arXiv subjects

Riccardo Borsato

Publications and source records attributed to Riccardo Borsato.

At least 19 recordsLinked to original sources

arXiv2026

A new perspective on non-commutative deformations of field and gauge theories

We construct non-commutative deformations of field and gauge theories based on star-products implemented by Drinfel'd twists. We are able to encompass a large family of twists, including those built out of conformal symmetries and supersymmetries. The main idea behind our construction is to work with twists constructed from symmetries of the undeformed theory, that are realised as active symmetry transformations. We argue that our construction amounts to a reformulation of known deformations of gauge theories, and that it significantly extends the range of applicable examples. To ensure consistency with gauge invariance, we also identify a unimodularity condition that is weaker than the one that is normally employed in the literature, so that we can apply twists that would otherwise be left out. Finally, we also prove a planar equivalence theorem stating that the Feynman diagrams of the deformed theories retain an undeformed internal structure, with the twist acting only on their external legs. All these results are important to identify and work with deformations of $\mathcal N=4$ super Yang-Mills that are proposed to be dual to homogeneous Yang-Baxter deformations of the $AdS_5\times S^5$ superstring, but the applicability of our construction and results goes beyond that.

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arXiv2026

Groenewold-Moyal twists, integrable spin-chains and AdS/CFT

We take the first steps to address via integrability the spectral problem of AdS/CFT dual pairs deformed by Groenewold-Moyal twists. In particular, we start by considering a twisted spin-chain that couples, through a Groenewold-Moyal twist deformation, two $\mathfrak{sl}(2)$-invariant spin-chains. We interpret this deformed spin-chain as a deformation of a subsector of the $AdS_3/CFT_2$ spin-chain, but the construction shares qualitative features also with the corresponding deformation of the $AdS_5/CFT_4$ spin-chain, for example. As in similar types of deformations, we show that there exists a certain basis in which the spin-chain Hamiltonian takes a Jordan-block form. At the same time, by working in the basis of eigenstates of the generators used to construct the Groenewold-Moyal twist, the Hamiltonian appears to be diagonalisable and with a deformed spectrum. Employing the method of the Baxter equation, we write down the energy of the ground state and of excited states in a perturbation of the deformation parameter. We then consider the string-theory side of the duality, where the twist is realised as a deformation of AdS of the type of Maldacena-Russo-Hashimoto-Itzhaki. We construct a deformation of the usual BMN classical solution, and in the large-$J$ limit we match the leading $\mathcal O(J^{-3})$ term of the energy of the spin-chain groundstate with a conserved charge of the string classical solution. Differently from the undeformed setup as well as similar kinds of deformations, we find that the general expression of this charge of the string sigma-model is non-local, and that it does not correspond to a standard isometry. Nevertheless, it can be computed from the monodromy matrix and it is part of the tower of conserved charges provided by integrability.

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arXiv2025

Non-commutative deformations of gauge theories via Drinfel'd twists of the scale symmetry

In this paper we consider gauge theories that are relativistic and scale-invariant, and we construct their deformed versions via suitable star products. In particular, the non-commutative structure is controlled by Drinfel'd twists that are built out of symmetry generators that include the scale transformation. To achieve this, we construct a twisted differential calculus that allows us to identify the proper gauge-covariant quantities. We also show that our construction is equivalent to twists where the symmetry generators are implemented as active transformations of fields. As a consequence of our construction, the deformed gauge theories possess a twisted version of the original symmetry group. Moreover, at the planar level, the deformation is encoded just on the external legs of Feynman diagrams, leaving then the amputated diagrams undeformed. This work extends previous constructions and allows us to define twist-deformations of $\mathcal N=4$ super Yang-Mills that are conjectured to be holographically dual to a class of homogeneous Yang-Baxter deformations of $AdS_5\times S^5$.

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arXiv2025

Jordanian deformation of the non-compact and $\mathfrak{sl}_2 $-invariant $XXX_{-1/2}$ spin-chain

Using a Drinfeld twist of Jordanian type, we construct a deformation of the non-compact and $\mathfrak{sl}_2$-invariant $XXX_{-1/2}$ spin-chain. Before the deformation, the seed model can be understood as a sector of the $\mathfrak{psu}(2,2|4)$-invariant spin-chain encoding the spectral problem of $\mathcal{N}=4$ super Yang-Mills at one loop in the planar limit. The deformation gives rise to interesting features because, while being integrable, the Hamiltonian is non-hermitian and non-diagonalisable, so that it only admits a Jordan decomposition. Moreover, the eigenvalues of the deformed Hamiltonian coincide with those of the original undeformed spin-chain. We use explicit examples as well as the techniques of the coordinate and of the algebraic Bethe ansatz to discuss the construction of the (generalised) eigenvectors of the deformed model. We also show that the deformed spin-chain is equivalent to an undeformed one with twisted boundary conditions, and that it may be derived from a scaling limit of the non-compact $U_q(\mathfrak{sl}_2)$-invariant $XXZ_{-1/2} $ spin-chain.

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arXiv2025

Particle production in a light-cone gauge fixed Jordanian deformation of $AdS_5\times S^5$

We consider a string on a Jordanian deformation of the $AdS_5\times S^5$ spacetime. This model belongs to the larger class of Homogeneous Yang-Baxter deformations, which preserve classical integrability in the sense that one can construct an explicit Lax connection. To study the scattering of bosonic worldsheet excitations, we fix light-cone gauge and expand around a pointlike classical solution that reduces to the BMN vacuum in the undeformed limit. Our analysis shows that the light-cone gauge-fixed Hamiltonian, under a perturbative field expansion, includes cubic terms that give rise to non-trivial cubic processes for physical particles. We discuss this unexpected result in relation to the property of Lax integrability of the sigma-model.

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arXiv2025

Integrability of the $λ$-deformation of the PCM with spectators

We construct a generalisation of the $λ$-deformation of the Principal Chiral Model (PCM) where we deform just a subgroup $F$ of the full symmetry group $G$. We find that demanding Lax integrability imposes a crucial restriction, namely that the coset $F\backslash G$ must be symmetric. Surprisingly, we also find that (when $F$ is non-abelian) integrability requires that the term in the action involving only the spectator fields should have a specific $λ$-dependence, which is a curious modification of the procedure expected from the known $F=G$ case. The resulting Lax connection has a novel analytical structure, with four single poles as opposed to the two poles of the cases of the PCM and of the standard $λ$-deformation. We also explicitly work out the example of $G=SU(2)$ and $F=U(1)$, discussing its renormalisation group flow to two loops.

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arXiv2024

Lecture notes on current-current deformations

These are pedagogical lecture notes discussing current-current deformations of 2-dimensional field theories. The deformations that are considered here are generated infinitesimally by bilinears of Noether currents corresponding to internal global symmetries of the "seed" theory. When the seed theory is conformal, these deformations are marginal and are often known as $J\bar J$-deformations. In this context, we review the criterion for marginal operators due to Chaudhuri and Schwartz. When the seed theory is an integrable $σ$-model (in the sense that it possesses a Lax connection), these deformations preserve the integrability. Here we review this fact by viewing the deformations as maps that leave the equations of motion and the Poisson brackets of the 2-dimensional $σ$-models invariant. The reinterpretation as undeformed theories with twisted boundary conditions is also discussed, as well as the effect of the deformation at the level of the S-matrix of the quantum theory. The finite (or integrated) form of the deformations is equivalent to sequences of T-duality--shift--T-duality transformations (TsT's), and here we review the $O(d,d)$-covariant formalism that is useful to describe them. The presentation starts with pedagogical examples of deformations of free massless scalars in 2 dimensions, and minimal prerequisites on conformal field theories or integrability are needed to understand later sections. Moreover, guided exercises are proposed to the reader. These notes were prepared for the Young Researchers Integrability School and Workshop (YRISW) held in Durham from 17 to 21 July 2023.

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arXiv2024

Inequivalent light-cone gauge-fixings of strings on $AdS_n \times S^n$ backgrounds

Light-cone gauge-fixed sigma-models on $AdS_n\times S^n$ backgrounds play an important role in the integrability formulation of the AdS/CFT correspondence. The string spectrum of the sigma-model is gauge-independent, however the Hamiltonian and scattering matrix of the transverse worldsheet fields are not. We study how these change for a large family of inequivalent light-cone gauges, which are interpreted as $T\bar{T}$, $\tilde{J}T_τ$, $JT_σ$ and $J^τ$ deformations. We investigate the moduli space of equivalent light-cone gauges and, specialising to $AdS_5 \times S^5$, compute the different light-cone gauge symmetry algebras, well-known to be $\mathfrak{psu}(2|2)^{\oplus 2} \oplus \mathfrak{u}(1)^{\oplus 2}$ for the standard gauge-fixing. Many integrable deformations require a non-standard light-cone gauge, hence our classification and analysis of inequivalent gauges will be important for analysing such models.

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arXiv2023

All Jordanian deformations of the $AdS_5 \times S^5$ superstring

We explicitly construct and classify all Jordanian solutions of the classical Yang-Baxter equation on $\mathfrak{psu}(2,2|4)$, corresponding to Jordanian Yang-Baxter deformations of the $AdS_5\times S^5$ superstring. Such deformations preserve the classical integrability of the underlying sigma-model and thus are a subclass of all possible integrable deformations. The deformations that we consider are divided into two families, unimodular and non-unimodular ones. The former ensure that the deformed backgrounds are still solutions of the type IIB supergravity equations. For the simplest unimodular solutions, we find that the corresponding backgrounds preserve a number $N<32$ of supercharges that can be $N=12,8,6,4,0$.

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arXiv2023

Semiclassical spectrum of a Jordanian deformation of $AdS_5 \times S^5$

We study a Jordanian deformation of the $AdS_5 \times S^5$ superstring that preserves 12 superisometries. It is an example of homogeneous Yang-Baxter deformations, a class that generalises TsT deformations to the non-abelian case. Many of the attractive features of TsT carry over to this more general class, from the possibility of generating new supergravity solutions to the preservation of worldsheet integrability. In this paper, we exploit the fact that the deformed $σ$-model with periodic boundary conditions can be reformulated as an undeformed one with twisted boundary conditions, to discuss the construction of the classical spectral curve and its semi-classical quantisation. First, we find global coordinates for the deformed background, and identify the global time corresponding to the energy that should be computed in the spectral problem. Using the curve of the twisted model, we obtain the one-loop correction to the energy of a particular solution, and we find that the charge encoding the twisted boundary conditions does not receive an anomalous correction. Finally, we give evidence suggesting that the unimodular version of the deformation (giving rise to a supergravity background) and the non-unimodular one (whose background does not solve the supergravity equations) have the same spectrum at least to one-loop.

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arXiv2022

On the Classical Integrability of Root-$T \overline{T}$ Flows

The Root-$T \overline{T}$ flow was recently introduced as a universal and classically marginal deformation of any two-dimensional translation-invariant field theory. The flow commutes with the (irrelevant) $T \overline{T}$ flow and it can be integrated explicitly for a large class of actions, leading to non-analytic Lagrangians reminiscent of the four-dimensional Modified-Maxwell theory (ModMax). It is not a priori obvious whether the Root-$T \overline{T}$ flow preserves integrability, like it is the case for the $T \overline{T}$ flow. In this paper we demonstrate that this is the case for a large class of classical models by explicitly constructing a deformed Lax connection. We discuss the principal chiral model and the non-linear sigma models on symmetric and semi-symmetric spaces, without or with Wess-Zumino term. We also construct Lax connections for the two-parameter families of theories deformed by both Root-$T \overline{T}$ and $T \overline{T}$ for all of these models.

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arXiv2022

Homogeneous Yang-Baxter deformations as undeformed yet twisted models

The homogeneous Yang-Baxter deformation is part of a larger web of integrable deformations and dualities that recently have been studied with motivations in integrable $σ$-models, solution-generating techniques in supergravity and Double Field Theory, and possible generalisations of the AdS/CFT correspondence. The $σ$-models obtained by the homogeneous Yang-Baxter deformation with periodic boundary conditions on the worldsheet are on-shell equivalent to undeformed models, yet with twisted boundary conditions. While this has been known for some time, the expression provided so far for the twist features non-localities (in terms of the degrees of freedom of the deformed model) that prevent practical calculations, and in particular the construction of the classical spectral curve. We solve this problem by rewriting the equation defining the twist in terms of the degrees of freedom of the undeformed yet twisted model, and we show that we are able to solve it in full generality. Remarkably, this solution is a local expression. We discuss the consequences of the twist at the level of the monodromy matrix and of the classical spectral curve, analysing in particular the concrete examples of abelian, almost abelian and Jordanian deformations of the Yang-Baxter class.

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arXiv2021

An Algebraic Classification of Solution Generating Techniques

We consider a two-fold problem: on the one hand, the classification of a family of solution-generating techniques in (modified) supergravity and, on the other hand, the classification of a family of canonical transformations of 2-dimensional $σ$-models giving rise to integrable-preserving transformations. Assuming a generalised Scherk-Schwarz ansatz, in fact, the two problems admit essentially the same algebraic formulation, emerging from an underlying double Lie algebra $\mathfrak d$. After presenting our derivation of the classification, we discuss in detail the relation to modified supergravity and the additional conditions to recover the standard (unmodified) supergravity. Starting from our master equation - that encodes all the possible continuous deformations allowed in the family of solution-generating techniques - we show that these are classified by the Lie algebra cohomologies $H^2(\mathfrak h,\mathbb R)$ and $H^3(\mathfrak h,\mathbb R)$ of the maximally isotropic subalgebra $\mathfrak h$ of the double Lie algebra $\mathfrak d$. {We illustrate our results with a non-trivial example, the bi-Yang-Baxter-Wess-Zumino model.

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arXiv2021

Supergravity solution-generating techniques and canonical transformations of $σ$-models from $O(D,D)$

Within the framework of the flux formulation of Double Field Theory (DFT) we employ a generalised Scherk-Schwarz ansatz and discuss the classification of the twists that in the presence of the strong constraint give rise to constant generalised fluxes interpreted as gaugings. We analyse the various possibilities of turning on the fluxes $H_{ijk}, F_{ij}{}^k, Q_i{}^{jk}$ and $R^{ijk}$, and the solutions for the twists allowed in each case. While we do not impose the DFT (or equivalently supergravity) equations of motion, our results provide solution-generating techniques in supergravity when applied to a background that does solve the DFT equations. At the same time, our results give rise also to canonical transformations of 2-dimensional $σ$-models, a fact which is interesting especially because these are integrability-preserving transformations on the worldsheet. Both the solution-generating techniques of supergravity and the canonical transformations of 2-dimensional $σ$-models arise as maps that leave the generalised fluxes of DFT and their flat derivatives invariant. These maps include the known abelian/non-abelian/Poisson-Lie T-duality transformations, Yang-Baxter deformations, as well as novel generalisations of them.

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arXiv2020

Quantum correction to generalized T-dualities

Poisson-Lie duality is a generalization of abelian and non-abelian T-duality, and it can be viewed as a map between solutions of the low-energy effective equations of string theory, i.e. at the (super)gravity level. We show that this fact extends to the next order in $α'$ (two loops in $σ$-model perturbation theory) provided that the map is corrected. The $α'$-correction to the map is induced by the anomalous Lorentz transformations of the fields that are necessary to go from a doubled $O(D,D)$-covariant formulation to the usual (super)gravity description.

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arXiv2020

The first $α'$-correction to homogeneous Yang-Baxter deformations using $O(d,d)$

We use the $O(d,d)$-covariant formulation of supergravity familiar from Double Field Theory to find the first $α'$-correction to (unimodular) homogeneous Yang-Baxter (YB) deformations of the bosonic string. A special case of this result gives the $α'$-correction to TsT transformations. In a suitable scheme the correction comes entirely from an induced anomalous double Lorentz transformation, which is needed to make the two vielbeins obtained upon the YB deformation equal. This should hold more generally, in particular for abelian and non-abelian T-duality, as we discuss.

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arXiv2020

Two-loop conformal invariance for Yang-Baxter deformed strings

The so-called homogeneous Yang-Baxter (YB) deformations can be considered a non-abelian generalization of T-duality--shift--T-duality (TsT) transformations. TsT transformations are known to preserve conformal symmetry to all orders in $α'$. Here we argue that (unimodular) YB deformations of a bosonic string also preserve conformal symmetry, at least to two-loop order. We do this by showing that, starting from a background with no NSNS-flux, the deformed background solves the $α'$-corrected supergravity equations to second order in the deformation parameter. At the same time we determine the required $α'$-corrections of the deformed background, which take a relatively simple form. In examples that can be constructed using, possibly non-commuting sequences of, TsT transformations we show how to obtain the first $α'$-correction to all orders in the deformation parameter by making use of the $α'$-corrected T-duality rules. We demonstrate this on the specific example of YB deformations of a Bianchi type II background.

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arXiv2019

Marginal deformations of WZW models and the classical Yang-Baxter equation

We show how so-called Yang-Baxter (YB) deformations of sigma models, based on an R-matrix solving the classical Yang-Baxter equation (CYBE), give rise to marginal current-current deformations when applied to the Wess-Zumino-Witten (WZW) model. For non-compact groups these marginal deformations are more general than the ones usually considered, since they can involve a non-abelian current subalgebra. We classify such deformations of the AdS(3) x S(3) string.

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