The calculus of many instantons

Abstract

We describe the modern formalism, ideas and applications of the instanton calculus for gauge theories with, and without, supersymmetry. Particular emphasis is put on developing a formalism that can deal with any number of instantons. This necessitates a thorough review of the ADHM construction of instantons with arbitrary charge and an in-depth analysis of the resulting moduli space of solutions. We review the construction of the ADHM moduli space as a hyper-Kähler quotient. We show how the functional integral in the semi-classical approximation reduces to an integral over the instanton moduli space in each instanton sector and how the resulting matrix partition function involves various geometrical quantities on the instanton moduli space: volume form, connection, curvature, isometries, etc. One important conclusion is that this partition function is the dimensional reduction of a higherdimensional gauged linear sigma model which naturally leads us to describe the relation of the instanton calculus to D-branes in string theory. Along the way we describe powerful applications of the calculus of many instantons to supersymmetric gauge theories including (i) the gluino condensate puzzle in N = 1 theories (ii) Seiberg-Witten theory in N = 2 theories; and (iii) the AdS/CFT correspondence in N = 2 and N = 4 theories. Finally, we brielfy review the modifications of the instanton calculus for a gauge theory defined on a non-commutative spacetime and we also describe a new method for calculating instanton processes using a form of localization on the instanton moduli space.

Stringy instantons at orbifold singularities

“…Of course, the ADS superpotential for this case can also be constructed along the same lines as section 3.2, see e.g. [18]. In fact, this derivation is somewhat simpler than the one for the SU(N) gauge group since there are no ADHM constraints at all in the one instanton case.…”

“…The above action is exactly the same which appears in the ADHM construction as reviewed in [18]. Gauge theory nodes are represented by round circles, instanton nodes by squares.…”

“…[18]. We thus quickly go to the result referring the reader to the above review for further details.…”

“…The last fermionic integration conspires to cancel the anti-holomorphic contributions and gives 15) which is just the expected ADS superpotential for N f = N c − 1, the only case where such non-perturbative contribution is generated by a genuine one-instanton effect and not by gaugino condensation. In (3.15) Λ is the SQCD strong coupling scale that is reconstructed by the combination of e −8π 2 /g 2 coming from the instanton action with various dimensional factors coming from the normalization of the instanton measure [18].…”

String theory dual of the β-deformed gauge theory

“…In section 2.1 we found that the Lunin-Maldacena expression for τ in Eq. (2.6) did not transform covariantly under the 7 Where T is a T-duality and s is a shift. 8 In particular, the dilaton-axion field τ in the Lunin-Maldacena background does not agree with the SYM instanton expression (4.10) unless β is small.…”