Andrea Cavaglià
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Andrea Cavaglià has a total of 33 co-authors. Here are the top co-authors by number of shared publications:
- Nikolay Gromov
Department of Mathematics King's College London The Strand WC2R 2LS London UK
19 shared publications - Roberto Tateo
Dipartimento di Fisica, Università di Torino and INFN, Sezione di Torino
15 shared publications - Michelangelo Preti
Nordita, KTH Royal Institute of Technology and Stockholm University, Stockholm, Sweden
9 shared publications - Julius Julius
Homi Bhabha National Institute
7 shared publications - Diego Bombardelli
Dipartimento di Fisica and INFN Università di Torino Via P. Giuria 1 10125 Torino Italy
6 shared publications - Fedor Levkovich-Maslyuk
University of London
4 shared publications
Latest Publications
- Open Access
- 25 May 2026
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2026 (5) , 270
On the quantum spectral curve for AdS3 × S3 × S3 × S1 strings and the 𝔡(2, 1; α) Q-system
A bstract: In this paper, we put forward and discuss a proposal for a Quantum Spectral Curve (QSC) describing the planar spectrum of the holographic CFT dual to strings on AdS 3 × S 3 × S 3 × S 1 , a theory with global symmetry 𝔡(2 , 1; α ) ⊕2 . We focus mainly on the case when the radii of the two spheres are the same, i.e. α = 1 / 2, where the symmetry reduces to 𝔬𝔰𝔭(4|2) ⊕2 . In this case, our proposal is based on two copies of an 𝔬𝔰𝔭(4|2) Q-system, glued through the branch cuts of the Q-functions in a minimal way. We study in detail the ensuing analytic properties of the Q-functions in this proposal. Focusing on purely massive excitations, we consider the large worldsheet limit in which the QSC leads to a set of Asymptotic Bethe Ansatz (ABA) equations, yielding strong constraints on the (so-far unfixed) dressing factors of the worldsheet S-matrix. In a ℤ 2 -symmetric sector, our proposal is consistent with all previous results on the worldsheet S-matrix. However, in the non-symmetric case, we found a subtle incompatibility between the analytic constraints arising from the proposed QSC, the crossing equations present in the literature, and braiding unitarity. We discuss possible explanations for this mismatch: either our minimal QSC proposal does not hold beyond the symmetric sector, or the crossing unitarity equations receive a nontrivial correction that needs to be understood. Finally, we also propose a generalisation of the Q-system for the case of α ≠ 1 / 2, corresponding to the superalgebra 𝔡(2 , 1; α ). This novel algebraic structure represents a significant step towards understanding the Quantum Spectral Curve of the entire theory.
- 25 May 2026
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2026 (5) , 270
Open AccessOn the quantum spectral curve for AdS3 × S3 × S3 × S1 strings and the 𝔡(2, 1; α) Q-system
A bstract: In this paper, we put forward and discuss a proposal for a Quantum Spectral Curve (QSC) describing the planar spectrum of the holographic CFT dual to strings on AdS 3 × S 3 × S 3 × S 1 , a theory with global symmetry 𝔡(2 , 1; α ) ⊕2 . We focus mainly on the case when the radii of the two spheres are the same, i.e. α = 1 / 2, where the symmetry reduces to 𝔬𝔰𝔭(4|2) ⊕2 . In this case, our proposal is based on two copies of an 𝔬𝔰𝔭(4|2) Q-system, glued through the branch cuts of the Q-functions in a minimal way. We study in detail the ensuing analytic properties of the Q-functions in this proposal. Focusing on purely massive excitations, we consider the large worldsheet limit in which the QSC leads to a set of Asymptotic Bethe Ansatz (ABA) equations, yielding strong constraints on the (so-far unfixed) dressing factors of the worldsheet S-matrix. In a ℤ 2 -symmetric sector, our proposal is consistent with all previous results on the worldsheet S-matrix. However, in the non-symmetric case, we found a subtle incompatibility between the analytic constraints arising from the proposed QSC, the crossing equations present in the literature, and braiding unitarity. We discuss possible explanations for this mismatch: either our minimal QSC proposal does not hold beyond the symmetric sector, or the crossing unitarity equations receive a nontrivial correction that needs to be understood. Finally, we also propose a generalisation of the Q-system for the case of α ≠ 1 / 2, corresponding to the superalgebra 𝔡(2 , 1; α ). This novel algebraic structure represents a significant step towards understanding the Quantum Spectral Curve of the entire theory.
- Open Access
- 27 February 2026
- Published by Stichting SciPost in SciPost Physics
- Vol. 20 (2) , 064
Orthogonality of Q-functions up to wrapping in planar $\mathcal{N}=4$ super Yang-Mills theoryWe construct orthogonality relations in the Separation of Variables framework for the \mathfrak{sl}(2) ???? ???? ( 2 ) sector of planar \mathcal{N}=4 ???? = 4 supersymmetric Yang-Mills theory. Specifically, we find simple universal measures that make Q-functions of operators with different spins vanish at all orders in perturbation theory, prior to wrapping corrections. To analyze this rank-one sector, we relax some of the assumptions thus far considered in the Separation of Variables framework. Our findings may serve as guidelines for extending this formalism to other sectors of the theory as well as other integrable models.
- 27 February 2026
- Published by Stichting SciPost in SciPost Physics
- Vol. 20 (2) , 064
Open AccessOrthogonality of Q-functions up to wrapping in planar $\mathcal{N}=4$ super Yang-Mills theoryWe construct orthogonality relations in the Separation of Variables framework for the \mathfrak{sl}(2) ???? ???? ( 2 ) sector of planar \mathcal{N}=4 ???? = 4 supersymmetric Yang-Mills theory. Specifically, we find simple universal measures that make Q-functions of operators with different spins vanish at all orders in perturbation theory, prior to wrapping corrections. To analyze this rank-one sector, we relax some of the assumptions thus far considered in the Separation of Variables framework. Our findings may serve as guidelines for extending this formalism to other sectors of the theory as well as other integrable models.
- Open Access
- 10 July 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (7) , 1-61
Regge trajectories and bridges between them in integrable AdS/CFT
A bstract: We study the analytic continuation in the spin of the planar spectrum of ABJM theory using the integrability-based Quantum Spectral Curve (QSC) method. Under some minimal assumptions, we classify the analytic properties of the Q-functions appearing in the QSC compatible with the spin being non-integer. In this way, we find not one but two distinct possibilities. While one is related to standard Regge trajectories, we show that the second choice can be used to build bridges which connect leading and subleading Regge trajectories, thus giving a shortcut to reach infinitely many sheets of the spin Riemann surface without going explicitly around the branch points in the complex spin plane. Moreover, the bridges are exact spin reflections of standard Regge trajectories. Together, these results reveal the existence of a hidden symmetry which we call “twist/co-twist symmetry”: every non-BPS local operator has an exact image — living below unitarity on a different Regge trajectory — with the same ∆ and the spin flipped by a Weyl reflection. We discuss how an analogous phenomenon, based on the same mechanism at the level of the QSC, also occurs at non-perturbative level in $$ \mathcal{N} $$ =4 SYM. This provides a framework to understand recent independent observations of the symmetry in this model at weak and strong coupling. We present numerical results for Regge trajectories in planar ABJM theory, in particular we compute exactly the coupling dependence of the position of the leading Regge pole in the correlator of four stress tensors. The shape of this leading trajectory shows a behaviour at weak coupling that strongly resembles the BFKL limit in $$ \mathcal{N} $$ =4 SYM.- 10 July 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (7) , 1-61
Open AccessRegge trajectories and bridges between them in integrable AdS/CFT
A bstract: We study the analytic continuation in the spin of the planar spectrum of ABJM theory using the integrability-based Quantum Spectral Curve (QSC) method. Under some minimal assumptions, we classify the analytic properties of the Q-functions appearing in the QSC compatible with the spin being non-integer. In this way, we find not one but two distinct possibilities. While one is related to standard Regge trajectories, we show that the second choice can be used to build bridges which connect leading and subleading Regge trajectories, thus giving a shortcut to reach infinitely many sheets of the spin Riemann surface without going explicitly around the branch points in the complex spin plane. Moreover, the bridges are exact spin reflections of standard Regge trajectories. Together, these results reveal the existence of a hidden symmetry which we call “twist/co-twist symmetry”: every non-BPS local operator has an exact image — living below unitarity on a different Regge trajectory — with the same ∆ and the spin flipped by a Weyl reflection. We discuss how an analogous phenomenon, based on the same mechanism at the level of the QSC, also occurs at non-perturbative level in $$ \mathcal{N} $$ =4 SYM. This provides a framework to understand recent independent observations of the symmetry in this model at weak and strong coupling. We present numerical results for Regge trajectories in planar ABJM theory, in particular we compute exactly the coupling dependence of the position of the leading Regge pole in the correlator of four stress tensors. The shape of this leading trajectory shows a behaviour at weak coupling that strongly resembles the BFKL limit in $$ \mathcal{N} $$ =4 SYM. - Open Access
- 17 June 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (6) , 1-60
Probing line defect CFT with mixed-correlator bootstrability
We continue our study of the defect CFT on a Maldacena-Wilson line in $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory using Bootstrability — the conformal bootstrap supplemented with exact integrability data. In this paper, we extend this program to charged sectors of the theory, considering a mixed-correlator setup first introduced by Liendo, Meneghelli, and Mitev. The exact spectrum in all channels is given by integrability at any coupling. Additionally, we use exact expressions for some structure constants fixed by localisation and leverage the exact discrete symmetries of the theory. We analyse the remaining data with the numerical bootstrap, developing an algorithm optimised to scan over a large multidimensional space of OPE coefficients and carve the allowed region with the “cutting surface” procedure. We compute upper and lower bounds for 12 OPE coefficients for several values of the coupling. Our results are sharp for the lowest states in each sector but become quite wide for the excited states due to their near degeneracy. This highlights the need for studying the system with non-protected external states and for further input from integrability in the form of integrated correlators of non-protected operators.- 17 June 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (6) , 1-60
Open AccessProbing line defect CFT with mixed-correlator bootstrability
We continue our study of the defect CFT on a Maldacena-Wilson line in $$ \mathcal{N} $$ = 4 Super-Yang-Mills theory using Bootstrability — the conformal bootstrap supplemented with exact integrability data. In this paper, we extend this program to charged sectors of the theory, considering a mixed-correlator setup first introduced by Liendo, Meneghelli, and Mitev. The exact spectrum in all channels is given by integrability at any coupling. Additionally, we use exact expressions for some structure constants fixed by localisation and leverage the exact discrete symmetries of the theory. We analyse the remaining data with the numerical bootstrap, developing an algorithm optimised to scan over a large multidimensional space of OPE coefficients and carve the allowed region with the “cutting surface” procedure. We compute upper and lower bounds for 12 OPE coefficients for several values of the coupling. Our results are sharp for the lowest states in each sector but become quite wide for the excited states due to their near degeneracy. This highlights the need for studying the system with non-protected external states and for further input from integrability in the form of integrated correlators of non-protected operators. - Top Cited PapersOpen Access
- 6 February 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (2) , 1-39
Computing four-point functions with integrability, bootstrap and parity symmetry
The combination of integrability and crossing symmetry has proven to give tight non-perturbative bounds on some planar structure constants in $$ \mathcal{N} $$ =4 SYM, particularly in the setup of defect observables built on a Wilson-Maldacena line. Whereas the precision is good for the low lying states, higher in the spectrum it drops due to the degeneracies at weak coupling when considering a single correlator. As this could be a clear obstacle in restoring higher point functions, we studied the problem of bounding directly a 4-point function at generic cross ratio, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. Another tool we are using to further narrow the bounds is a parity symmetry descending from the $$ \mathcal{N} $$ =4 SYM theory, which allowed us to reduce the number of parameters. We also give an interpretation for the parity in terms of the Quantum Spectral Curve at weak coupling. Our numerical bounds give an accurate determination of the 4-point function for physical values of the cross ratio, with at worst 5-6 digits precision at weak coupling and reaching more than 11 digits for ’t Hooft coupling $$ \frac{\sqrt{\lambda }}{4\pi}\sim 4 $$ .- 6 February 2025
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2025 (2) , 1-39
Top Cited PapersOpen AccessComputing four-point functions with integrability, bootstrap and parity symmetry
The combination of integrability and crossing symmetry has proven to give tight non-perturbative bounds on some planar structure constants in $$ \mathcal{N} $$ =4 SYM, particularly in the setup of defect observables built on a Wilson-Maldacena line. Whereas the precision is good for the low lying states, higher in the spectrum it drops due to the degeneracies at weak coupling when considering a single correlator. As this could be a clear obstacle in restoring higher point functions, we studied the problem of bounding directly a 4-point function at generic cross ratio, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. Another tool we are using to further narrow the bounds is a parity symmetry descending from the $$ \mathcal{N} $$ =4 SYM theory, which allowed us to reduce the number of parameters. We also give an interpretation for the parity in terms of the Quantum Spectral Curve at weak coupling. Our numerical bounds give an accurate determination of the 4-point function for physical values of the cross ratio, with at worst 5-6 digits precision at weak coupling and reaching more than 11 digits for ’t Hooft coupling $$ \frac{\sqrt{\lambda }}{4\pi}\sim 4 $$ . - Open Access
Computing Four-Point Functions with Integrability, Bootstrap and Parity Symmetry
The combination of integrability and crossing symmetry has proven to give tight non-perturbative bounds on some planar structure constants in $\mathcal{N}$=4 SYM, particularly in the setup of defect observables built on a Wilson-Maldacena line. Whereas the precision is good for the low lying states, higher in the spectrum it drops due to the degeneracies at weak coupling when considering a single correlator. As this could be a clear obstacle in restoring higher point functions, we studied the problem of bounding directly a 4-point function at generic cross ratio, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. Another tool we are using to further narrow the bounds is a parity symmetry descending from the $\mathcal{N}$=4 SYM theory, which allowed us to reduce the number of parameters. We also give an interpretation for the parity in terms of the Quantum Spectral Curve at weak coupling. Our numerical bounds give an accurate determination of the 4-point function for physical values of the cross ratio, with at worst 5-6 digits precision at weak coupling and reaching more than 11 digits for 't Hooft coupling $\frac{\sqrtλ}{4 π} \sim 4$.Open AccessComputing Four-Point Functions with Integrability, Bootstrap and Parity Symmetry
The combination of integrability and crossing symmetry has proven to give tight non-perturbative bounds on some planar structure constants in $\mathcal{N}$=4 SYM, particularly in the setup of defect observables built on a Wilson-Maldacena line. Whereas the precision is good for the low lying states, higher in the spectrum it drops due to the degeneracies at weak coupling when considering a single correlator. As this could be a clear obstacle in restoring higher point functions, we studied the problem of bounding directly a 4-point function at generic cross ratio, showing how to adapt for this purpose the numerical bootstrap algorithms based on semidefinite programming. Another tool we are using to further narrow the bounds is a parity symmetry descending from the $\mathcal{N}$=4 SYM theory, which allowed us to reduce the number of parameters. We also give an interpretation for the parity in terms of the Quantum Spectral Curve at weak coupling. Our numerical bounds give an accurate determination of the 4-point function for physical values of the cross ratio, with at worst 5-6 digits precision at weak coupling and reaching more than 11 digits for 't Hooft coupling $\frac{\sqrtλ}{4 π} \sim 4$. - Open Access
- 13 December 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (12) , 1-38
Exploring the Quantum Spectral Curve for AdS3/CFT2
Despite the rich and fruitful history of the integrability approach to string theory on the AdS3× S3× T4background, it has not been possible to extract many concrete predictions from integrability, except in a strict asymptotic regime of large quantum numbers, due to the severity of wrapping effects. The situation changed radically with two independent and identical proposals for the Quantum Spectral Curve (QSC) for this system in a background of pure Ramond-Ramond flux. In other integrable superstring backgrounds there is compelling evidence that this formulation captures all wrapping effects exactly and describes the full planar spectrum. This great success motivates us to study the new proposed QSC and develop methods to extract from it concrete predictions for spectral data. The AdS3× S3× T4case presents a significant novel feature and challenge compared to its higher-dimensional analogues — massless modes. It has been conjectured that these manifest themselves in a new property of this QSC: the non-quadratic nature of the branch-cut singularities of the QSC Q-functions. This feature implies new technical challenges in solving the QSC equations as compared to the well-studied case of $$ \mathcal{N} $$ = 4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for unprotected string excitations in the planar limit with finite quantum numbers and RR flux. We explain how to extract a systematic expansion around the analogue of the weak ’t Hooft coupling limit in $$ \mathcal{N} $$ = 4 SYM and also obtain high-precision numerical results. These concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT.- 13 December 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (12) , 1-38
Open AccessExploring the Quantum Spectral Curve for AdS3/CFT2
Despite the rich and fruitful history of the integrability approach to string theory on the AdS3× S3× T4background, it has not been possible to extract many concrete predictions from integrability, except in a strict asymptotic regime of large quantum numbers, due to the severity of wrapping effects. The situation changed radically with two independent and identical proposals for the Quantum Spectral Curve (QSC) for this system in a background of pure Ramond-Ramond flux. In other integrable superstring backgrounds there is compelling evidence that this formulation captures all wrapping effects exactly and describes the full planar spectrum. This great success motivates us to study the new proposed QSC and develop methods to extract from it concrete predictions for spectral data. The AdS3× S3× T4case presents a significant novel feature and challenge compared to its higher-dimensional analogues — massless modes. It has been conjectured that these manifest themselves in a new property of this QSC: the non-quadratic nature of the branch-cut singularities of the QSC Q-functions. This feature implies new technical challenges in solving the QSC equations as compared to the well-studied case of $$ \mathcal{N} $$ = 4 SYM. In this paper we resolve these difficulties and obtain the first ever predictions for unprotected string excitations in the planar limit with finite quantum numbers and RR flux. We explain how to extract a systematic expansion around the analogue of the weak ’t Hooft coupling limit in $$ \mathcal{N} $$ = 4 SYM and also obtain high-precision numerical results. These concrete data and others obtainable from the QSC could help to identify the so-far mysterious dual CFT. - Open Access
- 5 April 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (4) , 1-46
Integrated correlators from integrability: Maldacena-Wilson line in $ \mathcal{N} $ = 4 SYMWe present a systematic method for the derivation of a relation which connects the correlation function of operators on the straight Maldacena-Wilson line with the integrability data for the cusp anomalous dimension. As we show, the derivation requires very careful treatment of the UV divergences. Our method opens a way to derive infinitely many constraints on integrals of multi-point correlation functions, relating them with the integrability data for the generalised cusp anomalous dimension governed by the Quantum Spectral Curve. Such constraints have been shown recently to be very powerful in combination with the numerical conformal bootstrap, leading to very narrow non-perturbative bounds on conformal data beyond the spectrum.
- 5 April 2023
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2023 (4) , 1-46
Open AccessIntegrated correlators from integrability: Maldacena-Wilson line in $ \mathcal{N} $ = 4 SYMWe present a systematic method for the derivation of a relation which connects the correlation function of operators on the straight Maldacena-Wilson line with the integrability data for the cusp anomalous dimension. As we show, the derivation requires very careful treatment of the UV divergences. Our method opens a way to derive infinitely many constraints on integrals of multi-point correlation functions, relating them with the integrability data for the generalised cusp anomalous dimension governed by the Quantum Spectral Curve. Such constraints have been shown recently to be very powerful in combination with the numerical conformal bootstrap, leading to very narrow non-perturbative bounds on conformal data beyond the spectrum.
- Open AccessIntegrated correlators from integrability: Maldacena-Wilson line in $\mathcal{N}=4$ SYM
We present a systematic method for the derivation of a relation which connects the correlation function of operators on the straight Maldacena-Wilson line with the integrability data for the cusp anomalous dimension. As we show, the derivation requires very careful treatment of the UV divergences. Our method opens a way to derive infinitely many constraints on integrals of multi-point correlation functions, relating them with the integrability data for the generalised cusp anomalous dimension governed by the Quantum Spectral Curve. Such constraints have been shown recently to be very powerful in combination with the numerical conformal bootstrap, leading to very narrow non-perturbative bounds on conformal data beyond the spectrum.
Open AccessIntegrated correlators from integrability: Maldacena-Wilson line in $\mathcal{N}=4$ SYMWe present a systematic method for the derivation of a relation which connects the correlation function of operators on the straight Maldacena-Wilson line with the integrability data for the cusp anomalous dimension. As we show, the derivation requires very careful treatment of the UV divergences. Our method opens a way to derive infinitely many constraints on integrals of multi-point correlation functions, relating them with the integrability data for the generalised cusp anomalous dimension governed by the Quantum Spectral Curve. Such constraints have been shown recently to be very powerful in combination with the numerical conformal bootstrap, leading to very narrow non-perturbative bounds on conformal data beyond the spectrum.
- Open Access
- 24 May 2022
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2022 (5) , 1-59
Bootstrability in defect CFT: integrated correlators and sharper bounds
We continue to develop Bootstrability — a method merging Integrability and Conformal Bootstrap to extract CFT data in integrable conformal gauge theories such as $$ \mathcal{N} $$ = 4 SYM. In this paper, we consider the 1D defect CFT defined on a $$ \frac{1}{2} $$ -BPS Wilson line in the theory, whose non-perturbative spectrum is governed by the Quantum Spectral Curve (QSC). In addition, we use that the deformed setup of a cusped Wilson line is also controlled by the QSC. In terms of the defect CFT, this translates into two nontrivial relations connecting integrated 4-point correlators to cusp spectral data, such as the Bremsstrahlung and Curvature functions — known analytically from the QSC. Combining these new constraints and the spectrum of the 10 lowest-lying states with the Numerical Conformal Bootstrap, we obtain very sharp rigorous numerical bounds for the structure constant of the first non-protected state, giving this observable with seven digits precision for the ’t Hooft coupling in the intermediate coupling region $$ \frac{\sqrt{\lambda }}{4\pi}\sim 1 $$ , with the error decreasing quickly at large ’t Hooft coupling. Furthermore, for the same structure constant we obtain a 4-loop analytic result at weak coupling. We also present results for excited states.- 24 May 2022
- Published by Springer Nature in Journal of High Energy Physics
- Vol. 2022 (5) , 1-59
Open AccessBootstrability in defect CFT: integrated correlators and sharper bounds
We continue to develop Bootstrability — a method merging Integrability and Conformal Bootstrap to extract CFT data in integrable conformal gauge theories such as $$ \mathcal{N} $$ = 4 SYM. In this paper, we consider the 1D defect CFT defined on a $$ \frac{1}{2} $$ -BPS Wilson line in the theory, whose non-perturbative spectrum is governed by the Quantum Spectral Curve (QSC). In addition, we use that the deformed setup of a cusped Wilson line is also controlled by the QSC. In terms of the defect CFT, this translates into two nontrivial relations connecting integrated 4-point correlators to cusp spectral data, such as the Bremsstrahlung and Curvature functions — known analytically from the QSC. Combining these new constraints and the spectrum of the 10 lowest-lying states with the Numerical Conformal Bootstrap, we obtain very sharp rigorous numerical bounds for the structure constant of the first non-protected state, giving this observable with seven digits precision for the ’t Hooft coupling in the intermediate coupling region $$ \frac{\sqrt{\lambda }}{4\pi}\sim 1 $$ , with the error decreasing quickly at large ’t Hooft coupling. Furthermore, for the same structure constant we obtain a 4-loop analytic result at weak coupling. We also present results for excited states.