What can be learnt about Instantons in the CP(N-1) Model?
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In the two-dimensional CP(N-1) model one can parametrize exact many-instanton solutions via N `constituents' (called `zindons'). This parameterization allows, in principle, a complete `melting' of individual instantons. The model is therefore well suited to study whether dynamics prefers a dilute or a strongly overlapping ensemble of instantons. We study the statistical mechanics of instantons both analytically and numerically. We find that at N=2 the instanton system collapses into zero-size instantons. At N=3,4 we find that well-isolated instantons are dynamically preferred though 15-25% of instantons have a considerable overlap with others.Keywords:
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We analyze instantons in the very strongly coupled large-N limit (N → ∞ with g 2 fixed) of large-N gauge theories, where the effect of the instantons remains finite. By using the exact partition function of four-dimensional $ \mathcal{N} $ = 2* gauge theories as a concrete example, we demonstrate that each instanton sector in the very strongly coupled large-N limit is related to the one in the 't Hooft limit (N → ∞ with g 2 N fixed) through a simple analytic continuation. Furthermore we show the equivalence between the instanton partition functions of a pair of large-N gauge theories related by an orbifold projection. This can open up a new way to analyze the partition functions of low/non-supersymmetric theories. We also discuss implication of our result to gauge/gravity dualities for M-theory as well as a possible application to large-N QCD.
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We introduce a new approach to investigate the selection rules governing the contributions of fluxed M5-instantons to the F-theory four-dimensional effective action, with emphasis on the generation of charged matter F-terms. The structure of such couplings is unraveled by exploiting the perturbative and non-perturbative homological relations, introduced in our companion paper arXiv:1506.06764, which encode the interplay between the self-dual 3-form flux on the M5-brane, the background 4-form flux and certain fibral curves. The latter are wrapped by time-like M2-branes representing matter insertions in the instanton path integral. In particular, we clarify how fluxed M5-instantons detect the presence of geometrically massive $U(1)$s which are responsible for `hidden' selection rules. We discuss how for non-generic embeddings the M5-instanton can probe `locally massless' $U(1)$ symmetries if the rank of its Mordell-Weil group is enhanced compared to that of the bulk. As a phenomenological off-spring we propose a new type of non-perturbative corrections to Yukawa couplings which may change the rank of the Yukawa matrix. Along the way, we also gain new insights into the structure of massive $U(1)$ gauge fluxes in the stable degeneration limit.
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We study the Bogomol'nyi–Prasad–Sommerfeld (BPS) vortices in the (1+1)-dimensional supersymmetric U(1) gauged nonlinear sigma model. We use the moduli matrix approach to analytically construct the moduli space of solutions and solve numerically the BPS equations. We identify two topologically inequivalent types of magnetic vortices, which we call S and N vortices. Moreover, we discuss their relation to instantons (lumps) present in the ungauged case. In particular, we describe how a lump is split into a couple of component S–N vortices after gauging. We extend this analysis to the case of the extended Abelian Higgs model with two flavors, which is known to admit semi-local vortices. After gauging the relative phase between fields, semi-local vortices are also split into component vortices. We discuss interesting applications of this simple set-up. Firstly, the gauging of nonlinear sigma models reveals a semiclassical 'partonic' nature of instantons in 1+1 dimensions. Secondly, weak gauging provides for a new interesting regularization of the metric of semi-local vortices.
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We present new evidence for the conjecture that BPS correlation functions in the N=2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N=2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N=2 theory on R^4 with appropriately twisted boundary conditions. For the U(1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U(1) with all chiral couplings included) is given by the free energy of the topological string on CP^1. Gravitational descendants play an important role in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets.
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