Alessandro Torrielli

Dr Alessandro Torrielli


Academic and research departments

Department of Mathematics.



2018: Recipient of the EPSRC-SFI grant “Solving spins and strings” (link here) jointly with Dr. Marius de Leeuw (Trinity College Dublin). The total value to Surrey is £487,604 (with a commensurate amount from SFI to Trinity College). It includes money for a postdoc, travel, and 20% time for the PI. The project starts in September 2019 and runs for 3 years.

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

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

2011 - present: Member of staff, 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

I am collaborating with:

  • Bogdan Stefanski (City), Riccardo Borsato (Santiago), Ben Hoare (Berlin / ETH), Olof Ohlsson-Sax (Nordita), Antonio Pittelli (Uppsala), Andrea Prinsloo, Vidas Regelskis (Hertfordshire), Alessandro Sfondrini (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 (Hamburg) and Sara Pasquetti on integrable aspects of N=2 supersymmetric gauge theories
  • Georgios Itsios (Sao Paulo), Konstantinos Sfetsos (Athens) and Konstantinos Siampos (CERN), on integrable interpolations in sigma models.
  • Joakim Stroemwall and Andrea Fontanella (Madrid) on the AdS_3 / CFT_2 and AdS_2 / CFT_1 correspondences
  • Evgeny Sklyanin (York), Karol Kozlowski (Lyon) and Benoit Vicedo (York) on the quantisation of the KP equation


Quantum Mechanics MAT3039 for year 3 Mathematics

Relativistic Quantum Mechanics MATM054 for year 4 Mathematics

Departmental duties

SurreyLearn contact person

Summer projects and Undergraduate Research

Support staff for the Milano Bicocca - Surrey dual doctorate

Affiliations and memberships


Research interests

My publications


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 ELSEVIER SCIENCE BV
We construct Drinfeld?s second realization of the Yangian based on psu(2|2) É 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.
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 Research Institute for Mathematical Sciences, Kyoto University
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.
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) Institute of Physics
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.
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.
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.
Fontanella A, Torrielli A (2016) Massless sector of AdS(3) superstrings: A geometric interpretation,PHYSICAL REVIEW D 94 (6) ARTN 066008 AMER PHYSICAL SOC
Matsumoto T, Moriyama S, Torrielli A (2007) A secret symmetry of the AdS/CFT S-matrix,Journal of High Energy Physics 9 Institute of Physics
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.
Leeuw MD, Matsumoto T, Moriyama S, Regelskis V, Torrielli A (2012) Secret Symmetries in AdS/CFT,Physica Scripta: an international journal for experimental and theoretical physics 86 (2) Institute of Pysics
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.
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 Springer
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.
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) Institute of Physics
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.
Dorn H, Salizzoni M, Torrielli A (2006) D-branes in overcritical electric fields,PHYSICAL REVIEW D 73 (2) AMERICAN PHYSICAL SOC
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.
Torrielli A (2003) Unitarity of noncommutative field theories from string theory,Mod. Phys. Lett. A 18 pp. 2525-?
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.
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 American Physical Society
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.
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) ARTN 175202 IOP PUBLISHING LTD
Torrielli A (2009) Structure of the string R-matrix,JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 42 (5) IOP PUBLISHING LTD
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, Torrielli A (2009) The bound state S-matrix for AdS(5) x S-5 superstring,NUCLEAR PHYSICS B 819 (3) pp. 319-350 ELSEVIER SCIENCE BV
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.
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) Institute of Physics
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.
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 SPRINGER
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.
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 ELSEVIER SCIENCE BV
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.
Hoare B, Pittelli A, Torrielli A (2015) S-matrix algebra of the AdS(2) x S-2 superstring, PHYSICAL REVIEW D 93 (6) ARTN 066006 AMER PHYSICAL SOC
Torrielli A (2003) D-branes and Unitarity of Noncommutative Field Theories,Bled Workshops in Physics 4 (2-3) DMFA
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.
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 Elsevier
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.
Torrielli A (2011) Review of AdS/CFT Integrability. Chapter VI.2: Yangian Algebra,Letters in Mathematical Physics 99 (1-3) pp. 547-565 Springer
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.
Plefka J, Spill F, Torrielli A (2006) Hopf algebra structure of the AdS/CFT S-matrix,PHYSICAL REVIEW D 74 (6) AMERICAN PHYSICAL SOC
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.
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 pp. 049-?
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.
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.
Pittelli A, Torrielli Alessandro, Wolf Martin (2014) Secret symmetries of type IIB superstring theory on AdS(3) x S-3 x M-4,JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 47 (45) 455402 IOP PUBLISHING LTD
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 $R\mathcal{T}\mathcal{T}$-realisation of the Yangian in AdS3 backgrounds, thus establishing a new incarnation of the Beisert?de Leeuw construction.
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 Springer-Verlag
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.
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.
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.
Hoare B, Pittelli A, Torrielli A (2014) Integrable S-matrices, massive and massless modes and the AdS_2 x S^2
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.
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 ¨ m1¨ m2¨m1+m2 and merge ¨ m1+m2¨m1¨m2 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.
Bassetto A, De Pol G, Torrielli A, Vian F (2006) Area preserving diffeomorphisms and Yang-Mills theory in two noncommutative dimensions,NUCLEAR PHYSICS B-PROCEEDINGS SUPPLEMENTS 161 pp. 21-26 ELSEVIER SCIENCE BV
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.
Schubert C, Torrielli A (2010) Open string pair creation from worldsheet instantons,J PHYS A-MATH THEOR 43 (40) IOP PUBLISHING LTD
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.
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 Elsevier
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.
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.
Torrielli A (2011) The Hopf superalgebra of AdS/CFT,J. Geom. Phys. 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.
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.
Moriyama S, Torrielli A (2007) A Yangian double for the AdS/CFT classical r-matrix,Journal of High Energy Physics 2007 (6) Institute of Physics
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.
Arutyunov G, De Leeuw M, Torrielli A (2009) Universal blocks of the AdS/CFT$ AdS/CFT scattering matrix,Journal of High Energy Physics 5 Institute of Physics
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.
Dorn H, Torrielli A (2004) Loop Equation in Two-dimensional Noncommutative Yang-Mills Theory,Journal of High Energy Physics 01 Institute of Physics
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
Matsumoto T, Moriyama S, Torrielli A (2008) A secret symmetry of AdS/CFT S-matrix,INTERNATIONAL JOURNAL OF MODERN PHYSICS A 23 (14-15) pp. 2262-2263 WORLD SCIENTIFIC PUBL CO PTE LTD
Torrielli A (2005) Focus on quantum field theory,In: Kovras O (eds.), 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 ...
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) American Physical Society
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 2À/?» 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.
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.
Babichenko A, Torrielli A (2012) Multi-parametric R-matrix for the sl(2|1) Yangian, Journal of Mathematical Physics 53 (8) American Institute of Physics
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.
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 Springer
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.
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) Institute of Physics
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.
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 Springer
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.
Kozlowski K, Sklyanin E, Torrielli Alessandro (2017) Quantization of the Kadomtsev-Petviashvili equation,Theoretical and Mathematical Physics 192 (2) pp. 259-283 Springer Verlag
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 ¨ m1¨ m2¨m1+m2 and merge ¨ m1+m2¨m1¨m2 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.
Fontanella A, Torrielli A (2017) Massless AdS2 scattering and Bethe Ansatz,Journal of High Energy Physics 75 Springer Verlag
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.
Torrielli Alessandro (2017) On AdS2/CFT1 transfer matrices, Bethe ansatz and scale invariance,Journal of Physics A: Mathematical and Theoretical 51 (1) 015402 IOP Publishing
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.
Borsato R, Torrielli Alessandro (2018) q
Nuclear Physics B 928 pp. 321-355 Elsevier
We consider the exact S-matrix governing the planar spectral problem for strings on AdS5×
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 Riccardo, Strömwall Joakim, Torrielli Alessandro (2018) q-Poincaré invariance of the AdS3/CFT2 R-matrix,Physical Review D 97 066001 pp. 066001-1-066001-17 American Physical Society
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-Poincaré 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.
Bombardelli Diego, Stefanski, jr Bogdan, Torrielli Alessandro (2018) The low-energy limit of AdS3/CFT2 and its TBA,Journal of High Energy Physics Springer Verlag
We investigate low-energy string excitations in AdS3 × S
3 × T
. When the worldsheet is decompactified,
the theory has gapless modes whose spectrum at low energies is determined by massless
relativistic integrable S matrices of the type introduced by Al. B. Zamolodchikov. The S matrices
are non-trivial only for excitations with identical worldsheet chirality, indicating that the low-energy
theory is a CFT2. We construct a Thermodynamic Bethe Ansatz (TBA) for these excitations and
show how the massless modes? wrapping effects may be incorporated into the AdS3 spectral problem.
Using the TBA and its associated Y-system, we determine the central charge of the low-energy CFT2
to be c = 6 from calculating the vacuum energy for antiperiodic fermions - with the vacuum energy
being zero for periodic fermions in agreement with a supersymmetric theory - and find the energies
of some excited states.
Scientific abstract
In this dissertation we perform a series of study concerning dualities and integrability properties underlying the AdS3/CFT2 and AdS2/CFT1 correspondences. These are particularly interesting because symmetry do not constrain the dynamics of AdS3 and AdS3 superstrings in the same way as in the higher dimensional instances of AdS/CFT, allowing for novel phenomena such as the presence of massless worldsheet modes or non-coset fermions. We will investigate the self-duality of Green?Schwarz supercoset sigma models on AdSdåSdåSd (d = 2, 3), whose isometry supergroups are (d ? 1) copies of the exceptional Lie supergroup D(2, 1; µ). Our main finding is that additional complex T-dualities along one of the spheres Sd are needed to map the superstring action to itself. Importantly, this proves dual superconformal symmetry of their CFT duals via AdS/CFT. Dual superconformal symmetry is strictly related to integrability, which we study in depth for both AdS3 and AdS2 superstrings. Indeed, we will derive the exact S-matrix conjectured to be related to the massive modes of type IIB AdS2 å S2 å T6 superstrings. This S-matrix
is psuc(1|1) invariant and it was found to be in perfect agreement with the tree-level result following from string perturbation theory. We also unveil the Yangian algebra ensuring the integrability of the AdS2åS2åT6 superstring
in the planar limit, Y[psu(1|1)c], as well as its secret symmetries. By using the RTT realisation, we provide two diµerent representations of the Hopf algebra: one is reminiscent of the Yangian underlying AdS5/CFT4, but it is not of evaluation type. The other representation, obtained
from co-commutativity, is instead of evaluation type.
We explore two limits of the S-matrix for AdS2/CFT1: one is the classical r-matrix, which is the first non-trivial order in the 1/g expansion, with g being the eµective tension in the AdS2 å S2 å T6 superstring action. In this limit, corresponding to classical strings, we found
that secret symmetries not only are present, but also essential to formulate the classical rmatrix in a universal, representation independent form. On the other hand, the limit g ! 0, corresponding to the weakly coupled CFT1, shows that the dual integrable model is described
by an eµective theory of free fermions on a periodic spin-chain if g = 0, while one obtains a non-trivial spin chain of XYZ type if g 6= 0.
Finally, we investigate Yangian and secret symmetries for AdS3 type IIB superstring backgrounds, verifying the persistence of such structures in AdS3/CFT2. Especially, we find that the antipode map, related to crossing symmetry, exchanges in a non-trivial way left and right generators of Y[psu(1|1)2c], the Yangian underlying the integrability of AdS3 superstrings.
Keywords and AMS Classification Codes:
Fontanella A., Torrielli A. (2019) Geometry of Massless Scattering in Integrable Superstring,Journal of High Energy Physics 2019 (6) 116 pp. 1-34 Springer Verlag
We consider the action of the q-deformed Poincaré superalgebra on the massless non-relativistic R-matrix in ordinary (undeformed) integrable AdS‚×S²×Tv type IIB superstring theory. The boost generator acts non-trivially on the R-matrix, confirming the existence of a non-relativistic rapidity ³ with respect to which the R-matrix must be of difference form. We conjecture that from a massless AdS/CFT integrable relativistic R-matrix one can obtain the parental massless non-relativistic R-matrix simply by replacing the relativistic rapidity with ³. We check our conjecture in ordinary (undeformed) AdSn×Sn×T10?2n, n=2,3. In the case n=3, we check that the matrix part and the dressing factor - up to numerical accuracy for real momenta - obey our prescription. In the n=2 case, we check the matrix part and propose the non-relativistic dressing factor. We then start a programme of classifying R-matrices in terms of connections on fibre bundles. The conditions obtained for the connection are tested on a set of known integrable R-matrices.
Fontanella Andrea, Ohlsson Sax Olof, StefaDski Bogdan jr., Torrielli Alessandro (2019) The Effectiveness of Relativistic Invariance in AdS3,Journal of High Energy Physics 105 Springer Verlag
We use relativistic invariance to investigate two aspects of integrable AdS3 string theory. Firstly, we write down the all-loop TBA equations for the massless sector of the theory with R-R flux, using the recently discovered hidden relativistic symmetry. Secondly, for the low-energy relativistic limit of the theory with NS-NS flux we write down the S matrix, dressing factors and TBA. We find that the integrable system coincides with a restriction to AdS3 of the relativistic q-deformed AdS5 theory. We also comment on the relativistic limit of the small-k NS-NS theory.
Garcia Juan Miguel Nieto, Torrielli Alessandro (2020) Norms and scalar products for AdS3,Journal of Physics A: Mathematical and Theoretical 53 (14) 145401 IOP Publishing
We compute scalar products and norms of Bethe vectors in the massless sector of AdS3 integrable
superstring theories, by exploiting the general difference form of the S-matrix of massless excitations
in the pure Ramond-Ramond case, and the difference form valid only in the BMN limit in the mixedflux
case. We obtain determinant-like formulas for the scalar products, generalising a procedure
developed in previous literature for standard R-matrices to the present non-conventional situation.
We verify our expressions against explicit calculations using Bethe vectors for chains of small length,
and perform some computer tests of the exact formulas as far as numerical accuracy sustains us. This
should be the first step towards the derivation of integrable form-factors and correlation functions
for the AdS3 S-matrix theory.
Nieto Garcia Juan, Torrielli Alessandro, Wyss Leander (2020) Boost generator in AdS3 integrable superstrings for general braiding,Journal of High Energy Physics Springer Verlag
In this paper we find a host of boost operators for a very general choice of coproducts in AdS3-
inspired scattering theories, focusing on the massless sector, with and without an added trigonometric
deformation. We find that the boost coproducts are exact symmetries of the R-matrices we construct,
besides fulfilling the relations of modified Poincar´e-type superalgebras. In the process, we discover
an ambiguity in determining the boost coproduct which allows us to derive differential constraints on
our R-matrices. In one particular case of the trigonometric deformation, we find a non-coassociative
structure which satisfies the axioms of a quasi-Hopf algebra.