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# Pedro Gil Vieira

# Research

Generically, I am quite interested in String and Field theories.

Currently, I'm mostly focused on String/Gauge dualities, in particular on the integrable facets of AdS/CFT.

To have an idea of what this subject is about give a look at the INI conference held in Cambridge last December. In particular you can listen to my talk or those of my collaborators here and here.

_{(By the way, those who want to start studying this science and feel the field is
too vast and hard to dwell into (as I did in the beginning) don't
hesitate to mail me. Hopefully I might be able to say something helpful.)}

Here are some (nice) pictures illustrating some of my most recent research activity in this field. Obviously the following explanations will not be self-contained but simply pretend to provide a glimpse of the kind over the work I have been doing. If it seems too technical please open the papers and you will see it is not so.

_{(They are extracted from works done in collaboration with Nikolay Gromov, Vladimir Kazakov, Sakura Schafer-Nameki and Kazuhiro Sakai)}

**Quantum Wrapped Giant Magnons**

Even if we know how to describe a field theory in infinite volume exactly, that is to compute the S-matrix for example between the several asymptotic states, when we put our theory in finite volume new virtual processes, where virtual particles wind the world (see figure), might appear and lead to new physics. We studied such new quantum effects in the context of AdS/CFT allowing use to (start to) understand the quantum finite size corrections to the fundamental excitations (known as Giant Magnons) of light cone gauged string theory in AdS_5xS^5. |

**Generic features of Bethe ansatz and quantum integrable models.**

Nested Bethe ansatz equations appear when we diagonalize particles with isotopic degrees of freedom, that is particles which have some additional quantum numbers. In this case the Bethe equations are equations for at least two types of roots. We found a duality among the solutions of these Bethe equations. In particular, in the scaling limit where Riemann surfaces appear this duality simply amounts to exchanging the Riemann Sheets as represented in this picture |

In Bethe Ansatz some strange configurations can appear in the limit of a large number of Bethe Roots. Despite the appearances this surface is not more complicated than a simple two sheet Riemann surface with a standard square root cut. We call this configurations Zippers. |

Here are some real numerics. The Bethe roots do condense close to the solid lines, our analytical prediction. |

A key formula in the foundations of the analytic Bethe ansatz was
the Bazhanov-Reshetikhin formula (1990) which is a determinant
representation for the quantum transfer matrices in any representation
in terms of the transfer matrices in symmetric representations. Except
for some mathematical proofs, very hard to follow for a physicist,
there was no proof of this relation. We generalized and proved this relation by
translating this formula into some new identities for characters of
group elements. To prove these identities the developed a new
diagrammatic expansion exemplified in the picture to the right. Here w(z) is the generator of symmetric characters and D is a group left co-derivative and solid and dashed lines indicate the tensor structure. |

**Integrability in AdS/CFT - Semi-classical quantization of the string in AdS_5xS^5**

From the works of Kazakov, Marshakov, Minahan, Zarembo (for a particular string sector) and of Beisert, Kazakov, Sakai, Zarembo, ww know that the classical string motions can be mapped to Riemann surfaces (with 8 sheets).

We studied the semi-classical quantization of strings in a very non-trivial background (AdS_5xS^5) by quantizing these algebraic curves:

The several physical fluctuations in the string Bethe ansatz. The 16 elementary physical
excitations are the stacks (bound states) containing the middle node root. From the left to the
right we have four S_5 fluctuations, four AdS5 modes and eight fermionic excitations. The bosonic
(fermionic) stacks contain an even (odd) number of fermionic roots represented by a cross in the
psu(2, 2|4) Dynkin diagram in the left. In this language we could reproduce all previously known results derived by Aryutunov, Frolov, Tseytlin (see e.g. here) for rigid circular strings. |

When we expand the action around some classical solution we obtain a quadratic lagrangian, that is a (infinite) collection of harmonic oscillators.

The frequencies of these oscillators give us the semi-classical spectrum. We understood the analytical behavior of these functions in the n plane (n being the string mode number) and in the x plane (x being the complex variable of the algebraic curve corresponding to the classical solution around which we are expanding). |

When we sum over all fluctuation (with 1/2 and minus signs for fermions) we get the one loop shift. We understood how this quantity is reproduced in the string Bethe ansatz around any classical solution. We were able to identify which contributions are accounted by the so called anomaly terms (finite size corrections to the Bethe ansatz) and by the Hernandez-Lopez phase (a modification of the Bethe equations which was conjectured and which we derived in this way).

**Integrability in AdS/CFT - Quantization of the string in S^5 and classical limit of Bethe ansatz vs Finite Gap**

The quantization of the SO(6) sigma model is achieved by the Bethe ansatz techniques using the bootstrap S-matrices found by the Zamolodchikov brothers. We studied the classical (and conformal) limit of this theory and found that it is described by a particular set of Riemann surfaces. On the other hand we know (see above) that classical strings in the sphere are also described by some algebraic curves.

Via the Zhukosvsky map z=z+1/x we showed that these curves were exactly the same! Here is a picture of the same surface in the two languages (Bethe Ansatz - to the right - and Finite Gap - to the left) |