Explicit construction of Yang-Mills instantons on ALE spaces

Abstract

We describe the explicit construction of Yang-Mills instantons on Asymptotically Locally Euclidean (ALE) spaces, following the work of Kronheimer and Nakajima. For multicenter ALE metrics, we determine the abelian instanton connections which are needed for the construction in the non-abelian case.We compute the partition function of Maxwell theories on ALE manifolds and comment on the issue of electromagnetic duality. We discuss the topological characterization of the instanton bundles as well as the identification of their moduli spaces. We generalize the 't Hooft ansatz to SU(2) instantons on ALE spaces and on other hyper-Kähler manifolds. Specializing to the Eguchi-Hanson gravitational background, we explicitly solve the ADHM equations for SU(2) gauge bundles with second

Geometric construction of new Yang–Mills instantons over Taub-NUT space

“…Consider the Levi-Civitá connection which is locally represented by an so(4)-valued 1-form ω on T M. Because M is spin and four-dimensional, we can consistently lift this connection to the spin connection, locally given by ω S , on the spin bundle SM (which is a complex bundle of rank four) and can project it to the su(2) ± components. The projected connections A ± live on the chiral spinor bundles S ± M where the decomposition SM = S + M ⊕ S − M corresponds to the above splitting of Spin (4). One can raise the question what are the conditions on the metric g for either A + or A − to be self-dual (seeking for antiself-dual solutions is only a matter of reversing the orientation of M).…”

“…There are only very few SU(2) Yang-Mills instantons on Taub-NUT space found in the literature compared to the case of flat R 4 (see [18] and [1]) and in general to ALE spaces (see [21] and also [4]). There is a complete ADHM-Nahm data for the U(1)-invariant self-dual configuration on Taub-NUT space described in [7].…”

Multi-instanton calculus on ALE spaces

“…See [18] for an exhaustive description of these points. Starting from the datum Ξ, it is possible to build a two form analogous to (2.3), invariant under U(|Γ|) transformations.…”

“…The same holds true for the cases with lower supersymmetries in which large N geometries of the type AdS 5 × S 5 /Z 2 , AdS 5 × S 3 , AdS 5 × S 1 and AdS 5 × S 5 /Z p [14,15,16] were recovered. At last these examples were reconsidered in [17] from the D brane viewpoint: according to the type of probe used to test the space-time geometry, the AdS 5 × S 1 (pure N = 2 SYM), AdS 5 × S 5 /Z p (quiver case) and AdS 5 /Z p × S 5 (ALE case) cases were recovered at finite n. The present paper draws largely from this last reference and from [18] in which SYM theories on ALE instantons were studied as exact solutions of the four dimensional heterotic string equations of motion with constant dilaton and zero torsion. The techniques of the two last mentioned reference can be very profitably incorporated into the scheme of the localization described in [2,3].…”

“…Moreover the moduli space of gauge connections must be made compact by introducing a regularization, ζ. The blow down ALE is defined via an orbifold projection with Γ inside T ǫ [17,18]. This implies in particular that fixed points under T ǫ are automatically invariant under Γ and therefore the analysis on R 4 adapts easily to the ALE case.…”