Hirota solutions of TBA and NLIE Francesco Ravanini Cortona 2010 A.D. 1088

Exact tools for FSE2 Les Houches, 1 Mar 2010 Finite size effects Luscher Finite size effectsS-matrix

Exact tools for FSE3 Les Houches, 1 Mar 2010 Scaling functions M l E Cardy, Blote, Nightingale 1984 link with CFT at UV for the i-th excited state (0=vacuum)

Exact tools for FSE4 Les Houches, 1 Mar 2010 T.B.A. (Thermodynamic Bethe Ansatz) Al. Zamolodchikov 1990 Integrability (2D) Thermodynamic Bethe Ansatz R L

Exact tools for FSE5 Les Houches, 1 Mar 2010 Scaling function of the vacuum (Casimir effect) Compute f(r): Dynamics dictated by Bethe-Yang eqs. in the thermodynamic limit Form of S-matrix Diagonalize color transfer matrix by Bethe ansatz (Al. Zamolodchikov, 1991)

Exact tools for FSE6 Les Houches, 1 Mar 2010 Example: Sine-Gordon – S-matrix (Zam-Zam, 1979) has a dressing factor and a matrix part coinciding with the XXZ spin ½ R-matrix

Exact tools for FSE7 Les Houches, 1 Mar 2010 The color transfer matrix is diagonalized by the fully inhomogeneous XXZ spin ½ Bethe ansatz Full set of Bethe-Yang equations for Sine-Gordon given by eigenvalues coupled to the exp(iPL) term and Bethe equations for the Bethe roots (magnon excitations)

Exact tools for FSE8 Les Houches, 1 Mar 2010 String hypotesis for the Bethe roots: in the thermodynamic limit the Bethe roots tend to organize as follows:

Exact tools for FSE9 Les Houches, 1 Mar 2010 In the thermodynamic limit the number of roots tends to infinity and they become dense. Intorduce density for each type of n-string and for the corresponding holes Logs of Bethe-Yang equations give coupled integral eqs. for the densities

Exact tools for FSE10 Les Houches, 1 Mar 2010 From density: compute energy, entropy and free energy of the system. Minimum of free energy gives the conditions (TBA equations) If more particles, each type is coupled to an equation with its mass term

Exact tools for FSE11 Les Houches, 1 Mar 2010 Through Fourier transform, one can prove a useful identity valid for all ADE S-matrices, that brings TBA in the so-called Universal Form (Al. Zamolodchikov, 1991 without magnons) (FR, Tateo, Valleriani, 1992 and Quattrini, FR, Tateo, 1993 with magnons)

Exact tools for FSE12 Les Houches, 1 Mar 2010 Dynkin TBA or… the Y-system Equivalent to TBA if complemented by the asymptotic conditions Draw diagrams with adjacency G and H

Exact tools for FSE13 Les Houches, 1 Mar 2010 H diagram can be Dynkin or extended Dynkin (or something else ??? ) (Quattrini, FR, Tateo, 1993) FR, Tateo, Valleriani, 1993 (Dynkin TBA’s) FR, 1992 Hollowood, 1994 Al.Zam

Exact tools for FSE14 Les Houches, 1 Mar 2010 Sine-Gordon for general p rational → Continued Fraction Takahashi, Suzuki (1972) – Mezincescu, Nepomechie (1990) – Tateo (1994) Tateo snakes

Exact tools for FSE15 Les Houches, 1 Mar 2010 Bazhanov, Lukyanov, A.Zamolodchikov (1995) suggest that nodes on H direction are linked with internal symmetry algebra: j labels IRREPS Kuniba, Nakanishi, Suzuki (1996): a,j related to Young tableau of IRREPS But then what about diagrams D or D (1) ? Truncation of Q-group reps. when IRREPS are only up to j=p-1 plus two type II reps labelled by p, p+1 Adjacency: General rational p: Mezincescu, Nepomechie: it gives the same continued fraction decomposition as strings → Tateo snakes

Exact tools for FSE16 Les Houches, 1 Mar 2010 SU(2)xSU(2) Principal chiral model: Y-system y + y - = YY T-system TT = 1 + T + T - Finite difference equation of Hirota type, 2nd order. Solution by Lax pair, known as TQ-relation: TQ = Q + + Q -

Exact tools for FSE17 Les Houches, 1 Mar 2010 Baxter Q-operator Entire function with zeroes coincident with the Bethe roots. Up to prefactor not containing zeroes Im-Periodicity related to relevant perturbing field of UV CFT Al. Zamolodchikov (1991)

Exact tools for FSE18 Les Houches, 1 Mar 2010 Solution Hirota (1981) expresses all the T k ’s in terms of “Gauge” symmetry Leaves the T-system invariant. Y functions are invariant Auxiliary functions (gauge invariant)

Exact tools for FSE19 Les Houches, 1 Mar 2010 Satisfy Use logarithmic index theorem, Fourier transform and some algebraic manipulation to show that satisfies the NLIE takingthe DdV NLIE for Sine-Gordon is recovered

Exact tools for FSE20 Les Houches, 1 Mar 2010 Conclusions TBA applied to relativistic integrable QFT has given very good results on Finite Size Effects NLIE can be seen as a powerful tool, sort of Bethe ansatz for QFT in the continuum The two methods are related thanks to their functional forms: Y-systems and T-systems, and the Baxter Q-operator Applications to recent problems (AdS/CFT): resummation is still an issue (sum of massive nodes is difficult…) Other models can be dealt with this methods, like sausages, surely interesting for a group based in Bologna… Thank you