Moduli spaces of SO(8) instantons on smooth ALE spaces as Higgs branches of 4d \( \mathcal{N} \) = 2 supersymmetric theories

Yuji Tachikawa 0 0 Department of Physics, Faculty of Science, University of Tokyo , Bunkyo-ku, Tokyo 133-0022 , Japan Institute for the Physics and Mathematics of the Universe, University of Tokyo , Kashiwa, Chiba 277-8583, Japan The worldvolume theory of D3-branes probing four D7-branes and an O7-plane on C2/Z2 is given by a supersymmetric USp USp gauge theory. We demonstrate that, at least for a particular choice of the holonomy at infinity, we can go to a dual description of the gauge theory, where we can add a Fayet-Iliopoulos term describing the blowing-up of the orbifold to the smooth ALE space. This allows us to express the moduli space of SO(8) instantons on the smooth ALE space as a hyperkahler quotient of a flat space times the Higgs branch of a class S theory. We also discuss a generalization to C2/Z2n, and speculate how to extend the analysis to bigger SO groups and to ALE spaces of other types. 1 Introduction 2 3 4 SO(8) instantons on C2g/Z2 2.1 Basic mathematical facts 2.2 Analysis using string dualities 2.2.1 T-duality to the D4-D6-O6 system 2.2.2 Re-interpretation as a class S construction 2.2.3 Identification of the new contribution 2.3 Summary of the procedure 2.4 A more field-theoretical approach 2.4.1 The infrared dual of the USp(2k + 2) theory 2.4.2 The infrared dual of the total system SO(8) instantons on C2g/Z2n 3.1 String-theoretic analysis 3.2 Field-theoretic analysis Conclusions and speculations The objective of this paper is to revisit the problem of the gauge theory description of D3branes probing a D7-O7 system on smooth asymptotically-locally-Euclidean (ALE) spaces. Let us first recall what the difficulty was. It is by now well-known that the open-string description of k Dp-branes probing flat D(p + 4)-branes, with or without O(p + 4)-plane, realizes the ADHM construction of the moduli space of instantons. When the (p + 4)-branes are put on an orbifold C2/Z2, the world-volume theory of Dp-branes becomes a quiver gauge theory. Without any O(p + 4)plane in place, the gauge group is of the form U(k) U(k0), and the blow-up parameters of the orbifold are given by the FI terms of the theory. This reproduces Kronheimer-Nakajima construction [1, 2] of unitary instantons on the ALE space C^2/Z2, as first shown in [3]. There is a problem with an O(p + 4)-plane, however. The gauge group of the system, which can be found by quantizing open strings on the orbifold, is now USp(2k) USp(2k0), for which we cannot add any FI terms. Still, it is clear geometrically that we can still blow up the orbifold. We would like to understand this process better in the gauge theory language and to find a way to describe the moduli space of orthogonal instantons on smooth ALE spaces. U(k)U(k) gauge theory U(k)U(k) gauge theory + FI term USp(2k)USp(2k) gauge theory When p = 2, the gauge theory is three dimensional with N = 4 supersymmetry. FI terms cannot be added directly to the USp USp gauge theory, but we can use the 3d mirror description (see e.g. [46] for recent discussions), where the blow-up parameters are visible as hypermultiplet mass terms. The moduli space of orthogonal instantons on a smooth ALE space is then given as the quantum-corrected Coulomb branch of this mirror theory. In general, describing the Coulomb branch of 3d N = 4 theories is a difficult problem, and therefore this construction does not yet tell us much about the moduli space of orthogonal instantons on the smooth ALE spaces. When p = 5, the gauge theory is six dimensional with N = (1, 0) supersymmetry. Here, as noticed first in [7], the blow-up parameters are a part of hypermultiplets involved in the transition between the tensor branch and the Higgs branch, about which not much is understood yet either. In this paper, we take p = 3, so that the gauge theory is four dimensional with N = 2 supersymmetry. In the last few years, a significant progress has been made in the understanding of the duality of such systems. We will see that, at least for SO(8) instantons with a particular holonomy at infinity, we can go to a dual description of the original gauge theory, where we can add appropriate FI terms. This method will give a description of the moduli space of such instantons as a hyperkahler quotient of a flat space times the Higgs branch of a particular class S theory. When the instanton number is sufficiently small, the moduli space reduces to a hyperkahler quotient of a flat space times a nilpotent orbit. In the next section, we study SO(8) instantons on the blown-up ALE space C^2/Z2. We provide two complimentary approaches leading to the same conclusion: one uses the embedding to string theory directly, and the other uses a field-theoretical infrared duality recently discussed in [8, 9]. In section 3, a generalization to C^2/Z2n will be described. We conclude with a discussion in section 4, where we speculate how we can extend the analysis to larger SO groups and to ALE spaces of other types. SO(8) i (...truncated)