Dr Alessandro Torrielli

Senior Lecturer

Phone: Work: 01483 68 9253
Room no: 33 AA 04

Further information


I was voted Best Lecturer at the Durham Integrability School in 2015. The award was presented to me at IGST15 in London.

December 2012: Awarded EPSRC First grant "Exotic quantum groups, Lie superalgebras & integrable systems"

September 2011 - present: Lecturer in Mathematics, Department of Mathematics, University of Surrey; "Fields, Strings and Geometry" group  

2010-2011: Post-doc at the Mathematics Department of the University of York, UK; EPSRC grant of Dr.Niall MacKay and Prof. Evgeny Sklyanin

2008-2010: Post-doc at the Institute for Theoretical Physics and Spinoza Institute, Utrecht University, The Netherlands

2006-2008: Bruno Rossi" INFN-MIT postdoctoral fellow at the Massachusetts Institute of Technology, USA

2004-2006: Post-doc at the Humboldt University of Berlin

2003: 2-month DAAD fellowship to collaborate with Dr. Harald Dorn - Humboldt University of Berlin

2003-2004: Postdoc at the University of Padova and fellow of the Italian Institute for Nuclear Physics (INFN)

2002-2003: Four months of scientific collaboration with the University of Padova

2003: PhD dissertation (Physics) at the University of Padova, Italy [17 Feb] (Supervisor Prof. Antonio Bassetto, Collaborator Prof. Giuseppe Nardelli)

1999: Laurea 110/110 e lode (top marks cum laude) in Physics at the University of Genova, Italy [13 Oct] (Supervisor Prof. Carlo Maria Becchi, Co-supervisor Prof. Nicola Maggiore)

1994: Esame di Maturita' Classica with top marks (60/60) at the Lyceum of Classical Studies "G. Parodi", Acqui Terme, Italy

Research Interests

I am interested in the theory of integrable systems and quantum groups, especially the ones that emerge in the context of the AdS/CFT correspondence. For more details, please see my personal webpage.

Research Collaborations

At the moment I am collaborating with:

  • Bogdan Stefanski (City), Riccardo Borsato (Utrecht / Imperial), Ben Hoare (Berlin / ETH), Olof Ohlsson-Sax (Imperial / Nordita), Antonio Pittelli, Andrea Prinsloo, Vidas Regelskis, Alessandro Sfondrini (Berlin / ETH) and Martin Wolf, on alternating spin-chains based on superalgebras and the AdS_3 / CFT_2 and AdS_2 / CFT_1 correspondences
  • Fabrizio Nieri and Sara Pasquetti on integrable aspects of N=2 supersymmetric gauge theories
  • Georgios Itsios (Oviedo), Konstantinos Sftesos (Athens) and Konstantinos Siampos (Bern), on integrable interpolations in sigma models.


Journal articles

  • Borsato R, Torrielli A. (2018) 'q-Poincar´e supersymmetry in AdS5/CF T4'. Nuclear Physics B,
    [ Status: Accepted ]


    We consider the exact S-matrix governing the planar spectral problem for strings on AdS5× S 5 and N = 4 super Yang-Mills, and we show that it is invariant under a novel “boost” symmetry, which acts as a differentiation with respect to the particle momentum. This generator leads us also to reinterpret the usual centrally extended psu(2|2) symmetry, and to conclude that the S-matrix is invariant under a q-Poincar´e supersymmetry algebra, where the deformation parameter is related to the ’t Hooft coupling. We determine the twoparticle action (coproduct) that turns out to be non-local, and study the property of the new symmetry under crossing transformations. We look at both the strong-coupling (large tension in the string theory) and weak-coupling (spin-chain description of the gauge theory) limits; in the former regime we calculate the cobracket utilising the universal classical rmatrix of Beisert and Spill. In the eventuality that the boost has higher partners, we also construct a quantum affine version of 2D Poincar´e symmetry, by contraction of the quantum affine algebra Uq(slc2) in Drinfeld’s second realisation.

  • Borsato R, Strömwall J, Torrielli A. (2018) 'q-Poincaré invariance of the AdS3/CFT2 R-matrix'. Physical Review D,
    [ Status: Accepted ]


    We consider the exact R-matrix of AdS3/CFT2, which is the building block for describing the scattering of worldsheet excitations of the light-cone gauge-fixed backgrounds AdS3 × S3 × T4 and AdS3 × S3 × S3 × S1 with pure Ramond-Ramond fluxes. We show that R is invariant under a “deformed boost” symmetry, for which we write an explicit exact coproduct, i.e. its action on 2-particle states. When we include the boost, the symmetries of the R-matrix close into a q-Poincar´e superalgebra. Our findings suggest that the recently discovered boost invariance in AdS5/CFT4 may be a common feature of AdS/CFT systems that are treatable with the exact techniques of integrability. With the aim of going towards a universal formulation of the underlying Hopf algebra, we also propose a universal form of the AdS3/CFT2 classical r-matrix.

  • Torrielli A. (2017) 'On AdS2/CFT1 transfer matrices, Bethe ansatz and scale invariance'. Journal of Physics A: Mathematical and Theoretical, 51 (1) Article number 015402


    We explicitly calculate the AdS2×S2×T6 transfer-matrix eigenvalues in the massless sector using the exact integrable S-matrix, for up to 5 particles. This enables us to conjecture the general pattern. We use the conjectured form of the eigenvalues to write down a set of massless Bethe ansatz equations. The same procedure applies to the relativistic as well as to the non-relativistic situation. In the relativistic case, the right and left modes decouple. We speculate that the relativistic massless Bethe ansatz we obtain in that case might capture the integrable structure of an underlying 2D critical theory. We finally take advantage of some remarkable simplifications to make progress in the massive case as well.

  • Fontanella A, Torrielli A . (2017) 'Massless AdS2 scattering and Bethe Ansatz'. Journal of High Energy Physics, 75
    [ Status: Accepted ]


    We first analyse the integrable scattering theory describing the massless excitations of AdS2 x S2 x T6 superstrings in the relativistic limit. The matrix part of the S-matrix is obtained in the BMN limit from the conjectured exact expression, and compared to known S-matrices with N = 1 supersymmetry in 1 + 1 dimensions. A dressing factor, yet unknown for the complete theory, is here constructed based on relativistic crossing symmetry. We derive a Bethe-ansatz condition by employing a transfer-matrix technique based on the so-called free-fermion condition. This is known to overcome the problem of lack of a reference state. We then generalise the method to the massless non-relativistic case, and compare the resulting Bethe-ansatz condition with a simple massless limit of the one conjectured by Sorokin, Tseytlin, Wulff and Zarembo.

  • Kozlowski K, Sklyanin E, Torrielli A . (2017) 'Quantization of the Kadomtsev-Petviashvili equation'. Theoretical and Mathematical Physics, 192 (2), pp. 259-283.


    A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of non-relativistic one-dimensional bosons carrying masses m = 1, 2, . . . The Hamiltonian is Galilei-invariant and includes the split Ψ† m1Ψ† m2Ψm1+m2 and merge Ψ† m1+m2Ψm1Ψm2 terms for all combinations of particles with masses m1, m2 and m1 + m2, with a special choice of coupling constants. The Bethe eigenfunctions for the model are constructed. The consistency of the coordinate Bethe Ansatz, and therefore, the quantum integrability of the model is verified up to the mass M = 8 sector.

  • Baggio M, Sax O, Sfondrini A, Stefanski jr F, Torrielli A . (2017) 'Protected string spectrum in AdS3/CFT2 from worldsheet integrability'. Journal of High Energy Physics, 91


    We derive the protected closed-string spectra of AdS3/CFT2 dual pairs with 16 supercharges at arbitrary values of the string tension and of the three-form fluxes. These follow immediately from the all-loop Bethe equations for the spectra of the integrable worldsheet theories. Further, representing the underlying integrable systems as spin chains, we find that their dynamics involves length-changing interactions and that protected states correspond to gapless excitations above the Berenstein-Maldacena-Nastase vacuum. In the case of AdS3 × S 3 × T 4 the degeneracies of such operators precisely match those of the dual CFT2 and the supergravity spectrum. On the other hand, we find that for AdS3 × S 3 × S 3 × S 1 there are fewer protected states than previous supergravity calculations had suggested. In particular, protected states have the same su(2) charge with respect to the two three-spheres.

  • Borsato R, Ohlsson Sax O, Sfondrini A, Stefanski B, Torrielli A. (2016) 'On the Dressing Factors, Bethe Equations and Yangian Symmetry of Strings on AdS3 × S3 × T4'. Journal of Physics A: Mathematical and Theoretical, 50


    Integrability is believed to underlie the AdS3/CFT2 correspondence with sixteen supercharges. We elucidate the role of massless modes within this integrable framework. Firstly, we find the dressing factors that enter the massless and mixed-mass worldsheet S matrix. Secondly, we derive a set of all-loop Bethe Equations for the closed strings, determine their symmetries and weak-coupling limit. Thirdly, we investigate the underlying Yangian symmetry in the massless sector and show that it fits into the general framework of Yangian integrability. In addition, we compare our S matrix in the near-relativistic limit with recent perturbative worldsheet calculations of Sundin and Wulff.

  • Fontanella A, Torrielli A. (2016) 'Massless sector of AdS(3) superstrings: A geometric interpretation'. PHYSICAL REVIEW D, 94 (6) Article number ARTN 066008
  • Prinsloo A, Regelskis V, Torrielli A. (2015) 'Integrable open spin-chains in AdS3/CFT2 correspondences'. Phys. Rev. D, 92


    We study integrable open boundary conditions for d(2,1;\alpha)^2 and psu(1,1|2)^2 spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type IIB open superstrings ending on D-branes in the AdS_3 x S^3 x S^3 x S^1 and AdS_3 x S^3 x T^4 supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1- and D5-brane maximal giant gravitons.

  • Hoare B, Pittelli A, Torrielli A. (2015) 'S-matrix algebra of the AdS(2) x S-2 superstring'. PHYSICAL REVIEW D, 93 (6) Article number ARTN 066006
  • Rolph A, Torrielli A. (2014) 'Drinfeld basis for string-inspired Baxter operators'. Phys. Rev. D, 91


    We propose Drinfeld's second realisation of the quantum group relevant to the Lax-operator approach developed in the work of Bazhanov, Frassek, Lukowski, Meneghelli and Staudacher.

  • Itsios G, Sfetsos K, Siampos K, Torrielli A. (2014) 'The classical Yang-Baxter equation and the associated Yangian symmetry of gauged WZW-type theories'. Nucl. Phys. B, 889, pp. 64-86.


    We construct the Lax-pair, the classical monodromy matrix and the corresponding solution of the Yang--Baxter equation, for a two-parameter deformation of the Principal chiral model for a simple group. This deformation includes as a one-parameter subset, a class of integrable gauged WZW-type theories interpolating between the WZW model and the non-Abelian T-dual of the principal chiral model. We derive in full detail the Yangian algebra using two independent methods: by computing the algebra of the non-local charges and alternatively through an expansion of the Maillet brackets for the monodromy matrix. As a byproduct, we also provide a detailed general proof of the Serre relations for the Yangian symmetry.

  • Hoare B, Pittelli A, Torrielli A. (2014) 'Integrable S-matrices, massive and massless modes and the AdS_2 x S^2 superstring'.


    We derive the exact S-matrix for the scattering of particular representations of the centrally-extended psu(1|1)^2 Lie superalgebra, conjectured to be related to the massive modes of the light-cone gauge string theory on AdS_2 x S^2 x T^6. The S-matrix consists of two copies of a centrally-extended psu(1|1) invariant S-matrix and is in agreement with the tree-level result following from perturbation theory. Although the overall factor is left unfixed, the constraints following from crossing symmetry and unitarity are given. The scattering involves long representations of the symmetry algebra, and the relevant representation theory is studied in detail. We also discuss Yangian symmetry and find it has a standard form for a particular limit of the aforementioned representations. This has a natural interpretation as the massless limit, and we investigate the corresponding limits of the massive S-matrix. Under the assumption that the massless modes of the light-cone gauge string theory transform in these limiting representations, the resulting S-matrices would provide the building blocks for the full S-matrix. Finally, some brief comments are given on the Bethe ansatz.

  • Pittelli A, Torrielli A, Wolf M. (2014) 'Secret Symmetries of Type IIB Superstring Theory on AdS3 x S3 x M4'. J. Phys. A: Math. Theor., 47


    We establish features of so-called Yangian secret symmetries for AdS3 type IIB superstring backgrounds thus verifying the persistence of such symmetries to this new instance of the AdS/CFT correspondence. Specifically, we find two a priori different classes of secret symmetry generators. One class of generators, anticipated from the previous literature, is more naturally embedded in the algebra governing the integrable scattering problem. The other class of generators is more elusive, and somewhat closer in its form to its higher-dimensional AdS5 counterpart. All of these symmetries respect left-right crossing. In addition, by considering the interplay between left and right representations, we gain a new perspective on the AdS5 case. We also study the RTT-realisation of the Yangian in AdS3 backgrounds thus establishing a new incarnation of the Beisert-de Leeuw construction.

  • Nieri F, Pasquetti S, Passerini F, Torrielli A. (2013) '5D partition functions, q-Virasoro systems and integrable spin-chains'.


    We analyze N = 1 theories on S5 and S4 x S1, showing how their partition functions can be written in terms of a set of fundamental 5d holomorphic blocks. We demonstrate that, when the 5d mass parameters are analytically continued to suitable values, the S5 and S4 x S1 partition functions degenerate to those for S3 and S2 x S1. We explain this mechanism via the recently proposed correspondence between 5d partition functions and correlators with underlying q-Virasoro symmetry. From the q-Virasoro 3-point functions, we axiomatically derive a set of associated reflection coefficients, and show they can be geometrically interpreted in terms of Harish-Chandra c-functions for quantum symmetric spaces. We then link these particular c-functions to the types appearing in the Jost functions encoding the asymptotics of the scattering in integrable spin chains, obtained taking different limits of the XYZ model to XXZ-type.

  • Borsato R, Sax OO, Sfondrini A, Stefanski B, Torrielli A. (2013) 'Dressing phases of AdS3/CFT2'. Physical Review D, 88


    We determine the all-loop dressing phases of the AdS3/CFT2 integrable system related to type IIB string theory on AdS3 x S3 x T4 by solving the recently found crossing relations and studying their singularity structure. The two resulting phases present a novel structure with respect to the ones appearing in AdS5/CFT4 and AdS4/CFT3. In the strongly-coupled regime, their leading order reduces to the universal Arutyunov-Frolov-Staudacher phase as expected. We also compute their sub-leading order and compare it with recent one-loop perturbative results, and comment on their weak-coupling expansion.

  • Borsato R, Sax OO, Sfondrini A, Stefanski B, Torrielli A. (2013) 'The all-loop integrable spin-chain for strings on AdS_3 x S^3 x T^4: the massive sector'. The Journal of High Energy Physics, 43


    We bootstrap the all-loop dynamic S-matrix for the homogeneous psu(1,1|2)^2 spin-chain believed to correspond to the discretization of the massive modes of string theory on AdS_3 x S^3 x T^4. The S-matrix is the tensor product of two copies of the su(1|1)^2 invariant S-matrix constructed recently for the d(2,1;alpha)^2 chain, and depends on two antisymmetric dressing phases. We write down the crossing equations that these phases have to satisfy. Furthermore, we present the corresponding Bethe Ansatz, which differs from the one previously conjectured, and discuss how our construction matches several recent perturbative calculations.

  • Ohlsson Sax O, Stefanski BJ, Torrielli A. (2013) 'On the massless modes of the AdS3/CFT2 integrable systems'. The Journal of High Energy Physics, 03


    We make a proposal for incorporating massless modes into the spin-chain of the AdS3/CFT2 integrable system. We do this by considering the alpha to 0 limit of the alternating d(2,1;alpha)^2 spin-chain constructed in arXiv:1106.2558. In the process we encounter integrable spin-chains with non-irreducible representations at some of their sites. We investigate their properties and construct their R-matrices in terms of Yangians.

  • Babichenko A, Torrielli A. (2012) 'Multi-parametric R-matrix for the sl(2|1) Yangian'. Journal of Mathematical Physics, 53 (8)


    We study the Yangian of the sl(2|1) Lie superalgebra in a multi-parametric four-dimensional representation. We use Drinfeld's second realization to derive the R-matrix, the antiparticle representation, the crossing and unitarity condition. We consistently apply the Yangian antipode and its inverse to the individual particles involved in the scattering. We explicitly find a scalar factor solving the crossing and unitarity conditions, and study the analytic structure of the resulting dressed R-matrix. The formulas we obtain bear some similarities with those familiar from the study of integrable structures in the AdS/CFT correspondence, although they present obvious crucial differences.

  • Beisert N, Ahn C, Alday LF, Bajnok Z, Drummond JM, Freyhult L, Gromov N, Janik RA, Kazakov V, Klose T, Korchemsky GP, Kristjansen C, Magro M, McLoughlin T, Minahan JA, Nepomechie RI, Rej A, Roiban R, Schafer-Nameki S, Sieg C, Staudacher M, Torrielli A, Tseytlin AA, Vieira P, Volin D, Zoubos K. (2012) 'Review of AdS/CFT Integrability: An Overview'. Letters in Mathematical Physics, 99 (1-3), pp. 3-32.


    This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection we present an overview of the achievements and the status of this subject as of the year 2010.

  • de Leeuw M, Regelskis V, Torrielli A. (2012) 'The quantum affine origin of the AdS/CFT secret symmetry'. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 45 (17) Article number ARTN 175202
  • Torrielli A. (2011) 'Yangians, S-matrices and AdS/CFT'. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 44 (26)


    This review is meant to be an account of the properties of the infinite-dimensional quantum group (specifically, Yangian) symmetry lying behind the integrability of the AdS/CFT spectral problem. In passing, the chance is taken to give a concise anthology of basic facts concerning Yangians and integrable systems, and to store a series of remarks, observations and proofs the author has collected in a 5 year span of research on the subject. We hope this exercise will be useful for future attempts to study Yangians in field and string theories, with or without supersymmetry.

  • Torrielli A. (2011) 'Review of AdS/CFT Integrability. Chapter VI.2: Yangian Algebra'. Letters in Mathematical Physics, 99 (1-3), pp. 547-565.


    We review the study of Hopf algebras, classical and quantum R-matrices, infinite-dimensional Yangian symmetries and their representations in the context of integrability for the N = 4 vs AdS5 X S5 correspondence.

  • Schubert C, Torrielli A. (2010) 'Open string pair creation from worldsheet instantons'. J PHYS A-MATH THEOR, 43 (40)


    Worldline instantons provide a particularly elegant way to derive Schwinger’s well-known formula for the pair creation rate due to a constant electric field in quantum electrodynamics. In this note, we show how to extend this method to the corresponding problem of open string pair creation.

  • Arutyunov G, De Leeuw M, Torrielli A. (2010) 'On Yangian and long representations of the centrally extended su(2|2) superalgebra'. Journal of High Energy Physics, 6


    The centrally extended su(2|2) superalgebra is an asymptotic symmetry of the light-cone string sigma model on AdS5 × S5. We consider an evaluation representation of the conventional Yangian built over a particular 16-dimensional long representation of the centrally extended su(2|2). Interestingly, we find that S-matrices compatible with this evaluation representation do not exist. On the other hand, by requiring centrally extended su(2|2) invariance and explicitly solving the Yang-Baxter equation, we find a scattering matrix for long-short representations of the Lie superalgebra. We notice that this S-matrix is invariant under a different representation of non-evaluation type, induced from the tensor product of short representations. Our findings concern the conventional Yangian only, and are not applied to possible algebraic extensions of the latter.

  • Arutyunov G, de Leeuw M, Torrielli A. (2009) 'The bound state S-matrix for AdS(5) x S-5 superstring'. NUCLEAR PHYSICS B, 819 (3), pp. 319-350.


    We determine the S-matrix that describes scattering of arbitrary bound states in the light-cone string theory in AdS5 × S5. The corresponding construction relies on the Yangian symmetry and the superspace formalism for the bound state representations. The basic analytic structure supporting the S-matrix entries turns out to be the hypergeometric function 4F3. We show that for particular bound state numbers it reproduces all the scattering matrices previously obtained in the literature. Our findings should be relevant for the TBA and L¨uscher approaches to the finite-size spectral problem. They also shed some light on the construction of the universal R-matrix for the centrally-extended psu(2|2) superalgebra.

  • Spill F, Torrielli A. (2009) 'On Drinfeld's second realization of the AdS/CFT su(2|2) Yangian'. JOURNAL OF GEOMETRY AND PHYSICS, 59 (4), pp. 489-502.


    We construct Drinfeld’s second realization of the Yangian based on psu(2|2) ⋉ R3 symmetry. The second realization is traditionally more suitable for deriving the quantum double and the universal R-matrix with respect to the first realization, originally obtained by Beisert, and it is generically more useful in order to study finite dimensional representations. We show that the two realizations are isomorphic, where the isomorphism is almost the standard one given by Drinfeld for simple Lie algebras, but needs some crucial corrections to account for the central charges. We also evaluate the generators of the second realization on the fundamental representation, finding the interesting result that the rapidity variable for some generators gets boosted by the energy eigenvalue.

  • Torrielli A. (2009) 'Structure of the string R-matrix'. JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 42 (5)


    By requiring invariance directly under the Yangian symmetry, we rederive Beisert’s quantum R-matrix, in a form that carries explicit dependence on the representation labels, the braiding factors, and the spectral parameters ui. In this way, we demonstrate that there exist a rewriting of its entries, such that the dependence on the spectral parameters is purely of difference form. Namely, the latter enter only in the combination u1−u2, as indicated by the shift automorphism of the Yangian. When recasted in this fashion, the entries exhibit a cleaner structure, which allows to spot new interesting relations among them. This permits to package them into a practical tensorial expression, where the non-diagonal entries are taken care by explicit combinations of symmetry algebra generators.

  • Arutyunov G, De Leeuw M, Suzuki R, Torrielli A. (2009) 'Bound state transfer matrix for AdS5×S5 superstring'. Journal of High Energy Physics, 2009 (10)


    We apply the algebraic Bethe ansatz technique to compute the eigenvalues of the transfer matrix constructed from the general bound state S-matrix of the lightcone AdS5 × S5 superstring. This allows us to verify certain conjectures on the quantum characteristic function, and to extend them to the general case.

  • Arutyunov G, De Leeuw M, Torrielli A. (2009) 'Universal blocks of the AdS/CFT$ AdS/CFT scattering matrix'. Journal of High Energy Physics, 5


    We identify certain blocks in the S-matrix describing the scattering of bound states of the AdS5 × S5 superstring that allow for a representation in terms of universal R-matrices of Yangian doubles. For these cases, we use the formulas for Drinfeld’s second realization of the Yangian in arbitrary bound-state representations to obtain the explicit expressions for the corresponding R-matrices. We then show that these expressions perfectly match with the previously obtained S-matrix blocks.

  • Heckenberger I, Spill F, Torrielli A, Yamane H. (2008) 'Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid'. RIMS Kokyuroku Bessatsu, (B8), pp. 171-216.


    We obtain Drinfeld second realization of the quantum affine superalgebras associated with the affine Lie superalgebra $D^{(1)}(2,1;x)$. Our results are analogous to those obtained by Beck for the quantum affine algebras. Beck's analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid.

  • de Haro S, Ramgoolam S, Torrielli A. (2007) 'Large N expansion of q-deformed two-dimensional Yang-Mills theory and Hecke algebras'. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 273 (2), pp. 317-355.


    We derive the q-deformation of the chiral Gross-Taylor holomorphic string large N expansion of two dimensional SU(N) Yang-Mills theory. Delta functions on symmetric group algebras are replaced by the corresponding objects (canonical trace functions) for Hecke algebras. The role of the Schur-Weyl duality between unitary groups and symmetric groups is now played by q-deformed Schur-Weyl duality of quantum groups. The appearance of Euler characters of configuration spaces of Riemann surfaces in the expansion persists. We discuss the geometrical meaning of these formulae.

  • Torrielli A. (2007) 'Classical r matrix of the su(2|2) super Yang-Mills spin chain'. Physical Review D - Particles, Fields, Gravitation and Cosmology, 75 (10)


    In this note we straightforwardly derive and make use of the quantum R matrix for the su(2|2) super Yang-Mills spin chain in the manifest su(1|2)-invariant formulation, which solves the standard quantum Yang-Baxter equation, in order to obtain the correspondent (undressed) classical r matrix from the first order expansion in the “deformation” parameter 2π/√λ and check that this last solves the standard classical Yang-Baxter equation. We analyze its bialgebra structure, its dependence on the spectral parameters, and its pole structure. We notice that it still preserves an su(1|2) subalgebra, thereby admitting an expression in terms of a combination of projectors, which spans only a subspace of su(1|2)⊗su(1|2). We study the residue at its simple pole at the origin and comment on the applicability of the classical Belavin-Drinfeld type of analysis.

  • Bietenholz W, Bigarini A, Torrielli A. (2007) 'Area-preserving diffeomorphisms in gauge theory on a non-commutative plane: A lattice study'. Journal of High Energy Physics, 2007 (8)


    We consider Yang-Mills theory with the U(1) gauge group on a non-commutative plane. Perturbatively it was observed that the invariance of this theory under area-preserving diffeomorphisms (APDs) breaks down to a rigid subgroup SL(2,R). Here we present explicit results for the APD symmetry breaking at finite gauge coupling and finite non-commutativity. They are based on lattice simulations and measurements of Wilson loops with the same area but with a variety of different shapes. Our results are consistent with the expected loss of invariance under APDs. Moreover, they strongly suggest that non-perturbatively the SL(2,R) symmetry does not persist either.

  • Matsumoto T, Moriyama S, Torrielli A. (2007) 'A secret symmetry of the AdS/CFT S-matrix'. Journal of High Energy Physics, 9


    We find a new quantum Yangian symmetry of the AdS/CFT S-matrix, which complements the original (2|2) symmetry to (2|2) and does not have a Lie algebra analog. Our finding is motivated by the Yangian double structure discovered at the classical level.

  • Moriyama S, Torrielli A. (2007) 'A Yangian double for the AdS/CFT classical r-matrix'. Journal of High Energy Physics, 2007 (6)


    We express the classical r-matrix of AdS/CFT in terms of tensor products involving an infinite family of generators, which takes a form suggestive of the universal expression obtained from a Yangian double. This should provide an insight into the structure of the infinite dimensional symmetry algebra underlying the integrability of the model, and give a clue to the construction of its universal R-matrix. We derive the commutation relations under which the algebra of these new generators close.

  • Plefka J, Spill F, Torrielli A. (2006) 'Hopf algebra structure of the AdS/CFT S-matrix'. PHYSICAL REVIEW D, 74 (6)


    We formulate the Hopf algebra underlying the su(2|2) worldsheet S-matrix of the AdS5 × S5 string in the AdS/CFT correspondence. For this we extend the previous construction in the su(1|2) subsector due to Janik to the full algebra by specifying the action of the coproduct and the antipode on the remaining generators. The nontriviality of the coproduct is determined by length-changing effects and results in an unusual central braiding. As an application we explicitly determine the antiparticle representation by means of the established antipode.

  • Salizzoni M, Torrielli A, Yang HS. (2006) 'ALE spaces from noncommutative U (1) instantons via exact Seiberg-Witten map'. PHYSICS LETTERS B, 634 (4), pp. 427-433.


    The exact Seiberg–Witten (SW) map of a noncommutative (NC) gauge theory gives the commutative equivalent as an ordinary gauge theory coupled to a field dependent effective metric. We study instanton solutions of this commutative equivalent whose self-duality equation turns out to be the exact SW map of NC instantons. We derive general differential equations governing U(1) instantons and we explicitly get an exact solution corresponding to the single NC instanton. Remarkably the effective metric induced by the single U(1) instanton is related to the Eguchi–Hanson metric—the simplest gravitational instanton. Surprisingly the instanton number is not quantized but depends on an integration constant. Our result confirms the expected non-perturbative breakdown of the SW map. However, the breakdown of the map arises in a consistent way: the instanton number plays the role of a parameter giving rise to a one-parameter family of Eguchi–Hanson metrics.

  • Dorn H, Salizzoni M, Torrielli A. (2006) 'D-branes in overcritical electric fields'. PHYSICAL REVIEW D, 73 (2)


    We collect some arguments for treating a D-brane with overcritical electric field as a well-posed initial condition for a D-brane decay. Within the field theoretical toy model of Minahan and Zwiebach we give an estimate for the condensates of the related infinite tower of tachyonic excitations.

  • Sieg C, Torrielli A. (2005) 'Wrapping interactions and the genus expansion of the 2-point function of composite operators'. Nuclear Physics B, 723 (1-2), pp. 3-32.


    We perform a systematic analysis of wrapping interactions for a general class of theories with color degrees of freedom, including N=4 SYM. Wrapping interactions arise in the genus expansion of the 2-point function of composite operators as finite size effects that start to appear at a certain order in the coupling constant at which the range of the interaction is equal to the length of the operators. We analyze in detail the relevant genus expansions, and introduce a strategy to single out the wrapping contributions, based on adding spectator fields. We use a toy model to demonstrate our procedure, performing all computations explicitly. Although completely general, our treatment should be particularly useful for applications to the recent problem of wrapping contributions in some checks of the AdS/CFT correspondence.

  • Bassetto A, Pol GD, Torrielli A, Vian F. (2005) 'On the invariance under area preserving diffeomorphisms of noncommutative Yang-Mills theory in two dimensions'. Journal of High Energy Physics, (05)


    We present an investigation on the invariance properties of noncommutative Yang-Mills theory in two dimensions under area preserving diffeomorphisms. Stimulated by recent remarks by Ambjorn, Dubin and Makeenko who found a breaking of such an invariance, we confirm both on a fairly general ground and by means of perturbative analytical and numerical calculations that indeed invariance under area preserving diffeomorphisms is lost. However a remnant survives, namely invariance under linear unimodular tranformations.

  • Torrielli A, Fortunato L. (2005) 'Theory of light emission in sonoluminescence based upon transitions in confined atoms'. Eur. Phys. J. D, 33
  • Bassetto A, Torrielli A, Valandro R. (2004) 'One-loop unitarity of string theories in a constant external background and their Seiberg-Witten limit'. Journal of High Energy Physics, (01)


    Perturbative spectra and related factorization properties of one-loop open string amplitudes in the presence of a constant external background B are analysed in detail. While the pattern of the closed string spectrum, obtained after a careful study of the properly symmetrized amplitudes, turns out to be unaffected by the presence of B, a series of double open-string poles, which would be absent when B is turned off, can couple owing to a partial symmetry loss. These features are studied first in a bosonic setting and then generalized in the more satisfactory superstring context. When the background is of an ``electric'' type, a classical perturbative instability is produced beyond a critical value of the electric field. In the Seiberg-Witten limit this instability is the origin of the unphysical tachyonic cut occurring in the non-planar amplitudes of the corresponding noncommutative field theories.

  • Dorn H, Torrielli A. (2004) 'Loop Equation in Two-dimensional Noncommutative Yang-Mills Theory'. Journal of High Energy Physics, 01


    The classical analysis of Kazakov and Kostov of the Makeenko-Migdal loop equation in two-dimensional gauge theory leads to usual partial differential equations with respect to the areas of windows formed by the loop. We extend this treatment to the case of U(N) Yang-Mills defined on the noncommutative plane. We deal with all the subtleties which arise in their two-dimensional geometric procedure, using where needed results from the perturbative computations of the noncommutative Wilson loop available in the literature. The open Wilson line contribution present in the non-commutative version of the loop equation drops out in the resulting usual differential equations. These equations for all N have the same form as in the commutative case for N to infinity. However, the additional supplementary input from factorization properties allowing to solve the equations in the commutative case is no longer valid.

  • Torrielli A. (2003) 'Cutting rules and perturbative unitarity of noncommutative electric-type field theories from string theory'. Phys.Rev. D, 67


    We discuss the breakdown of perturbative unitarity of noncommutative quantum field theories in electric-type background in the light of string theory. We consider the analytic structure of string loop two-point functions using a suitable off-shell continuation and then study the zero slope limit of Seiberg and Witten. In this way we pick up how the unphysical tachyonic branch cut appears in the noncommutative field theory. We briefly discuss discontinuities and cutting rules for the full string theory amplitude and relate them to the noncommutative field theoretical results, and also discuss the insight one gains into the magnetic case too.

  • Bassetto A, Nardelli G, Torrielli A. (2002) 'Scaling properties of the perturbative Wilson loop in two-dimensional non-commutative Yang-Mills theory'. Phys.Rev. D, 66


    Commutative Yang-Mills theories in 1+1 dimensions exhibit an interesting interplay between geometrical properties and U(N) gauge structures: in the exact expression of a Wilson loop with $n$ windings a non trivial scaling intertwines $n$ and $N$. In the non-commutative case the interplay becomes tighter owing to the merging of space-time and ``internal'' symmetries in a larger gauge group $U(\infty)$. We perform an explicit perturbative calculation of such a loop up to ${\cal O}(g^6)$; rather surprisingly, we find that in the contribution from the crossed graphs (the genuine non-commutative terms) the scaling we mentioned occurs for large $n$ and $N$ in the limit of maximal non-commutativity $\theta=\infty$. We present arguments in favour of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series.

  • Bassetto A, Nardelli G, Torrielli A. (2001) 'Perturbative Wilson loop in two-dimensional non-commutative Yang-Mills theory'. Nuclear Physics B, 617 (1-3), pp. 308-320.


    We perform a perturbative ${\cal O}(g^4)$ Wilson loop calculation for the U(N) Yang-Mills theory defined on non-commutative one space - one time dimensions. We choose the light-cone gauge and compare the results obtained when using the Wu-Mandelstam-Leibbrandt ($WML$) and the Cauchy principal value ($PV$) prescription for the vector propagator. In the $WML$ case the $\theta$-dependent term is well-defined and regular in the limit $\theta \to 0$, where the commutative theory is recovered; it provides a non-trivial example of a consistent calculation when non-commutativity involves the time variable. In the $PV$ case, unexpectedly, the result differs from the $WML$ one only by the addition of two singular terms with a trivial $\theta$-dependence. We find this feature intriguing, when remembering that, in ordinary theories on compact manifolds, the difference between the two cases can be traced back to the contribution of topological excitations.

Conference papers

  • Leeuw MD, Matsumoto T, Moriyama S, Regelskis V, Torrielli A. (2012) 'Secret Symmetries in AdS/CFT'. Institute of Pysics Physica Scripta: an international journal for experimental and theoretical physics, Stockholm, Sweden: Nordita: Exact Results in Gauge-String Dualities 86 (2)


    We discuss special quantum group (secret) symmetries of the integrable system associated to the AdS/CFT correspondence. These symmetries have by now been observed in a variety of forms, including the spectral problem, the boundary scattering problem, n-point amplitudes, the pure-spinor formulation and quantum affine deformations.

  • Torrielli A. (2011) 'The Hopf superalgebra of AdS/CFT'. J. Geom. Phys., Leiden, The Netherlands: The Interface of Integrability and Quantization 61, pp. 230-236.


    We describe the unconventional infinite-dimensional Hopf superalgebra related to the integrable S-matrix of the AdS/CFT correspondence, and discuss its typical and atypical representations.

  • Matsumoto T, Moriyama S, Torrielli A. (2008) 'A secret symmetry of AdS/CFT S-matrix'. WORLD SCIENTIFIC PUBL CO PTE LTD INTERNATIONAL JOURNAL OF MODERN PHYSICS A, Osaka City Univ, Osaka, JAPAN: International Workshop on the Progress of String Theory and Quantum Field Theory 23 (14-15), pp. 2262-2263.
  • Torrielli A, Bietenholz W, Bigarini A, Nishimura J, Susaki Y, Volkholz J. (2007) 'Simulation Results for U(1) Gauge Theory on Non-Commutative Spaces'. PoS LATTICE2007, Regensburg, Germany: 25th International Symposium on Lattice Field Theory (Lattice 2007)


    We present numerical results for U(1) gauge theory in 2d and 4d spaces involving a non-commutative plane. Simulations are feasible thanks to a mapping of the non-commutative plane onto a twisted matrix model. In d=2 it was a long-standing issue if Wilson loops are (partially) invariant under area-preserving diffeomorphisms. We show that non-perturbatively this invariance breaks, including the subgroup SL(2,R). In both cases, d=2 and d=4, we extrapolate our results to the continuum and infinite volume by means of a Double Scaling Limit. In d=4 this limit leads to a phase with broken translation symmetry, which is not affected by the perturbatively known IR instability. Therefore the photon may survive in a non-commutative world.

  • Bassetto A, De Pol G, Torrielli A, Vian F. (2006) 'Area preserving diffeomorphisms and Yang-Mills theory in two noncommutative dimensions'. ELSEVIER SCIENCE BV NUCLEAR PHYSICS B-PROCEEDINGS SUPPLEMENTS, Cairns, AUSTRALIA: 8th Workshop on Light-Cone QCD and Nonperturbative Hadron Physics 161, pp. 21-26.


    We present some evidence that noncommutative Yang-Mills theory in two dimensions is not invariant under area preserving diffeomorphisms, at variance with the commutative case. Still, invariance under linear unimodular maps survives, as is proven by means of a fairly general argument.

  • Torrielli A. (2003) 'D-branes and Unitarity of Noncommutative Field Theories'. Ljubljana : DMFA Bled Workshops in Physics, Portoroz, Slovenia: Euroconference on Symmetries Beyond the Standard Model 4 (2-3)


    We review the original result we have obtained in the analysis of the breaking of perturbative unitarity in space-time noncommutative field theories in the light of their relations to D-branes in electric backgrounds.

  • Torrielli A. (2003) 'Unitarity of noncommutative field theories from string theory'. Mod. Phys. Lett. A, Vietri sul Mare, Italy: Conference on Spacetime and Fundamental Interactions: Quantum Aspects (In honor of A.P. Balachandran's 65th Birthday) 18


    We improve the study of the lack of perturbative unitarity of noncommutative space-time quantum field theories derived from open string theory in electric backgrounds, enforcing the universality of the mechanism by which a tachyonic branch cut appears when the Seiberg-Witten limit freezes the string in an unstable vacuum. The main example is realized in the context of the on-shell four-tachyon amplitude of the bosonic string, and the dependence of the phenomenon on the brane-worldvolume dimension is analysed. We discuss the possibility of a proof in superstring theory, and finally mention the NCOS limit in this framework.

Book chapters

  • Torrielli A. (2005) 'Focus on quantum field theory'. in Kovras O (ed.) Focus on quantum field theory Nova Science Pub Inc


    Editor: O. Kovras. pp. 83-1 13 © 2005 Nova Science Publishers, ... the Example of noncommutative field theories Alessandro Torrielli* Dipartimento di Fisica ...

Working Papers

  • Kozlowski KK, Sklyanin E, Torrielli A. (2016) Quantisation of Kadomtsev-Petviashvili equation. arXiv arXiv,


    A quantisation of the KP equation on a cylinder is proposed that is equivalent to an infinite system of non-relativistic one-dimensional bosons carrying masses m = 1, 2, . . . The Hamiltonian is Galilei-invariant and includes the split Ψ† m1Ψ† m2Ψm1+m2 and merge Ψ† m1+m2Ψm1Ψm2 terms for all combinations of particles with masses m1, m2 and m1 + m2, with a special choice of coupling constants. The Bethe eigenfunctions for the model are constructed. The consistency of the coordinate Bethe Ansatz, and therefore, the quantum integrability of the model is verified up to the mass M = 8 sector.


Quantum Mechanics for year 3 Mathematics 

Quantum Field Theory for year 4 Mathematics

Departmental Duties

Helping John Rayman with Placement matters

Visiting Tutor for Placement Year

Local organiser of the TMT undergraduate conference

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