Instantons on Calabi–Yau cones

Abstract

The Hermitian Yang-Mills equations on certain vector bundles over Calabi-Yau cones can be reduced to a set of matrix equations; in fact, these are Nahm-type equations. The latter can be analysed further by generalising arguments of Donaldson and Kronheimer used in the study of the original Nahm equations. Starting from certain equivariant connections, we show that the full set of instanton equations reduce, with a unique gauge transformation, to the holomorphicity condition alone.

Instantons on Calabi-Yau and hyper-Kähler cones

“…As explained, for instance, in [21], the condition F (0,2) α = 0 introduces a holomorphic structure on the vector bundle E. Since we have three holomorphic structures arising, the bundle becomes tri-holomorphic. Denote the space of tri-holomorphic connections as…”

“…In this section we describe the space of connections over a hyper-Kähler manifold M 4m and show that it is equipped with a (formal) hyper-Kähler structure, which is induced from M 4m . This account is inspired from the analogous implication for the space of connections over Kähler manifolds, for which we refer to [21,33,34].…”

“…The underlying base for a hyper-Kähler cone is a 3-Sasakian space, while a Calabi-Yau cone starts from a Sasaki-Einstein base. Instantons on certain conical extensions of G-manifolds have been considered, for instance, in [12][13][14][15][16][17][18][19][20][21][22]. In all the references, the instanton equations have been reduced to a set of matrix equations by a certain equivariant ansatz.…”

“…In all the references, the instanton equations have been reduced to a set of matrix equations by a certain equivariant ansatz. The resulting matrix equations for Calabi-Yau cones over arbitrary Sasaki-Einstein manifolds have been discussed in [21] for one choice of boundary conditions. The aim of the present paper is twofold: firstly, to extend the discussion on the Calabi-Yau cones by considering different boundary conditions, which appear to be more physically relevant.…”

“…Interestingly, the instanton matrix equations resulting from the equivariant reduction can be viewed as generalised Nahm's equations, called Nahm-type equations in [21]. Recalling the prominent role of Nahm's equations and nilpotent orbits for BPS boundary conditions for 4-dimensional N =4 super Yang-Mills theories [23,24], the constructions of 4-dimensional N =1 theories by compactifying 6dimensional theories [25,26] or assigning 1/4 BPS boundary conditions [27,28] on 4-dimensional N =4 super-Yang-Mills led to the appearance of generalised Nahm's equations.…”