Tópicos Teoria Quântica de Campos: Advanced matrix models

Código: PFIS2095
Curso: Mestrado em Física
Créditos: 4
Carga horária: 60
Ementa: 1. The O(n) loop model on random maps. Maps and loop configurations. Statistical weights and partition functions. Phase diagram and critical points.
Combinatorial solution: nested loop approach, loop equations, transfer matrix, functional equations. O(n) matrix model. Spectral curve, critical behaviour and
relation to Liouville gravity.

2. SOS, RSOS and A-D-E models on planar graphs. Statistical weights and partition functions. Formulation in terms of loop gas.
Combinatorial solution by nested loop approach. Spectral curve, critical behaviour and relation to Liouville gravity. The A-D-E matrix models
as minimal models of 2D quantum gravity. The SOS model on random maps and loop ensembles associated with affine Lie algebras of A-D-E type.

3. The six-vertex model on random maps. Statistical weights and partition functions. Reformulation as a loop model and combinatorial solution.
The six-vertex matrix model. Critical behaviour and relation to Matrix Quantum Mechanics.
Bibliografia: [1] M. Gaudin and I. Kostov, O(n) model on a fluctuating random lattice: some exact results. Phys. Lett., B220:200, 1989.

[2] I. Kostov. Strings with discrete target space. Nucl. Phys., B376:539–598, 1992, http://arxiv.org/abshep-th/9112059.

[3] I. Kostov, Strings with discrete target space, Nucl.Phys. B376, 539–598(1992), arXiv:hep-th/9112059, http://arxiv.org/abs/hep-th/9112059

[4] I. Kostov, Exact solution of the six-vertex model on a random lattice, Nucl.Phys., B575:513–534, 2000, http://arxiv.org/abs/hep-th/9911023.

[5] I. Kostov, B. Ponsot, and D. Serban, Boundary liouville theory and 2d quantum gravity. Nucl.Phys., B683:309–362, 2004, http://arxiv.org/abs/hep-th/0307189

[6] I. K. Kostov, Boundary Loop Models and 2D Quantum Gravity, Lecture notes of the summer school on Exact methods in low-dimensional statistical
physics and quantum computing, Les Houches, June 30 – August 1, 2008, Oxford Univ. Press, 2008.

[7] G. Borot, B. Eynard, Enumeration of maps with self avoiding loops and the O(n) model on random lattices of all topologies, J. Stat. Mech. 2011, P01010
(2011), arXiv:0910.5896

[8] A. Elvey Price and P. Zinn-Justin. The six-vertex model on random planar maps revisited. Journal of Combinatorial Theory, Series A, 196:105739,
2023, arXiv:2007.07928

[9] G. Borot, J. Bouttier, and B. Duplantier, Nesting statistics in the O(n) loop model on random planar maps, Communications in Mathematical Physics, 404(3):1125–1229, 2023