Branes, Instantons, And Taub-NUT Spaces

TL;DR

This work connects brane constructions to hyper‑Kähler quotient realizations of Taub–NUT spaces and their instanton moduli. By using a D3‑brane probe and T‑duality, Witten derives Taub–NUT as a finite‑dimensional hyper‑Kähler quotient and extends the ADHM framework to instantons on TN_k, including a precise accounting of line bundles, Chern classes, and monodromies. It develops a detailed map between NS5–D5–D3 brane data and topological invariants of instanton bundles on TN_k, including the first and second Chern classes and their infinity monodromies, and provides an M‑theory interpretation via a product of Taub–NUT spaces. The results unify brane physics with mathematical descriptions of instantons on ALF/ALE spaces, and illuminate how T‑duality and Nahm’s equations encode the underlying geometry and topology. These insights advance understanding of brane constructions of instantons and their role in M‑theory dualities.

Abstract

ALE and Taub-NUT (or ALF) hyper-Kahler four-manifolds can be naturally constructed as hyper-Kahler quotients. In the ALE case, this construction has long been understood in terms of D-branes; here we give a D-brane derivation in the Taub-NUT case. Likewise, instantons on ALE spaces and on Taub-NUT spaces have ADHM-like constructions related to hyper-Kahler quotients. Here we refine the analysis in the Taub-NUT case by making use of a D-brane probe, and give an application to M-theory.

Figures (8)

• Figure 1: (a) Nonintersecting curves $l_\sigma$, $\sigma=1,\dots,k$, connecting the points $x_\sigma\in{\mathbb{R}}^3$ to infinity (in the figure, $k=3$). Their inverse images in ${\rm TN}_{\bf k}$ are the noncompact cycles $C_\sigma$. (b) A curve $l_{\sigma \sigma'}$ connecting two of the points $x_\sigma,x_{\sigma'}$, and not meeting any of the others (sketched here for $\sigma,\sigma'=1,2$). Its inverse image is a two-cycle $C_{\sigma\sigma'}\subset{\rm TN}_{\bf k}$ that is topologically $S^2$. $l_{\sigma\sigma'}$ is homologous to the difference $l_\sigma-l_{\sigma'}$ (which represents a curve from $x_\sigma$ to infinity and back to $x_{\sigma'}$).
• Figure 2: The intersection pairings of the cycles $C_{\sigma,\sigma+1 }$ have a natural interpretation in terms of the Dynkin diagram of the group ${\mathrm A}_{k-1}$. More precisely, the cycles $C_{01},C_{12},\dots,C_{k-2,k-1}$, which are arranged here in the horizontal row at the bottom, correspond to the nodes of the ordinary Dynkin diagram of ${\mathrm A}_{k-1}$. The diagonal elements of the intersection matrix are The cycle $C_{k-1,0}$, depicted here at the top, then represents the additional node of the extended Dynkin diagram of this group.
• Figure 3: The horizontal direction in this and later figures represents $S^1$, and the vertical direction symbolically represents the six-dimensions of ${\mathbb{R}}^3_{{\vec{X}}}\times {\mathbb{R}}^3_{\vec{Y}}$. Horizontal solid lines represent D3-branes, vertical solid lines represent NS5-branes, and vertical dotted lines represent D5-branes. (a) An NS5-brane with $r_+$ D3-branes ending on its right and $r_-$ on its left. We set $\delta r=r_+-r_-$. (b) If there are additionally $m$ D5-branes to the right of the given NS5-brane, then its linking number is $\ell=\delta r - m$. In the figure, $m=2$.
• Figure 4: This is part of a larger picture, possibly with additional fivebranes to the left and the right of those that are drawn. Let $\ell$ and $\ell'$ be the linking numbers of the two NS5-branes on the leftmost and rightmost fivebrane in the picture. There is one D5-brane between them; it contributes $-1$ to $\ell$ and 0 to $\ell'$. Any additional D5-branes to the right of the picture contribute $-1$ to both $\ell$ and $\ell'$. So the difference $\Delta\ell=\ell'-\ell$ depends only on the branes that are depicted; its value is $\Delta\ell=1+\delta r'-\delta r$, where $\delta r'=r_4-r_3$ and $\delta r=r_2-r_1$ are the net numbers of D3-branes ending on the two D5-branes.
• Figure 5: After we remove from $S^1$ a point -- called the basepoint -- which does not coincide with the position of any fivebrane, the brane configuration can be mapped to ${\mathbb{R}}$. (We have done this without discussion in all of the previous figures!) In this paper, we choose the basepoint to be just to the left of the lefftmost NS5-brane, so that when we map to ${\mathbb{R}}$, the leftmost fivebrane is of NS type. We label the positions of the NS5-branes from left to right as $y_\sigma$, $\sigma=1,\dots,k$.