In the string theory book by Ibanez and Uranga (click here for the Google books excerpt), the four-dimensional multi-center Taub-NUT metric is written as

$$ds^2 = \frac{V(\textbf{x})}{4}d\textbf{x}^2 + \frac{V(\textbf{x})^{-1}}{4}(dx^{10} + \omega\cdot d\textbf{x})^2$$ $$V(\textbf{x}) = 1 + \sum_{a = 1}^{N}\frac{1}{|\textbf{x}-\textbf{x}_a|}, \qquad \nabla\times\omega = \nabla V(\textbf{x})$$

where $\omega$ the 3D vector potential for $N$ Dirac magnetic monopoles (equivalently $D6$-branes) in $\mathbb{R}^3$ and $\textbf{x} \in R^3$ parametrize the 3D space transverse to the D6 branes.

The authors say

Around each degenerate fiber over a a center $\textbf{x}_a$, the geometry supports a normalizable harmonic $2$-form $\omega_a$. The component of the M-theory 3-form along it,

$$ C_3 = \omega_a \wedge A_1^a$$

produces a $U(1)$ gauge boson, interpreted in the type-IIA picture as the $D6$-brane worldvolume $U(1)$.

Question 1: From simply the solution written above, is it obvious that there is a harmonic 2-form at each $\textbf{x}_a$?

I know that $V(\textbf{x})$ is harmonic in $\mathbb{R}^3$. And at first, it seemed tempting to connect $\omega$ the vector to $\omega_a$ the normalizable 2-form. But that does not seem right because one should have $N$ 2-forms according to the quoted paragraph.

So I dug a bit deeper and found Gubser's TASI lecture notes where, on page 24, he says [for $n+1$ parallel $D6$-branes]

..there are $n$ homologically non-trivial cycles for the $(n+1)$-center Taub-NUT geometry: topoligically this is identical to a resolved $A_n$ singularity. Thus there exist $n$ harmonic, normalizable two-forms, call them $\omega^i$. These forms are localized near the centers of the Taub-NUT space, and they are the cohomological forms dual to the $n$ non-trivial homology cycles. Furtheremore, there is one additional normalizable $2$-form on the Taub-NUT geometry, which can be constructed explicitly for $n = 0$, but which owes its existence to no particular topological property. Let us call this form $\omega^0$. ..we expand the Ramond-Ramond three-form of type-IIA as $$ C_{(3)} = \sum_{i=0}^{n}\omega^i \wedge A_i + \cdots$$

In contrast to the book, where the notion of 2-cycles is described later (and is not seemingly used to assert the existence of normalizable harmonic 2-forms), Gubser's argument seems to suggest that the normalizable harmonic 2-forms exist due to the presence of non-trivial homology cycles.


Question 2: Why does $\omega^0$ not "owe its existence to any particular topological property"? What is the significance of this "additional" normalizable 2-form on the Taub-NUT geometry?

I would have thought that all D6-brane centers would be treated the same way -- if we have $(n+1)$ centers, then why do we seem to make a distinction between $n$ and the $0^{th}$ one?


2 Answers 2

  1. The existence of normalizable harmonic 2-forms in a multi-center Taub-NUT geometry is indeed not obvious at all. The specific form of these forms may be found in Sen's "Dynamics of Multiple Kaluza-Klein Monopoles in M- and String Theory", who gives them (adapted to the notation in the question) as \begin{align} \omega_i & = \mathrm{d}\xi_i \tag{1a}\\ \xi_i & = V^{-1} V_i (\mathrm{d}x^{10} + \omega\cdot\mathrm{d}x) - \omega_i \cdot \mathrm{d}x,\tag{1b}\end{align} where $V_i = \frac{m}{x-x_i}$. Sen, in turn, cites Ruback's "The motion of Kaluza-Klein monopoles" as the origin of these formulae, where it turns out that the exterior derivative in eq. (1a) is only meant to work as the derivative on the coordinates of Taub-NUT space that are not $x^{10}$, if I am reading Ruback's notation correctly. Ruback, in turn, helpfully cites "Page, D.N.: Private communication; Yuille, A.L., PhD. Thesis, University of Combridge (1980) unpublished" as the origin of these formulae, so here the trail ends.

  2. $\omega^0$ in the quote from Gubser does not "owe its existence to any particular property" because it's not associated to the basis of 2-cycles that arises from the lines between the center of the Taub-NUT monopoles - you'll note that applying Poincaré duality to these $n-1$ cycles gives us only $n-1$ compactly supported (and hence normalizable) modes. Note also that these modes are not the $\omega_i$ from eq. (1a), since although normalizable, they are not compactly supported. So what Gubser means is that $\omega^0$ is an additional normalizable mode that does not show up through Poincaré duality, while the rest of the normalizable modes have "compactly supported versions" we can see through the homological argument. Conveniently, this explains a discrepancy between the Taub-NUT version of gauge enhancement and the type IIa $D_6$-brane gauge enhancement:

If you look at the type IIa approach, you'll note that the total gauge group there is $\mathrm{U}(N) = \mathrm{SU}(N)\times\mathrm{U}(1)$, but that the Taub-NUT approach looking at the cycles only delivers $\mathrm{SU}(N)$. The additional normalizable mode not arising from the cycles is precisely the explanation for the $\mathrm{U}(1)$.

  • $\begingroup$ Nice answer. Wrt to point 2, is it possible that this distinction between compact support vs $L^2$ cohomology is what is called 'geometrical' vs 'topological' homology/cohomology towards the end of section 2.4.1 in arxiv.org/abs/0902.0948 ? $\endgroup$
    – jj_p
    Commented Jul 10, 2020 at 14:09
  • $\begingroup$ @jj_p Yes, this appears to be the same line of reasoning. $\endgroup$
    – ACuriousMind
    Commented Jul 13, 2020 at 15:20

I am going to offer a small bone here. I am somewhat interested in the role of Taub-NUT spacetimes, and so contributing will help me to track this in order to read other contributions. Thanks for the Gubser paper.

From a more physical perspective I will just throw out something with magnetic monopoles, which are related to Taub-NUT spacetimes that have a gravi-magnetic monopole content. I like to try to put advanced topics in the context of simple ideas. The Dirac monopole is a semi-infinite solonoid, where the end of the solonoid is approximately a magnetic monopole. This means the other end is sufficiently far away that the dipole field is not locally apparent near this opening. We then have the phase induced on a wave function that passes the solonoid $$ \psi \rightarrow e^{i\frac{e}{\hbar}\oint\vec A\cdot d{\vec x}}\psi. $$ This is the Aharanov- Bohm effect. If we want to elimnate the solonoid we have to in addition eliminate the phase so that $\frac{e}{\hbar}\oint\vec A\cdot d{\vec x} = 2\pi N$, for $N \in \mathbb Z$. The Stokes rule means $$ \frac{e}{\hbar}\oint\vec A\cdot d{\vec x} = \frac{e}{\hbar}\int\int\nabla\times \vec A\cdot d{\vec a} = 2\pi N. $$ This integration of the magnetic field is the magnetic charge $g = \int\int\vec B\cdot d\vec x$ which gives the Bohr-Sommerfeld relationship $eg = 2\pi N\hbar$ presented by Olive and Montenen.

This is a result with a holonomy or phase due to a gauge connection or field flux through a loop or area enclosed by this. The generalization to Taub-NUT spaces concerns the flux through Dp-branes and a duality with gravit-magnetic charge or NUT parameter. The homological cycles define the AB-phase that have discrete structure. I think this is reflected on page 172 of the book with the $A_{k-1}$ singularities on $\mathbb C^2/\mathbb Z_k$ with $(z_1, z_2)\rightarrow (e^{2\pi i/k}z_1, e^{2\pi i/k}z_2)$ that are $SU(2)$ and $\frac{1}{2}$ supersymmetric.

This only partially addresses this, and I am mostly interested in seeing what other activity occurs with this interesting question.

  • 1
    $\begingroup$ Thanks for your reply Lawrence, but how does this observation explain the existence of the 2-forms, and also the zeroth 2-form? $\endgroup$ Commented May 30, 2016 at 14:31
  • 1
    $\begingroup$ I will have to read up on this. This is a bit odd, for the $\frac{1}{{\bf x} - {\bf x}_n}$ suggests several coordinate singular or actual singular regions. The Taub-NUT spacetime is sort of a black hole, but instead of the radius as the horizon region, it is time. These regions where ${\bf x} = {\bf x}_n$ appear to have topological content and are a source of these cycles. $\endgroup$ Commented May 30, 2016 at 23:16
  • $\begingroup$ In what sense are the holomorphic two-forms "normalizable"? $\endgroup$ Commented Jun 8, 2016 at 8:53

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