals that are similar to the eigenvalue integrals of the random matrix, but with important differences. Using ideas from the Fermi Gas formalism and the Clustering method, we evaluate their large $N$ limit exactly as a function of the 't Hooft coupling. The results are given by simple integrals of polynomials that resemble the $Q$-functions of the Quantum Spectral Curve, with integration measures depending on the number of insertions. Next, we study the correlation functions of fluctuations on the probe D3 and D5 branes in AdS. We compute a selection of three- and four-point functions from perturbation theory on the D-branes, and show that they agree with the results of localization when restricted to supersymmetric kinematics. We also explain how the difference of the internal geometries of the D3 and D5 branes manifests itself in the localization computation.

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arXiv2020

The $O(N)$ Model in $4<d<6$: Instantons and Complex CFTs

We revisit the scalar $O(N)$ model in the dimension range $4<d<6$ and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the $1/N$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In $d$ dimensions and for large $N$, we find that they are of order $e^{-N f(d)}$, where, remarkably, the function $f(d)$ equals the sphere free energy of a conformal scalar in $d-2$ dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $N$, the $O(N)$ models in $4<d<6$ may be thought of as complex CFTs. When $N$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional $O(N)$ models may be studied using the techniques of numerical bootstrap.

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arXiv2020

Wilson loops in $\mathcal{N}=4$ $SO(N)$ SYM and D-Branes in $AdS_5\times \mathbb{RP}^5$

We study the half-BPS circular Wilson loop in ${\cal N}=4$ super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of $SO(N)$. We focus on the large $N$ limit and present some simple quantitative tests of the duality with type IIB string theory in $AdS_5\times \mathbb{RP}^5$. In particular, we show that the strong coupling limit of the expectation value of the Wilson loop in the spinor representation of the gauge group precisely matches the classical action of the dual string theory object, which is expected to be a D5-brane wrapping a $\mathbb{RP}^4$ subspace of $\mathbb{RP}^5$. We also briefly discuss the large $N$, large $λ$ limits of the $SO(N)$ Wilson loop in the symmetric/antisymmetric representations and their D3/D5-brane duals. Finally, we use the D5-brane description to extract the leading strong coupling behavior of the "bremsstrahlung function" associated to a spinor probe charge, or equivalently the normalization of the two-point function of the displacement operator on the spinor Wilson loop, and obtain agreement with the localization prediction.

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arXiv2020

Loop Equation and Exact Soft Anomalous Dimension in N=4 Super Yang-Mills

BPS Wilson loops in supersymmetric gauge theories have been the subjects of active research since they are often amenable to exact computation. So far most of the studies have focused on loops that do not intersect. In this paper, we derive exact results for intersecting 1/8 BPS Wilson loops in N=4 supersymmetric Yang-Mills theory, using a combination of supersymmetric localization and the loop equation in 2d gauge theory. The result is given by a novel matrix-model-like representation which couples multiple contour integrals and a Gaussian matrix model. We evaluate the integral at large N, and make contact with the string worldsheet description at strong coupling. As an application of our results, we compute exactly a small-angle limit (and more generally near-BPS limits) of the cross anomalous dimension which governs the UV divergence of intersecting Wilson lines. The same quantity describes the soft anomalous dimension of scattering amplitudes of W-bosons in the Coulomb branch.

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arXiv2019

Prismatic Large $N$ Models for Bosonic Tensors

We study the $O(N)^3$ symmetric quantum field theory of a bosonic tensor $ϕ^{abc}$ with sextic interactions. Its large $N$ limit is dominated by a positive-definite operator, whose index structure has the topology of a prism. We present a large $N$ solution of the model using Schwinger-Dyson equations to sum the leading diagrams, finding that for $2.81 < d < 3$ and for $d<1.68$ the spectrum of bilinear operators has no complex scaling dimensions. We also develop perturbation theory in $3-ε$ dimensions including eight $O(N)^3$ invariant operators necessary for the renormalizability. For sufficiently large $N$, we find a "prismatic" fixed point of the renormalization group, where all eight coupling constants are real. The large $N$ limit of the resulting $ε$ expansions of various operator dimensions agrees with the Schwinger-Dyson equations. Furthermore, the $ε$ expansion allows us to calculate the $1/N$ corrections to operator dimensions. The prismatic fixed point in $3-ε$ dimensions survives down to $N\approx 53.65$, where it merges with another fixed point and becomes complex. We also discuss the $d=1$ model where our approach gives a slightly negative scaling dimension for $ϕ$, while the spectrum of bilinear operators is free of complex dimensions.

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