### Dr Alessandro Torrielli

### Biography

### Biography

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.

### Teaching

Quantum Mechanics for year 3 Mathematics

Quantum Field Theory for year 4 Mathematics

### Departmental dutiesSurreyLearn contact person

SurreyLearn contact person

### Research

### Research interests

Integrable systems, quantum groups, AdS/CFT

### My publications

### Publications

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.

analysis uses heavily the (extended) affine Weyl groups of the affine Lie algebras. In our approach the structures are based on a Weyl groupoid.

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.

Lax-operator approach developed in the work of Bazhanov, Frassek, Lukowski,

Meneghelli and Staudacher.

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.

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.

massive sector, The Journal of High Energy Physics 43 Springer

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.

field theories from string theory, Phys.Rev. D 67 American Physical Society

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.

non-commutative Yang-Mills theory, Phys.Rev. D 66 American Physical Society

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.

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.

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.

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.

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.

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.

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.

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.

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.

superstring,

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.

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.

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.

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.

of gauged WZW-type theories, Nucl. Phys. B 889 pp. 64-86

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.

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.

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.

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.

of noncommutative field theories Alessandro Torrielli* Dipartimento di Fisica

...

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.

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.

based upon transitions in confined atoms, Eur. Phys. J. D 33 EDP Sciences

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.

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.

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.

-Poincaré

supersymmetry

in

Ad

S

5

/CF

T

4, Nuclear Physics B 928 pp. 321-355 Elsevier

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.

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.

3 × T

4

. When the worldsheet is decompactified,

the theory has gapless modes whose spectrum at low energies is determined by massless

relativistic integrable S matrices of the type introduced by Al. B. Zamolodchikov. The S matrices

are non-trivial only for excitations with identical worldsheet chirality, indicating that the low-energy

theory is a 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.