Confinement and Mayer cluster expansions

Abstract

In these notes, we study a class of grand-canonical partition functions with a kernel depending on a small parameter . This class is directly relevant to Nekrasov partition functions of N = 2 SUSY gauge theories on the 4d Ω-background, for which is identified with one of the equivariant deformation parameter. In the NekrasovShatashvili limit → 0, we show that the free energy is given by an on-shell effective action. The equations of motion take the form of a TBA equation. The free energy is identified with the Yang-Yang functional of the corresponding system of Bethe roots. We further study the associated canonical model that takes the form of a generalized matrix model. Confinement of the eigenvalues by the short-range potential is observed. In the limit where this confining potential becomes weak, the collective field theory formulation is recovered. Finally, we discuss the connection with the alternative expression of instanton partition functions as sums over Young

Fermions and scalars in N=4 Wilson loops at strong coupling and beyond

“…A tree cluster T n is a connected cluster containing exactly the minimal number of links, namely n − 1. The crucial remark is that all the connected clusters contribute to the same order, see the discussion in [55,56]. Indeed, although one would naively expect f (u ij ) to contribute at order 2 , more subtly for small distances a link is proportional (in a distributional sense) to a Dirac δ-function, so that f (u ij ) ∼ δ(u ij ), i.e.…”

“…Then, the long-range potential will be treated via a Hubbard-Stratonovich transformation. Finally, the two potentials will combined into the full partition function: we will find a representation for Z as a sum over bound states of instantons [55,56], and a TBA equation in the NS limit.…”

Exact quantization conditions for the relativistic Toda lattice

“…Alternatively, one can write these quantization conditions in terms of integral equations of the TBA type [8]. These TBA equations are obtained by resumming the instanton expansion of the partition function (see [18,19] for a detailed derivation), and as shown in [20] they are equivalent to the quantization conditions in [1][2][3][4].…”

Strong Wilson polygons from the lodge of free and bound mesons

“…This pattern perfectly reproduces that used in [42] to re-sum the series to a TBA-like system (with Yang-Yang functional in place of the usual free energy). This mimics somehow the so-called Nekrasov-Shatashvili (NS) limit 2 → 0 [44], which corresponds to our strong coupling g ∼ i/ 2 → +∞: indeed, in NS limit the leading contribution is an infinite sum over instantons and their bound states which gives rise to a TBA-like equation [53,54]. In addition, we have noticed that contributions from unbound fermions are sub-leading; in this sense fermions disappear from the spectrum at infinite coupling and give rise to what we suggestively called confinement as they contribute only via their bound states (mesons of above).…”

“…The entire partition function is a sum over the number, N , of instantons, each interacting with the others and an external potential. In the so-called Nekrasov-Shatashvili (NS) limit 2 → 0, which corresponds to our strong coupling limit, the leading contribution was worked out extensively by [53,54] and resulted in a sum over instantons and their bound states so to give rise to a TBA-like equation. In a nutshell, the limiting series shares the spirit of the meson sector with the hexagonal Wilson loop in [42].…”