Abstract
We present a range of exact techniques within two-dimensional conformal field theory (CFT), using the Q-state Potts and the O(n) models as exploratory tools. Both are equivalent to models of oriented loops, which act as level lines of a height model. The height model can be treated via a geometrical Coulomb gas construction, giving access to exact bulk and boundary properties. We detail the derivation of critical exponents and relate their physical interpretation to properties of clusters and loops. The underlying Temperley-Lieb algebra is discussed, and we show how the various topological sectors should be combined to yield exact continuum limit partition functions. These give access to probabilistic results, in particular to crossing formulae in percolation. Finally, we discuss how these results can be extended to classes of new conformally invariant boundary conditions, in which the weights of boundary-touching loops are modified.
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Notes
- 1.
This implies that the dual of \( M\left( G \right) \) is a quadrangulation \(\widehat{G}\), which is however different from the quadrangulation G ∪ G ∗. See Fig. 4.1c. The Potts model admits yet another representation, namely as a height model—or RSOS model—on \(\widehat{G}\).
- 2.
The same construction can be taken over for the O(n) loop model (4.18) on condition that the roots be located on the dual lattice, and by allowing for other obvious modifications.
- 3.
We consider the scalar curvature in a generalised sense, so that delta function contributions may be located at the boundaries. Implicitly, we are just applying the Gauss-Bonnet theorem.
- 4.
A similar effect could be obtained by adding a surface magnetic field, but here we do not wish to break the symmetry of the model [typically O(n) in applications to loop models].
- 5.
This should not (as is sometimes seen in the literature) be confused with imposing fixed boundary conditions, which would rather correspond to an infinite symmetry-breaking field applied on the boundary (and is sometimes referred to as normal transition).
- 6.
It makes sense to think of this in the radial quantisation, or transfer matrix, picture. The theories are initially considered on a semi-infinite cylinder (resp. a strip) with specified transverse boundary conditions (periodic, resp. non-periodic) and unspecified longitudinal boundary conditions. This gives access to the transfer matrix eigenvalues. To access the fine structure, such as amplitudes of the eigenvalues, one must impose periodic longitudinal boundary conditions and take the length of the cylinder (resp. strip) to be finite.
- 7.
Technically speaking this is the mixed ordinary-special transition, but we have simplified the terminology according to the above remarks.
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Jacobsen, J.L. (2012). Loop Models and Boundary CFT. In: Henkel, M., Karevski, D. (eds) Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution. Lecture Notes in Physics, vol 853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27934-8_4
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