Dr Alessandro Torrielli
Lecturer
Email: a.torrielli@surrey.ac.uk
Phone: Work: 01483 68 2955
Room no: 42 AA 04
Further information
Biography
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:
Evgeny Sklyanin (University of York) and Benoit Vicedo (University of Hertfordshire) on non-ultralocality in integrable systems.
Bogdan Stefanski (City University London), Riccardo Borsato / Olof Ohlsson-Sax / Alessandro Sfondrini (Utrecht University) on alternating spin-chains based on superalgebras and the AdS_3 / CFT_2 correspondence
Publications
Journal articles
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(2012) 'On the massless modes of the AdS3/CFT2 integrable systems'. Submitted to Journal of High Energy Physics,
[ Status: Accepted ] - .
(2012) 'Multi-parametric R-matrix for the sl(2|1) Yangian'. American Institute of Physics Journal of Mathematical Physics, 53 (8)doi: 10.1063/1.4740022
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(2012) 'The quantum affine origin of the AdS/CFT secret symmetry'. IOP PUBLISHING LTD JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 45 (17) Article number ARTN 175202 Full text is available at: http://epubs.surrey.ac.uk/315149/
- . (2012) 'Review of AdS/CFT Integrability: An Overview'. Letters in Mathematical Physics, 99 (1-3), pp. 3-32.
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(2011) 'Yangians, S-matrices and AdS/CFT'. IOP PUBLISHING LTD JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 44 (26)Full text is available at: http://epubs.surrey.ac.uk/37215/
Abstract
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.
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(2011) 'Review of AdS/CFT Integrability. Chapter VI.2: Yangian Algebra'. Springer Letters in Mathematical Physics, 99 (1-3), pp. 547-565.Full text is available at: http://epubs.surrey.ac.uk/37214/
Abstract
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.
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(2010) 'Open string pair creation from worldsheet instantons'. IOP PUBLISHING LTD J PHYS A-MATH THEOR, 43 (40)Full text is available at: http://epubs.surrey.ac.uk/72393/
Abstract
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.
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(2010) 'On Yangian and long representations of the centrally extended su(2|2) superalgebra'. Springer Journal of High Energy Physics, 6Full text is available at: http://epubs.surrey.ac.uk/72391/
Abstract
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.
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(2009) 'The bound state S-matrix for AdS(5) x S-5 superstring'. ELSEVIER SCIENCE BV NUCLEAR PHYSICS B, 819 (3), pp. 319-350.Full text is available at: http://epubs.surrey.ac.uk/72397/
Abstract
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.
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(2009) 'On Drinfeld's second realization of the AdS/CFT su(2|2) Yangian'. ELSEVIER SCIENCE BV JOURNAL OF GEOMETRY AND PHYSICS, 59 (4), pp. 489-502.Full text is available at: http://epubs.surrey.ac.uk/72389/
Abstract
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.
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(2009) 'Structure of the string R-matrix'. IOP PUBLISHING LTD JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 42 (5)Full text is available at: http://epubs.surrey.ac.uk/72396/
Abstract
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.
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(2009) 'Universal blocks of the AdS/CFT$ AdS/CFT scattering matrix'. Institute of Physics Journal of High Energy Physics, 5Full text is available at: http://epubs.surrey.ac.uk/72398/
Abstract
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.
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(2009) 'Bound state transfer matrix for AdS5×S5 superstring'. Institute of Physics Journal of High Energy Physics, 2009 (10)Full text is available at: http://epubs.surrey.ac.uk/72380/
Abstract
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.
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(2008) 'Drinfeld second realization of the quantum affine superalgebras of $D^{(1)}(2,1;x)$ via the Weyl groupoid'. Research Institute for Mathematical Sciences, Kyoto University RIMS Kokyuroku Bessatsu, (B8), pp. 171-216.Full text is available at: http://epubs.surrey.ac.uk/72384/
Abstract
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.
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(2007) 'Large N expansion of q-deformed two-dimensional Yang-Mills theory and Hecke algebras'. SPRINGER COMMUNICATIONS IN MATHEMATICAL PHYSICS, 273 (2), pp. 317-355.Full text is available at: http://epubs.surrey.ac.uk/72386/
Abstract
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.
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(2007) 'A secret symmetry of the AdS/CFT S-matrix'. Institute of Physics Journal of High Energy Physics, 9Full text is available at: http://epubs.surrey.ac.uk/72376/
Abstract
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.
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(2007) 'Area-preserving diffeomorphisms in gauge theory on a non-commutative plane: A lattice study'. Institute of Physics Journal of High Energy Physics, 2007 (8)Full text is available at: http://epubs.surrey.ac.uk/72379/
Abstract
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.
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(2007) 'Classical r matrix of the su(2|2) super Yang-Mills spin chain'. American Physical Society Physical Review D - Particles, Fields, Gravitation and Cosmology, 75 (10)Full text is available at: http://epubs.surrey.ac.uk/72381/
Abstract
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.
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(2007) 'A Yangian double for the AdS/CFT classical r-matrix'. Institute of Physics Journal of High Energy Physics, 2007 (6)Full text is available at: http://epubs.surrey.ac.uk/72377/
Abstract
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.
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(2006) 'Hopf algebra structure of the AdS/CFT S-matrix'. AMERICAN PHYSICAL SOC PHYSICAL REVIEW D, 74 (6)Full text is available at: http://epubs.surrey.ac.uk/72385/
Abstract
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.
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(2006) 'ALE spaces from noncommutative U (1) instantons via exact Seiberg-Witten map'. ELSEVIER SCIENCE BV PHYSICS LETTERS B, 634 (4), pp. 427-433.Full text is available at: http://epubs.surrey.ac.uk/72378/
Abstract
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.
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(2006) 'D-branes in overcritical electric fields'. AMERICAN PHYSICAL SOC PHYSICAL REVIEW D, 73 (2)Full text is available at: http://epubs.surrey.ac.uk/72383/
Abstract
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.
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(2005) 'Wrapping interactions and the genus expansion of the 2-point function of composite operators'. Elsevier Nuclear Physics B, 723 (1-2), pp. 3-32.Full text is available at: http://epubs.surrey.ac.uk/72399/
Abstract
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.
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(2005) 'On the invariance under area preserving diffeomorphisms of noncommutative Yang-Mills theory in two dimensions'. Institute of Physics Journal of High Energy Physics, (05)Full text is available at: http://epubs.surrey.ac.uk/72390/
Abstract
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.
- . (2005) 'Theory of light emission in sonoluminescence based upon transitions in confined atoms'. EDP Sciences Eur. Phys. J. D, 33
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(2004) 'Loop Equation in Two-dimensional Noncommutative Yang-Mills Theory'. Institute of Physics Journal of High Energy Physics, 01Full text is available at: http://epubs.surrey.ac.uk/72387/
Abstract
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.
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(2004) 'One-loop unitarity of string theories in a constant external background and their Seiberg-Witten limit'. Institute of Physics Journal of High Energy Physics, (01)Full text is available at: http://epubs.surrey.ac.uk/72392/
Abstract
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.
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(2003) 'Cutting rules and perturbative unitarity of noncommutative electric-type
field theories from string theory'. American Physical Society Phys.Rev. D, 67Full text is available at: http://epubs.surrey.ac.uk/72382/
Abstract
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.
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(2002) 'Scaling properties of the perturbative Wilson loop in two-dimensional
non-commutative Yang-Mills theory'. American Physical Society Phys.Rev. D, 66Full text is available at: http://epubs.surrey.ac.uk/72395/
Abstract
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.
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(2001) 'Perturbative Wilson loop in two-dimensional non-commutative Yang-Mills theory'. Elsevier Nuclear Physics B, 617 (1-3), pp. 308-320.Full text is available at: http://epubs.surrey.ac.uk/72394/
Abstract
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
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(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)Full text is available at: http://epubs.surrey.ac.uk/487902/
Abstract
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.
- . (2011) 'The Hopf superalgebra of AdS/CFT'. J. Geom. Phys., Leiden, The Netherlands: The Interface of Integrability and Quantization 61, pp. 230-236.
- . (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.
- . (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)
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(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.Full text is available at: http://epubs.surrey.ac.uk/72375/
Abstract
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.
- . (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)
- . (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
Book chapters
- . (2005) 'Focus on quantum field theory'. in Kovras O (ed.) Focus on quantum field theory Nova Science Pub Inc
Theses and dissertations
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(2003) Noncommutative perturbative quantum field theory: Wilson loop
in two-dimensional Yang-Mills, and unitarity from string theory. Full text is available at: http://epubs.surrey.ac.uk/75487/
Teaching
Personal tutor to 11 Mathematics students (years 1-2)
Quantum Mechanics for year 3 Mathematics
Departmental Duties
Summer Projects Coordinator
Visiting Tutor for Placement Year
