arXiv subjects

Diego Bombardelli

Publications and source records attributed to Diego Bombardelli.

At least 19 recordsLinked to original sources

arXiv2018

The low-energy limit of AdS$_3$/CFT$_2$ and its TBA

We investigate low-energy string excitations in AdS$_3\times$S$^3\times$T$^4$. When the worldsheet is decompactified, the theory has gapless modes whose spectrum at low energies is determined by massless relativistic integrable S matrices of the type introduced by Al. B. Zamolodchikov. The S matrices are non-trivial only for excitations with identical worldsheet chirality, indicating that the low-energy theory is a CFT$_2$. We construct a Thermodynamic Bethe Ansatz (TBA) for these excitations and show how the massless modes' wrapping effects may be incorporated into the AdS$_3$ spectral problem. Using the TBA and its associated Y-system, we determine the central charge of the low energy CFT$_2$ to be $c=6$ from calculating the vacuum energy for antiperiodic fermions - with the vacuum energy being zero for periodic fermions in agreement with a supersymmetric theory - and find the energies of some excited states.

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arXiv2018

Exploring the spectrum of planar $AdS_4/CFT_3$ at finite coupling

The Quantum Spectral Curve (QSC) equations for planar $\mathcal{N}=6$ super-conformal Chern-Simons (SCS) are solved numerically at finite values of the coupling constant for states in the $\mathfrak{sl}(2|1)$ sector. New weak coupling results for conformal dimensions of operators outside the $\mathfrak{sl}(2)$-like sector are obtained by adapting a recently proposed algorithm for the QSC perturbative solution. Besides being interesting in their own right, these perturbative results are necessary initial inputs for the numerical algorithm to converge on the correct solution. The non-perturbative numerical outcomes nicely interpolate between the weak coupling and the known semiclassical expansions, and novel strong coupling exact results are deduced from the numerics. Finally, the existence of contour crossing singularities in the TBA equations for the operator $\textbf{20}$ is ruled out by our analysis. The results of this paper are an important test of the QSC formalism for this model, open the way to new quantitative studies and provide further evidence in favour of the conjectured weak/strong coupling duality between $\mathcal{N}=6$ SCS and type IIA superstring theory on $AdS_4 \times CP^3$. Attached to the arXiv submission, a Mathematica implementation of the numerical method and ancillary files containing the numerical results are provided.

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arXiv2018

The full Quantum Spectral Curve for $AdS_4/CFT_3$

The spectrum of planar N=6 superconformal Chern-Simons theory, dual to type IIA superstring theory on $AdS_4 \times CP^3$, is accessible at finite coupling using integrability. Starting from the results of [arXiv:1403.1859], we study in depth the basic integrability structure underlying the spectral problem, the Quantum Spectral Curve. The new results presented in this paper open the way to the quantitative study of the spectrum for arbitrary operators at finite coupling. Besides, we show that the Quantum Spectral Curve is embedded into a novel kind of Q-system, which reflects the OSp(4|6) symmetry of the theory and leads to exact Bethe Ansatz equations. The discovery of this algebraic structure, more intricate than the one appearing in the $AdS_5/CFT_4$ case, could be a first step towards the extension of the method to $AdS_3/CFT_2$.

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arXiv2016

Lectures on S-matrices and Integrability

In these notes we review the S-matrix theory in (1+1)-dimensional integrable models, focusing mainly on the relativistic case. Once the main definitions and physical properties are introduced, we discuss the factorization of scattering processes due to integrability. We then focus on the analytic properties of the two-particle scattering amplitude and illustrate the derivation of the S-matrices for all the possible bound states using the so-called bootstrap principle. General algebraic structures underlying the S-matrix theory and its relation with the form factors axioms are briefly mentioned. Finally, we discuss the S-matrices of sine-Gordon and SU(2), SU(3) chiral Gross-Neveu models. This is part of a collection of lecture notes for the Young Researchers Integrability School, organized by the GATIS network at Durham University on 6-10 July 2015.

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arXiv2016

An Integrability Primer for the Gauge-Gravity Correspondence: an Introduction

We introduce a series of articles reviewing various aspects of integrable models relevant to the AdS/CFT correspondence. Topics covered in these reviews are: classical integrability, Yangian symmetry, factorized scattering, the Bethe ansatz, the thermodynamic Bethe ansatz, and integrable structures in (conformal) quantum field theory. In the present article we highlight how these concepts have found application in AdS/CFT, and provide a brief overview of the material contained in this series.

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arXiv2015

12 loops and triple wrapping in ABJM theory from integrability

Adapting a method recently proposed by C. Marboe and D. Volin for ${\cal N}$=4 super-Yang-Mills, we develop an algorithm for a systematic weak coupling expansion of the spectrum of anomalous dimensions in the $sl(2)$-like sector of planar $\mathcal{N}$=6 super-Chern-Simons. The method relies on the Quantum Spectral Curve formulation of the problem and the expansion is written in terms of the interpolating function $h(λ)$, with coefficients expressible as combinations of Euler-Zagier sums with alternating signs. We present explicit results up to 12 loops (six nontrivial orders) for various twist L=1 and L=2 operators, corresponding to triple and double wrapping terms, respectively, which are beyond the reach of the Asymptotic Bethe Ansatz as well as Lüscher's corrections. The algorithm works for generic values of L and S and in principle can be used to compute arbitrary orders of the weak coupling expansion. For the simplest operator with L=1 and spin S=1, the Padé extrapolation of the 12-loop result nicely agrees with the available Thermodynamic Bethe Ansatz data in a relatively wide range of values of the coupling. A Mathematica notebook with a selection of results is attached.

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arXiv2013

A next-to-leading Luescher formula

We propose a next-to-leading Luescher-like formula for the finite-size corrections of the excited states energies in integrable theories. We conjecture the expressions of the corrections for both the energy and the particles' rapidities by interpreting the excited states as momenta-dependent defects. We check the resulting formulas in some simple relativistic model and conjecture those for the AdS5/CFT4 case.

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arXiv2012

Exact S-matrices for AdS_3/CFT_2

We propose exact $S$-matrices for the AdS_3/CFT_2 duality between Type IIB strings on AdS_3 x S^3 x M_4 with M_4=S^3 x S^1 or T^4 and the corresponding two-dimensional conformal field theories. We fix the complete two-particle S-matrices for both those cases of AdS_3/CFT_2, on the basis of the symmetries su(1|1) and su(1|1) x su(1|1), respectively preserved by their vacua. A crucial justification comes from the derivation of the all-loop Bethe ansatz matching exactly the recent conjecture proposed by [1] and [2].

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arXiv2012

Real and Virtual Bound States in Lüscher Corrections for CP3 Magnons

We study classical and quantum finite-size corrections to giant magnons in AdS_4 x CP^3 using generalised Lüscher formulae. Lüscher F-terms are organised in powers of the exponential suppression factor exp(-Delta/2h)^m, and we calculate all terms in this series, matching one-loop algebraic curve results from our previous paper arXiv:1006.2174. Starting with the second term, the structure of these terms is different to those in AdS_5 x S^5 thanks to the appearance of heavy modes in the loop, which can here be interpreted as two-particle bound states in the mirror theory. By contrast, physical bound states can represent dyonic giant magnons, and we also calculate F-terms for these solutions. Lüscher mu-terms, suppressed by exp(-Delta/E), instead give at leading order the classical finite-size correction. For an elementary dyonic giant magnon, we recover the correction given by arXiv:0903.3365. We then extend this to calculate the next term in 1/h, giving a one-loop prediction. Finally we also calculate F-terms for the various composite giant magnons, RP^3 and `big', again finding agreement to all orders.

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arXiv2012

Finite-size effects of beta-deformed AdS_5/CFT_4 at strong coupling

We compute both classical and quantum finite-size corrections at leading order in the strong coupling limit for the (dyonic) giant magnon in the Lunin-Maldacena background. Based on the exact S-martix conjectured for the deformed theory, we generalize the Luscher formula to include twisted boundary conditions and show that the results match with those derived both by finite-size classical solutions of the giant magnon and by algebraic curve analysis.

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arXiv2011

TBA, NLO Luscher correction, and double wrapping in twisted AdS/CFT

The ground-state energy of integrably-twisted theories is analyzed in finite volume. We derive the leading and next-to-leading order (NLO) Lüscher-type corrections for large volumes of the vacuum energy for integrable theories with twisted boundary conditions and twisted S-matrix. We then derive the twisted thermodynamic Bethe ansatz (TBA) equations to describe exactly the ground state, from which we obtain an untwisted Y-system. The two approaches are compared by expanding the TBA equations to NLO, and exact agreement is found. We give explicit results for the O(4) model and for the three-parameter family of $γ$-deformed (non-supersymmetric) planar AdS/CFT model, where the ground-state energy can be nontrivial and can acquire finite-size corrections. The NLO corrections, which correspond to double-wrapping diagrams, are explicitly evaluated for the latter model at six loops.

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arXiv2011

Twisted Bethe equations from a twisted S-matrix

All-loop asymptotic Bethe equations for a 3-parameter deformation of AdS5/CFT4 have been proposed by Beisert and Roiban. We propose a Drinfeld twist of the AdS5/CFT4 S-matrix, together with c-number diagonal twists of the boundary conditions, from which we derive these Bethe equations. Although the undeformed S-matrix factorizes into a product of two su(2|2) factors, the deformed S-matrix cannot be so factored. Diagonalization of the corresponding transfer matrix requires a generalization of the conventional algebraic Bethe ansatz approach, which we first illustrate for the simpler case of the twisted su(2) principal chiral model. We also demonstrate that the same twisted Bethe equations can alternatively be derived using instead untwisted S-matrices and boundary conditions with operatorial twists.

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arXiv2010

Quantum Strings and the AdS4/CFT3 Interpolating Function

The existence of a nontrivial interpolating function h(λ) is one of the novel features of the new AdS4/CFT3 correspondence involving ABJM theory. At strong coupling, most of the investigation of semiclassical effects so far has been for strings in the AdS4 sector. Several cutoff prescriptions have been proposed, leading to different predictions for the constant term in the expansion h(λ)=\sqrt{λ/2} + c + ... . We calculate quantum corrections for giant magnons, using the algebraic curve, and show by comparing to the dispersion relation that the same prescriptions lead to the same values of c in this CP3 sector. We then turn to finite-J effects, where a comparison with the Luescher F-term correction shows a mismatch for one of the three sum prescriptions. We also compute some dyonic and higher F-terms for future comparisons.

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arXiv2010

Finite-size effect for four-loop Konishi of the beta-deformed N=4 SYM

We propose that certain twists of the su(2|2) S-matrix elements describe the beta-deformation of N=4 supersymmetric Yang-Mills theory. We compute the perturbative four-loop anomalous dimension of the Konishi operator of the deformed gauge theory from the Luscher formula based on these twisted S-matrix elements. The result agrees exactly with the perturbative gauge theory computations.

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arXiv2009

TBA and Y-system for planar $AdS_4/CFT_3$

We conjecture the set of asymptotic Bethe Ansatz equations for the {\it mirror} model of the $\text{AdS}_4\times\mathbb{CP}^3$ string theory, corresponding to the planar $\mathcal{N}=6$ superconformal Chern-Simons gauge theory in three dimensions. Hence, we derive the (vacuum energy) thermodynamic Bethe Ansatz equations and the Y-system describing the {\it direct} $\text{AdS}_4/\text{CFT}_3$ string theory.

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arXiv2009

Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal

Moving from the mirror theory Bethe-Yang equations proposed by Arutyunov and Frolov, we derive the thermodynamic Bethe Ansatz equations which should control the spectrum of the planar $\text{AdS}_5/\text{CFT}_4$ correspondence. The associated set of universal functional relations (Y-system) satisfied by the exponentials of the TBA pseudoenergies is deduced, confirming the structure inferred by Gromov, Kazakov and Vieira.

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arXiv2009

Finite-Size Corrections of the $\mathbb{CP}^3$ Giant Magnons: the Lüscher terms

We compute classical and first quantum finite-size corrections to the recently found giant magnon solutions in two different subspaces of $\mathbb{CP}^3$. We use the Lüscher approach on the recently proposed exact S-matrix for $\mathcal{N}=6$ superconformal Chern-Simons theory. We compare our results with the string and algebraic curve computations and find agreement, thus providing a non-trivial test for the new $AdS_4/CFT_3$ correspondence within an integrability framework.

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arXiv2008

Large spin corrections in ${\cal N}=4$ SYM sl(2): still a linear integral equation

Anomalous dimension and higher conserved charges in the $sl(2)$ sector of ${\cal N}=4$ SYM for generic spin $s$ and twist $L$ are described by using a novel kind of non-linear integral equation (NLIE). The latter can be derived under typical situations of the SYM sectors, i.e. when the scattering need not depend on the difference of the rapidities and these, in their turn, may also lie on a bounded range. Here the non-linear (finite range) integral terms, appearing in the NLIE and in the dimension formula, go to zero as $s\to \infty$. Therefore they can be neglected at least up to the $O(s^0)$ order, thus implying a linear integral equation (LIE) and a linear dimension/charge formula respectively, likewise the 'thermodynamic' (i.e. infinite spin) case. Importantly, these non-linear terms go faster than any inverse logarithm power $(\ln s)^{-n}$, $n>0$, thus extending the linearity validity.

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