Physics on manifolds with exotic differential structures

Ulrich Chiapi-Ngamako¹ ulrich.chiapi.ngamako@umontreal.ca M. B. Paranjape^1,2,3 paranj@lps.umontreal.ca ¹Groupe de physique des particules, Département de physique, Université de Montréal, Campus MIL, 1375 Av Thérèse-Lavoie-Roux,, Montréal, Québec, Canada, H3B 2V4 ² Centre de recherche mathématiques and Institut Courtois, Université de Montréal. ³ Department of Physics, University of Auckland, Auckland, New Zealand, 1010

Abstract

A given topological manifold can sometimes be endowed with inequivalent differential structures. Physically this means that what is meant by a differentiable function (smooth) is simply different for observers using inequivalent differential structures. Specifically, that means that one observer will say that this is a smooth function while the other observer will say it is not smooth. On the other hand, the notion of a continuous function is the same for both, defined by the common topology. In this paper, we examine the import of inequivalent differential structures on the physics of fields obeying the Dirac equation on the 7-sphere. The 7-sphere was the first, celebrated example, found by Milnor, of a topological manifold which can be endowed with a finite number, 28, of inequivalent differential structures. We find that the spectra of these operators are dependent on the choice of differential structure and hence the identical topological manifolds have different physical laws.

pacs:

12.60.Jv,11.27.+d

^†^†preprint: UdeM-GPP-TH-25-304

I Introduction

Almost all of physics relies on being able to take the derivative of some relevant real-valued function. For a general, $n$ dimensional manifold, the notion of what is a differentiable function on the manifold depends on the set of charts (continous, invertible maps (homeomorphisms) from open sets in $\mathbb{R}^{n}$ to the manifold) that cover (each point in the manifold is the image of a point in some chart) the manifold. A function that is defined on the manifold, is pulled-back (i.e. defined by the composition of the map corresponding to the chart with the function on the manifold) to a local set in $\mathbb{R}^{n}$ , and the derivative is accordingly defined by the derivative of the pulled-back function on the local set in $\mathbb{R}^{n}$ .

However, a manifold is only completely defined by the union of open charts that cover the topological space. Where two charts intersect, we can define a function from $\mathbb{R}^{n}\to\mathbb{R}^{n}$ , the so-called transition functions, using one chart to go to a point in the intersection on the manifold and then using the inverse of the second chart to return to $\mathbb{R}^{n}$ . One can impose conditions on these transition functions. A topological manifold requires only that the transition functions be continous. A smooth manifold requires that the transitions functions be infinitely differentiable with infinitely differentiable inverse. An atlas of a $C^{N}$ manifold consists of the union of all charts such that the transition functions and their inverses are $C^{N}$ , i.e. $N$ times continuously differentiable. We say that the manifold admits a $C^{N}$ differentiable structure. It is then clear that a $C^{0}$ manifold, i.e. simply a topological manifold, admits a much larger atlas than a $C^{\infty}$ manifold, the transition functions need only be continous. Indeed then, it is not impossible to imagine that inequivalent subsets of the charts of a topological manifold could give rise to inequivalent $C^{\infty}$ structures, i.e. give rise to different, $C^{\infty}$ atlases that cannot be combined while maintaining the $C^{\infty}$ of each other.

MilnorMilnor (1956) gave the first example of such a case for the 7-sphere. Subsequently Milnor and Kervaire Kervaire and Milnor (1963) analyzed the possibility of inequivalent differentiable structures on all possible finite dimensional manifolds. These examples were mathematical oddities and did not seem very relevant to physics. However, in the 80s, Freedman’s analysis Freedman (1982); Freedman and Taylor (1986) and Donaldson’s subsequent analysis Donaldson (1983) of the moduli spaces of instantons on $\mathbb{R}^{4}$ made the shocking discovery that $\mathbb{R}^{4}$ admits inequivalent differentiable structures, and that $\mathbb{R}^{4}$ is very special in that respect, all other $\mathbb{R}^{N}$ s admit only one differentiable structure. This prompted an intriguing speculation by Taubes Taubes (1984) about how physical systems choose the differentiable structure and what would be import of the inequivalent differentaible structures on the physics. We make some inroads into answering this sort of question by studying physics on the original, exotic 7-spheres of Milnor. Although there has been some work done on physics on exotic 7-spheres, and exotic manifolds in general, see these references for a partial list Yamagishi (1984); Freund (1985); Witten (1985); Asselmeyer (1997); Cavenaghi and Grama (2024); Sladkowski (1996); Schettini Gherardini (2023); Brans and Randall (1993) and the references within, we find that the nature of these mathematical oddities is not generally understood in the theoretical physics community. A very recent article that mirrors our analysis closely, especially concerning the Kaluza-Klein approach, is available here Berman et al. (2024).

II The Exotic 7-Spheres of Milnor

II.1 Manifolds homeomorphic to the 7-sphere

The standard, unit 7-sphere is defined by the set of points in $\mathbb{R}^{8}$ with Cartesian coordinates $(x_{1},x_{2},\cdots,x_{8})$ such that

x_{1}^{2}+x_{2}^{2}+\cdots+x_{8}^{2}=1

(1)

and the differential structure is that induced by the unique, differential structure of $\mathbb{R}^{8}$ . To obtain the exotic 7-spheres, Milnor used the generalizations of the Hopf fibering that gives the 7-sphere as an $\mathbb{S}^{3}$ bundle over $\mathbb{S}^{4}$ .

The standard Hopf fibering of the 7-sphere corresponds to using two fundamental charts to describe the manifold. We use the coordinates

(u,v)\in\mathbb{S}^{7}\ni u\in\mathbb{S}^{4},\quad v\in\mathbb{S}^{3}.

(2)

Then it is convenient to use the quaternions, $u\in\mathbb{H}$ where $\mathbb{H}$ corresponds to the set

	$\displaystyle u=u_{0}+iu_{1}+ju_{2}+ku_{3},\quad u_{0},\cdots u_{3}\in\mathbb{R}$
	$\displaystyle i^{2}=j^{2}=k^{2}=-1$
	$\displaystyle ij=k,jk=1,ki=j$
	$\displaystyle ij=-ji,jk=-kj,ki=-ik$		(3)

The quaternions form a non-commutative field, $|u|=\sqrt{\Sigma_{i}u_{i}^{2}}$ and with the definition $\bar{u}=u_{0}-iu_{1}-ju_{2}-ku_{3}$ the inverse is given by $\frac{1}{u}=\frac{\bar{u}}{|u|^{2}}$ .

Topologically $\mathbb{H}$ corresponds to $\mathbb{R}^{4}$ , hence the quaternionic coordinates can be thought of as the coordinates coming from stereographic projection of $\mathbb{S}^{4}$ onto $\mathbb{R}^{4}=\mathbb{H}$ . $v$ the coordinate on $\mathbb{S}^{3}$ can be identified with the set of unit quaternions, $v=v_{0}+iv_{1}+jv_{2}+kv_{3}$ with $v_{0},\cdots v_{3}$ restricted to a three ball of unit radius and $v_{0}=\pm\sqrt{1-(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})}$ . The fundamental set of charts are given by

(u,v)\quad{\rm and}\quad(u^{\prime},v^{\prime})

(4)

where the coordinates $u$ correspond to stereographic projection from the north pole of $\mathbb{S}^{4}$ along with the cartesian product of the coordinates $v$ on $\mathbb{S}^{3}$ while the coordinates $u^{\prime}$ correspond to stereographic projection from the south pole of $\mathbb{S}^{4}$ , again with a cartesian product with coordinates $v^{\prime}$ on $\mathbb{S}^{3}$ . The transition functions, corresponding to the (generalized) Hopf fibration, are then defined in terms of quaternions,

(u^{\prime},v^{\prime})=\left(\frac{u}{|u|^{2}},\frac{u^{h}vu^{l}}{|u|^{h+l}}\right)

(5)

or inversely

(u,v)=\left(\frac{u^{\prime}}{|u^{\prime}|^{2}},|u^{\prime}|^{(h+l)}(u^{\prime})^{-h}v^{\prime}(u^{\prime})^{-l}\right).

(6)

This standard Hopf fibration corresponds to $h=1$ , $l=0$ and gives rise to the 7-sphere analogously to the standard Hopf fibration of $S^{1}$ on $S^{2}$ giving rise to the 3-sphere. However, for other values of $h$ and $l$ , generalized fibre bundles with transition functions defined by Eqn. (5) and Eqn. (6) give rise to new 7-dimensional manifolds. Note that arbitrary powers, including inverse powers, of quaternions make perfect sense, $h$ or $l$ can be positive or negative.

Amazingly, for the case $h+l=1$ , the manifolds are topologically homeomorphic to the standard 7-sphere. For this case, the transition functions become

	$\displaystyle(u^{\prime},v^{\prime})$	$\displaystyle=$	$\displaystyle\left(\frac{u}{\|u\|^{2}},\frac{u^{h}vu^{l}}{\|u\|}\right)$
	$\displaystyle(u,v)$	$\displaystyle=$	$\displaystyle\left(\frac{u^{\prime}}{\|u^{\prime}\|^{2}},{\|u^{\prime}\|}(u^{\prime})^{-h}v^{\prime}(u^{\prime})^{-l}\right).$		(7)

To prove this, Milnor Milnor (1956) invoked Morse theory Morse (1934) and specifically Reeb’s theorem Reeb (1946) which states if a function can be defined on a $d$ -dimensional, compact manifold which has exactly two, non-degenerate critical points, then the manifold is homeomorphic to a d-dimensional sphere. Morse theory relates the critical points of a function to the minima, maxima and topological handles (minimaxes) on the manifold. For a compact manifold with exactly two critical points, these critical points have to be the global minimum and the global maximum, there can be no handles. Reeb’s theorem then states that the manifold has to be topologically a sphere. For the case $h+l=1$ , Milnor Milnor (1956) exhibited the following Morse function

f(u,v)=\frac{{\cal R}(v)}{\sqrt{1+|u|^{2}}}

(8)

where ${\cal R}(v)$ stands for the real part of $v$ , and showed that it has exactly two critical points. ${\cal R}(v)=v_{0}=\pm{\sqrt{1-(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})}}$ . We can see this by calculating the derivatives of the Morse function in the coordinate system given by the $u_{i}$ with $i=1,2,3,4$ and $v_{i}$ with $i=1,2,3$ . For a critical point we need

	$\displaystyle\frac{\partial}{\partial u_{i}}f(u,v)=\frac{-{\cal R}(v)}{(1+\|u\|^{2})^{3/2}}u_{i}=0$
	$\displaystyle\frac{\partial}{\partial v_{i}}f(u,v)=\frac{-v_{i}}{{\cal R}(v){\sqrt{1+\|u\|^{2}}}}=0$		(9)

which means $u_{i}=0$ and $v_{i}=0$ , which implies $(u,v)=(0,\pm 1)$ . These are the only two critical points in the northern patch. For the southern patch, we have

$\displaystyle f(u^{\prime},v^{\prime})$	$\displaystyle=$	$\displaystyle\frac{{\cal R}(\|u^{\prime}\|{u^{\prime}}^{-h}v^{\prime}{u^{\prime}}^{-l})}{\sqrt{1+1/\|u^{\prime}\|^{2}}}$
	$\displaystyle=$	$\displaystyle\frac{{\cal R}({\|u^{\prime}\|}{u^{\prime}}^{-(h+l)}v^{\prime})}{\sqrt{1+1/\|u^{\prime}\|^{2}}}$
	$\displaystyle=$	$\displaystyle\frac{\|u^{\prime}\|{\cal R}({u^{\prime}}^{-1}v^{\prime})}{\sqrt{1+1/\|u^{\prime}\|^{2}}}$

where we have used $h+l=1$ and that ${\cal R}$ is cyclic. Then using ${\cal R}(q^{-1})={\cal R}(\bar{q}/|q|^{2})={\cal R}(q/|q|^{2})$ for any quaternion $q$ and $|v^{\prime-1}u^{\prime}|=|u^{\prime}|$ as $v^{\prime}$ is a unit quaternion, we have

	$\displaystyle f(u^{\prime},v^{\prime})$	$\displaystyle=$	$\displaystyle\frac{\|u^{\prime}\|{\cal R}(v^{\prime-1}u^{\prime})}{\|u^{\prime}\|^{2}\sqrt{1+1/\|u^{\prime}\|^{2}}}$		(11)
		$\displaystyle=$	$\displaystyle\frac{{\cal R}(u^{\prime}v^{\prime-1})}{\sqrt{1+\|u^{\prime}v^{\prime-1}\|^{2}}}.$		(11)

Now $u^{\prime}v^{\prime-1}$ is a perfectly general, independent quaternion, call it $u^{\prime\prime}=u^{\prime\prime}_{0}+iu^{\prime\prime}_{1}+ju^{\prime\prime}_{2}+ku^{\prime\prime}_{3}$ . Then

f(u^{\prime},v^{\prime})=\frac{{\cal R}(u^{\prime\prime})}{\sqrt{1+|u^{\prime\prime}|^{2}}}=\frac{u^{\prime\prime}_{0}}{\sqrt{1+{u^{\prime\prime}_{0}}^{2}+{u^{\prime\prime}_{1}}^{2}+{u^{\prime\prime}_{2}}^{2}+{u^{\prime\prime}_{3}}^{2}}}

(12)

It is easy to see that the derivative of this function with respect to $u^{\prime\prime}_{0}$ never vanishes

$\displaystyle\frac{\partial f(u^{\prime},v^{\prime})}{\partial u^{\prime\prime}_{0}}$	$\displaystyle=$	$\displaystyle\frac{1}{\sqrt{1+{u^{\prime\prime}_{0}}^{2}+{u^{\prime\prime}_{1}}^{2}+{u^{\prime\prime}_{2}}^{2}+{u^{\prime\prime}_{3}}^{2}}}-$
	$\displaystyle-$	$\displaystyle\frac{{u^{\prime\prime}_{0}}^{2}}{(1+{u^{\prime\prime}_{0}}^{2}+{u^{\prime\prime}_{1}}^{2}+{u^{\prime\prime}_{2}}^{2}+{u^{\prime\prime}_{3}}^{2})^{3/2}}$
	$\displaystyle=$	$\displaystyle\frac{(1+{u^{\prime\prime}_{1}}^{2}+{u^{\prime\prime}_{2}}^{2}+{u^{\prime\prime}_{3}}^{2})}{(1+{u^{\prime\prime}_{0}}^{2}+{u^{\prime\prime}_{1}}^{2}+{u^{\prime\prime}_{2}}^{2}+{u^{\prime\prime}_{3}}^{2})^{3/2}}>0.$

Hence the function has no critical points in the southern patch and exactly two critical points in the northern patch, i.e. two critical points that are easily seen to be non-degenerate. Hence by Morse theory and specifically Reeb’s theorem, the manifold is homeomorphic to the standard $\mathbb{S}^{7}$ . Let us call the manifolds $M^{7}_{k}$ where $h+l=1$ but $h-l=k$ .

II.2 Existence of diffeomorphically inequivalent 7-spheres

Then the proof that some of these fibre bundles are not diffeomorphic to the standard $\mathbb{S}^{7}$ follows from the Hirzebruch signature theorem Hirzebruch (1995). One assumes that $M^{7}_{k}$ are indeed diffeomorphic to the standard $\mathbb{S}^{7}$ and then we obtain a contradiction.

An integer valued, modulo 7, topological invariant, $\lambda(M^{7}_{k})$ , can be defined for the manifolds $M^{7}_{k}$ . First we construct a smooth, 8-dimensional manifold, $B^{8}$ , whose boundary is given by $M^{7}_{k}$ . $B^{8}$ always exists by a theorem of Thom Thom (1954) given $M^{7}_{k}$ is closed, oriented and with vanishing 3rd and 4th cohomology groups. That these cohomology groups vanish is clear because $M^{7}_{k}$ is homeomorphic to the 7-sphere, and the 7-sphere only has non-vanishing cohomology classes $H^{0}(\mathbb{S}^{7})$ and $H^{7}(\mathbb{S}^{7})$ . The standard 7-sphere is the boundary of the standard 8-disc $D^{8}$ . As $M^{7}_{k}$ is homeomorphic to the standard $\mathbb{S}^{7}$ , and now we assume diffeomorphic, we can smoothly glue together $D^{8}$ to $B^{8}$ on their boundary to form a smooth, closed 8-dimensional manifold which we will call $W^{8}_{k}$ . Then the Hirzebruch signature theorem says

\sigma(W^{8}_{k})=\frac{1}{45}(7p_{2}(W^{8}_{k})-p_{1}^{2}(W^{8}_{k}))

(14)

where $p_{1}$ and $p_{2}$ are the first and second Pontrjagin class respectively. The signature $\sigma(W^{8}_{k})=\pm 1$ , choose +1, then we have

45+p_{1}^{2}(W^{8}_{k})=0\,\,\,{\rm modulo}\,\,\,7.

(15)

Then it is incumbent on us to compute only $p_{1}^{2}(W^{8}_{k})$ , which is found by Milnor, Milnor (1956), to be $4k^{2}$ . Thus we get the equation

45+4k^{2}=0\,\,\,{\rm modulo}\,\,\,7\quad{\it{i.e.}}\quad 3+4k^{2}=0\,\,\,{\rm modulo}\,\,\,7.

(16)

$k=\pm 1$ obviously is a solution, but $k=2,3,4,5$ are easily seen not to satisfy this equation, which is a contradiction.

2\,\,{\rm and}\,\,5=\pm 2\,\,\,{\rm modulo}\,\,\,7\Rightarrow 3+4\cdot 4=19=5\neq 0\,\,\,{\rm modulo}\,\,\,7

(17)

and

3\,\,{\rm and}\,\,4=\pm 3\,\,\,{\rm modulo}\,\,\,7\Rightarrow 3+4\cdot 9=39=4\neq 0\,\,\,{\rm modulo}\,\,\,7.

(18)

Therefore, the assumption that we made, that $M^{7}_{k}$ is diffeomorphic to the standard $\mathbb{S}^{7}$ has to be false for the cases $k=2,3,4,5\,\,{\rm modulo}\,\,7$ and as such there exist exotic 7-spheres that are homeomorphic to the standard $\mathbb{S}^{7}$ , topologically the same, but that cannot be diffeomorphic to the standard $\mathbb{S}^{7}$ .

This result is rather astonishing. Two manifolds which have the same notion of continuous functions do not have the same notion of differentiable functions. The fundamental question arises, what part of physical reality depends only on the notion of continuity, and not on the notion of differentiabilty. All kinds of physical phenomena do not depend on the global differential structure of the manifold on which the phenomena occurs. The diffeomorphically inequivalent 7-spheres all admit smooth metrics, which give a notion of length scale. All phenomena which occurs esentially locally, such as crystal growth or any biological phenomena for example, are simply identical in any spacetime that is smooth, but where the length scale of the physical phenomena is small compared to the length scale over which the differential structures varies. Our diffeomorphically inequivalent 7-spheres are of course locally flat when equipped with a metric, and the inequivalent differential structures occur only because of global obstructions. Hence, physical phenomena which occur over length scales small compared to the length scale of the variation of the differential structure are bound to be identical.

However, all of classical or quantum mechanics depends on the notion of differentiability. Hence there will clearly be criteria by which one could physically discern between topologically equivalent manifolds which are not diffeomorphic. This is what we endeavour to find in the rest of this paper. We will look at the spectrum of the Dirac operator on the different, exotic 7-spheres compared with the operator defined on the standard 7-sphere. The spectrum of the operator, especially for the low-lying modes will clearly be of physical importance and will give a tangible criterion with which to discern between exotic and standard 7-spheres. The metric on the 7-spheres can be chosen to correspond to a Kaluza-Klein reduction. This does not affect the global topology nor the differential structure. In this reduction, the metric on the $\mathbb{S}^{3}$ is taken so that the size of the 3-sphere is very small compared to the size of the base, $\mathbb{S}^{4}$ . Then the effective theory we are left with is a Einstein-Yang-Mills theory on the $\mathbb{S}^{4}$ base. Such a theory could be quite relevant to our 4-dimensional physical world.

III Kaluza-Klein reduction

Having established that the manifolds $M^{7}_{k}$ , for $h+l=1$ are all homeomorphic to the 7-sphere, we will imagine the Kaluza-Klein reduction of the manifolds, Straumann (1986). Such a reduction maintains the topology and the differential structure of the manifold. As all the manifolds $M^{7}_{k}$ are fibre bundles of $\mathbb{S}^{3}$ over $\mathbb{S}^{4}$ , the Kaluza-Klein reduction means that we choose a metric such that the size of the fibre $\mathbb{S}^{3}$ becomes very small compared to the size of the base $\mathbb{S}^{4}$ . Ideally, then in the low energy dynamics on the base, the fibre should be a simple $\mathbb{S}^{3}$ of infinitesimal radius and only its isometries can have any impact on the low energy dynamics taking place in the ambient space given by the base, $\mathbb{S}^{4}$ . It is not consistent in this limit to think of deformations of the fibre, these would correspond only to very high energy excitations. The only degree of freedom left is the liberty to rotate the fibre arbitrarily by the group transformations that are symmetries (isometries) of the fibre as we move along the base manifold. This gives rise to a gauge degree of freedom, the gauge group being the group of isometries of the fibre, in this case ${\rm SO}(4)$ .

The low energy dynamics coming from an assumed Einsteinian dynamics on the original 7 dimensional manifold then simply reduces to 4 dimensional Einstein gravity on $\mathbb{S}^{4}$ coupled to ${\rm SO}(4)$ gauge fields with Yang-Mills dynamics. However, most importantly, due to the exotic differential structure, these gauge fields have to be connections on topologically non-trivial fibre bundles that are distinct from the standard Hopf fibring that gives rise to the standard 7-sphere. This means that they must have topological invariants that are distinct from those of gauge fields that would be defined on the standard 7-sphere also in the Kaluza-Klein limit.

The metric on the corresponding fibre bundle can be written as Straumann (1986):

g=g_{\mathbb{S}^{4}}+k_{ij}(\theta^{i}-K^{i}_{a}A^{a})\otimes(\theta^{j}-K^{j}_{b}A^{b})

(19)

where $g_{\mathbb{S}^{4}}$ is the metric on the base $\mathbb{S}^{4}$ , $k_{ij}$ are the components of the metric on the fibre $\mathbb{S}^{3}$ , $K^{i}_{a}$ are the components of the Killing vectors that describe the isometries of the fibre, ${\rm SO}(4)$ , $\theta^{i}$ are the components of the dreibein (triad) one forms on the fibre and $A^{a}$ are the components of a Yang-Mills gauge field corresponding to the gauge group given by the isometries of the fibre, ${\rm SO}(4)$ . The gauge field is necessarily present as the manifold is a non-trivial fibre bundle of $\mathbb{S}^{3}$ over $\mathbb{S}^{4}$ . If the gauge field were absent, the manifold would simply be just the Cartesian product of $\mathbb{S}^{3}$ with $\mathbb{S}^{4}$ , which is not even a standard 7-sphere.

The metric on $\mathbb{S}^{4}$ , $g_{\mathbb{S}^{4}}$ , can be arbitrary, the simplest to take is the constant curvature metric. The metric on $\mathbb{S}^{3}$ , which would be $k_{ij}\theta^{i}\otimes\theta^{j}$ if the gauge field $A^{a}$ were absent, is also one of constant curvature. To make a 7-sphere, the $\mathbb{S}^{3}$ fibre has to be twisted as it goes around the equator of $\mathbb{S}^{4}$ . It is the gauge fields that capture the topologically non-trivial structure inherent in the normal and exotic 7-spheres, and as such impose global constraints on the possible gauge fields. In the Kaluza-Klein reduction of the manifold, the base manifold is topologically and differentiably $\mathbb{S}^{4}$ , but it has locally a direct product with a tiny $\mathbb{S}^{3}$ associated with each point of the $\mathbb{S}^{4}$ . This $\mathbb{S}^{3}$ twists as it is defined over the $\mathbb{S}^{4}$ . These twistings, are defined by the generalized Hopf fibrings defined by Eq.(7), for $h+l=1$ .

The metric can be defined in terms of the vierbeins $e^{A}_{\mu}$ , $g_{\mu\nu}=\eta_{AB}e^{A}_{\mu}\otimes e^{B}_{\nu}$ , then the spin connection is defined by the equation $de^{A}+\Omega^{A}_{\,\,B}\wedge e^{B}=0$ and the curvature 2-form is defined by $R^{A}_{\,\,B}=d\Omega^{A}_{\,\,B}+\Omega^{A}_{\,\,C}\wedge\Omega^{C}_{\,\,B}=\frac{1}{2}R^{A}_{\,\,BCD}e^{C}\wedge e^{C}$ where all indices $\mu$ and $A$ go from 1 to 10. It is well understood Straumann (1986), that with the metric of the form Eqn.(19), the scalar curvature is simply given by

R=R_{\mathbb{S}^{4}}+R_{\mathbb{S}^{3}}+{\cal L}_{YM}

(20)

where $R_{\mathbb{S}^{4}}$ is the scalar curvature of $g_{\mathbb{S}^{4}}$ on $\mathbb{S}^{4}$ , $R_{\mathbb{S}^{3}}$ is the scalar curvature of $k_{ab}$ on $\mathbb{S}^{3}$ and ${\cal L}_{YM}$ is the Yang-Mills Lagrangian for the gauge field $A^{a}$ on $\mathbb{S}^{4}$ .

The gauge field must be consistent with the bundle structure defined by $h$ and $l$ . This means that the transition functions for the gauge fields between the northern patch and the southern patch must reflect the values of $h$ and $l$ . Specifically, the action on the fibre from Eqn.(7)

v^{\prime}=\hat{u}^{h}v\hat{u}^{l}.

(21)

The bundle is an $\mathbb{S}^{3}$ bundle over $\mathbb{S}^{4}$ , the isometry group of $\mathbb{S}^{3}$ being ${\rm SO}(4)$ . Therefore we actually construct an ${\rm SO}(4)$ principal bundle over $\mathbb{S}^{4}$ . The defining representation consists of $4\times 4$ dimensional matrices acting on four dimensional vector representation in $\mathbb{R}^{4}$ . The general quaternionic transformation

x^{\prime}=\hat{q}x\hat{r}

(22)

with $\hat{q}=\cos\theta+\sin\theta\hat{\theta}\cdot\vec{i}$ and $\hat{r}=\cos\zeta+\sin\zeta\hat{\zeta}\cdot\vec{i}$ where $\vec{i}\equiv(i,j,k)$ of the vector of the fundmental quaternions, can be written as

{x^{\prime}}^{\mu}=(R_{L}R_{R}^{T})^{\mu}_{\,\,\,\nu}x^{\nu}

(23)

where $R_{R}^{T}$ is the transpose (hence the inverse) of the orthogonal matrix $R_{R}$ and $\mu\,,\,\nu\in\,1,2,3,4$ . $R_{L}$ and $R_{R}$ are respectively the left and right isoclinic decompositions of the fundamental representation of ${\rm SO}(4)$ . Here we can take the explicit representations,

R_{L}=\cos\theta+\sin\theta\hat{\theta}\cdot\vec{T}_{L}\quad R_{R}=\cos\zeta+\sin\zeta\hat{\zeta}\cdot\vec{T}_{R}

(24)

with $\theta\,\zeta\in(0,\pi)$ to fully cover the unit quaternions, and

T^{1}_{L}=-i{\mathbb{I}}\otimes\tau^{2}\quad T_{L}^{2}=-i\tau^{2}\otimes\tau^{3}\quad T_{L}^{3}=-i\tau^{2}\otimes\tau^{1}

(25)

and

T^{1}_{R}=i\tau^{3}\otimes\tau^{2}\quad T_{R}^{2}=-\tau^{2}\otimes{\mathbb{I}}\quad T_{R}^{3}=i\tau^{1}\otimes\tau^{2}

(26)

where $\tau^{i}$ are the Pauli matrices. The generators $T^{i}_{L}$ and $T^{i}_{R}$ mutually commute and provide a $4\times 4$ representation of the fundamental quaternions. Furthermore, $T^{i}_{L}/2$ and $T^{i}_{R}/2$ are the generators of two independent, reducible representations of ${\rm SU}(2)$ , the representation $\frac{1}{2}\oplus\frac{1}{2}$ .

For our purposes, from Eqn.(21), we have $\hat{q}\to\hat{u}^{h}=\cos(h\theta)+\sin(h\theta)\hat{\theta}\cdot\vec{i}$ while $\hat{r}\to\hat{u}^{l}=\cos(l\zeta)+\sin(l\zeta)\hat{\zeta}\cdot\vec{i}$ . Then with $R=R_{L}R^{T}_{R}$ we can take the gauge field to be zero in the northern patch, and which satisfies at the equator

A^{\prime}=R^{T}(A+d)R

(27)

and $A^{\prime}$ is simply switched of to zero as we go the the south pole. Such a gauge field will not be a solution of the Yang-Mills equations, not have any particular symmetry property, however, it will be consistent with the topological constraints imposed by the bundle structure. Indeed, the topological number $h-l$ then shows up through the topological invariant called the Pontrjagin number of the gauge field (which is anti-hermitean), $p(A)$ :

$\displaystyle p(A)$	$\displaystyle=$	$\displaystyle\frac{-1}{16\pi^{2}}\int_{\mathbb{S}^{4}}\epsilon^{\mu\nu\sigma\tau}Tr\left(F_{\mu\nu}F_{\sigma\tau}\right)$	(28)
	$\displaystyle=$	$\displaystyle\frac{-1}{16\pi^{2}}\int_{\partial S_{4}=\mathbb{S}^{3}}d\sigma_{\mu}\epsilon^{\mu\nu\sigma\tau}Tr\left(A^{\prime}_{\nu}\partial_{\sigma}A^{\prime}_{\tau}+\frac{2}{3}A^{\prime}_{\nu}A^{\prime}_{\sigma}A^{\prime}_{\tau}\right)$
	$\displaystyle=$	$\displaystyle\frac{1}{24\pi^{2}}\int_{\partial S_{4}=\mathbb{S}^{3}}d^{3}x\epsilon^{ijk}Tr\left((R^{T}\partial_{i}R)(R^{T}\partial_{j}R)(R^{T}\partial_{k}R)\right)=2(h-l)$

The factor of two occurs simply because we have a direct sum of two fundamental spin $\frac{1}{2}$ representations in both the left handed and the right handed sectors. Then the integral projects to an integral only over the equatorial 3-sphere, which is just the winding number of the map defined by $R$ , the left handed part giving $2h$ and the right handed part giving $-2l$ . .

With the Kaluza-Klein reduction of the exotic 7-spheres, we are able to analyze the spectrum of the Dirac operator for $\mathbb{S}^{4}$ symmetric gauge fields which are of course consistent with the bundle structure, which we do in the next section.

IV Spherically Symmetric Instantons and the Dirac Spectrum

Any gauge field is consistent with the bundle structure, as long as it satisfies the constraint coming from the global topology, as the example we have chosen above. However, there was much work done on “spherically” symmetric gauge fields which actually automatically solve the Yang-Mills equations of motion fo the gauge field, and hence are nominally spherically symmetric instantons (i.e. exact solutions fo the Yang-Mills equations). Such gauge field configurations are useful since it is well understood how to find the eigenvalues of the Dirac operator in their presence. It is these eigenvalues that give a tangible difference to the physics on the 7-spheres with exotic differential structure and hence give us a handle on how the physics can be different on topologically identical manifolds but with inequivalent differential structures.

The general $G-$ symmetric multi-instantons on symmetric spaces $G/H$ were studied by A. N. Schellekens Schellekens (1985, 1984). Here, we present an explicit construction in the case of the $4-$ sphere $\mathbb{S}^{4}\cong{\rm SO}(5)/{\rm SO}(4)$ . We mostly follow the conventions and presentation of Dolan (2003) with some precisions in the case of $\mathbb{S}^{4}$ . Using a decomposition $\mathfrak{so}(4)=\mathfrak{s}u(2)\oplus\mathfrak{s}u(2)$ , each of the multi-instantons will be composed of a left $SU(2)-$ multi-instanton and a right $SU(2)-$ multi-instanton.

IV.1 Coset construction of ${\rm S}^{4}$

The $4-$ sphere will be seen as a coset space $\mathbb{S}^{4}\cong{\rm SO}(5)/{\rm SO}(4)$ . The 10 generators of ${\rm SO}(5)$ are labeled by $M,N,P,Q=1,\dots,10$ and the 6 generators of ${\rm SO}(4)$ are labeled by $a,b,c,d=1,\dots,6$ , which of course is a closed subgroup of ${\rm SO}(5)$ . Coordinate indices of the base manifold $\mathbb{S}^{4}$ are labeled by $\mu,\nu,\rho,\sigma=1,\dots,4$ , and vierbein indices of $\mathbb{S}^{4}$ are labeled by $m,n,p,q=1,\dots,4$ . The (anti-Hermitian) generators of ${\rm SO}(5)$ will be denoted by

\displaystyle\{T_{M}\,,\,M=1,\dots,10\}

(29)

and its (totally anti-symmetric) structure constants $\{C^{P}_{MN}\quad|\quad\,P,M,N=1,\dots,10\}$ are defined by

\displaystyle[T_{M},T_{N}]=C^{P}_{MN}T_{P}\,.

(30)

We fix a set of generators of $\mathfrak{so}(4)$ as

\displaystyle\{T_{a}\,,\,a=1,\dots,6\}\,.

(31)

The remaining generators span the tangent space of $\mathbb{S}^{4}$ at a fixed point, $T({\rm SO}(5)/{\rm SO}(4))$ , and are denoted by

\displaystyle\{T_{\mu}\,,\,\mu=1,\dots,4\}\,.

Irreducible representations of ${\rm SO}(5)$ are labeled by two integers, $p,q$ , $p\geq q\geq 0$ , with the corresponding representation noted as $(p,q)_{5}$ . Since ${\rm SO}(5)$ is compact, then in any representation $(p,q)_{5}$ there exist orthogonal generators $T_{M}((p,q)_{5})$ satisfying

\displaystyle\mbox{Tr}[T_{M}((p,q)_{5})T_{N}((p,q)_{5})]=-C_{1}^{{\rm SO}(5)}((p,q)_{5})\,\delta_{MN}\,

(32)

$C_{1}^{{\rm SO}(5)}(R)$ is called the (second order) Dynkin index for the representation $(p,q)_{5}$ . It then follows, from the definition of the generators of ${\rm SO}(4)$ , that $(p,q)_{5}$ induces a (possibly reducible) representation $R$ of ${\rm SO}(4)$ , so that

\displaystyle\mbox{Tr}[T_{a}(R)T_{b}(R)]=-C_{1}^{{\rm SO}(4)}(R)\,\delta_{ab}=-C_{1}^{{\rm SO}(5)}((p,q)_{5}))\,\delta_{ab},

(33)

which defines the normalizations of the generators of ${\rm SO}(5)$ and ${\rm SO}(4)$ . The quadratic Casimir operator in the representation $(p,q)_{5}$ is defined by

\displaystyle\sum_{M}T_{M}((p,q)_{5})T_{M}((p,q)_{5}):=C_{2}^{{\rm SO}(5)}((p,q)_{5})\,{\mathbb{I}}.

(34)

It is related to $C_{1}^{{\rm SO}(5)}((p,q)_{5})$ by

\displaystyle C_{2}^{{\rm SO}(5)}((p,q)_{5})=\frac{10}{\dim((p,q)_{5}))}\,C_{1}^{{\rm SO}(5)}((p,q)_{5})\,.

(35)

The quadratic Casimirs and the dimensions of the irreducible representations are well known and given respectively by

	$\displaystyle C_{2}^{{\rm SO}(5)}((p,q)_{5})=\frac{p^{2}+q^{2}}{2}+2p+q$		(36)
	$\displaystyle\dim((p,q)_{5})=\frac{1}{6}(p+q+3)(p-q+1)(p+2)(q+1)\,.$		(37)

We have the following expression for the structure constants of ${\rm SO}(5)$ , Schellekens (1984):

	$\displaystyle C^{a}_{bc}\quad\text{ are the structure constants of ${\rm SO}(4)$}$
	$\displaystyle C^{\mu}_{ab}=0\quad\text{ by closure of ${\rm SO}(4)$}$
	$\displaystyle C^{a}_{\mu\nu}=\begin{cases}-\frac{1}{\sqrt{2}}\eta^{a}_{\mu\nu}\quad\text{if}\quad a=1,2,3\\ -\frac{1}{\sqrt{2}}\overline{\eta}^{(a-3)}_{\mu\nu}\quad\text{if}\quad a=4,5,6\end{cases}$
	$\displaystyle C^{\mu}_{\nu\gamma}=0\quad\text{since ${\rm SO}(5)/{\rm SO}(4)$ is a symmetric coset space}\,.$

where we have used the ’t Hooft symbols, ’t Hooft (1976a, b)

\displaystyle\eta_{\mu\nu}^{i}:=\epsilon_{\mu\nu i4}+\delta_{\mu i}\delta_{\nu 4}-\delta_{\mu 4}\delta_{\nu i},\quad

\displaystyle\overline{\eta}_{\mu\nu}^{i}:=\epsilon_{\mu\nu i4}-(\delta_{\mu i}\delta_{\nu 4}-\delta_{\mu 4}\delta_{\nu i})\,.

which are used in the expression for the exact solution instanton gauge fields that he found for the group ${\rm SU}(2)$ .

IV.2 ${\rm SO}(5)$ invariant metric on $\mathbb{S}^{4}$ and construction of spherically symmetric instantons on $\mathbb{S}^{4}$

For completeness, we record the $SO(5)$ invariant metric on ${\rm SO}(4)$ . On the $4-$ sphere $\mathbb{S}^{4}\cong{\rm SO}(5)/{\rm SO}(4)$ , we put the standard ${\rm SO}(5)-$ invariant Riemannian metric, the generators of ${\rm SO}(5)$ are the Killing vectors and the holonomy group is ${\rm SO}(4)$ . The metric is obtained as follows. First, the $4-$ sphere in $\mathbb{R}^{5}$ is defined by

\displaystyle\mathbb{S}^{4}:=\big{\{}\,(z_{1},z_{2},\dots,z_{5})\quad|\quad z_{1}^{2}+z_{2}^{2}+\dots+z_{5}^{2}=1\,\big{\}}\,.

Consider the following local parametrization of $\mathbb{S}^{4}$ in polar coordinates:

	$\displaystyle z_{1}:=\sin\xi\sin\chi\sin\theta\cos\phi$
	$\displaystyle z_{2}:=\sin\xi\sin\chi\sin\theta\sin\phi$
	$\displaystyle z_{3}:=\sin\xi\sin\chi\cos\theta\qquad\qquad$	$\displaystyle\text{where}\qquad 0\leq\xi\leq\pi\,,\,0\leq\chi\leq\pi\,,\,0\leq\theta\leq\pi\,,\,0\leq\phi<2\pi\,.$
	$\displaystyle z_{4}:=\sin\xi\cos\chi$
	$\displaystyle z_{5}:=\cos\xi\,.$

The standard ${\rm SO}(5)-$ invariant Riemannian metric on $\mathbb{S}^{4}$ in these coordinates is

	$\displaystyle g_{\mathbb{S}^{4}}$	$\displaystyle:=\mathrm{d}\xi\otimes\mathrm{d}\xi+\sin^{2}\xi\,(\,\mathrm{d}\chi\otimes\mathrm{d}\chi+\sin^{2}\chi\,\mathrm{d}\theta\otimes\mathrm{d}\theta+\sin^{2}\chi\sin^{2}\theta\,\mathrm{d}\phi\otimes\mathrm{d}\phi\,)$
		$\displaystyle\equiv\sum_{m}e^{m}\otimes e^{m}\,,$

where $\{e^{m},m=1,\dots,4\}$ is the standard vierbein basis for this metric. The corresponding volume form is

\mathrm{d}vol_{\mathbb{S}^{4}}=\mathrm{d}z_{1}\wedge\mathrm{d}z_{2}\wedge\dots\wedge\mathrm{d}z_{5}=\sin^{3}\xi\sin^{2}\chi\sin\theta\,\mathrm{d}\xi\wedge\mathrm{d}\chi\wedge\mathrm{d}\theta\wedge\mathrm{d}\phi.

The spin connection of $\mathbb{S}^{4}$ is defined by the equation $de^{m}+\omega^{m}_{\,\,\,\,n}\wedge e^{n}=0$ and the curvature 2-form is defined by $R^{m}_{\,\,\,\,n}=d\omega^{m}_{\,\,\,\,n}+\omega^{m}_{\,\,\,\,p}\wedge\omega^{p}_{\,\,\,\,n}$ . In the standard vierbein basis $\{e^{m},m=1,\dots,4\}$ , it is given by

\omega_{mn}=\frac{1}{1+z_{5}}(z_{m}\mathrm{d}z_{n}-z_{n}\mathrm{d}z_{m})\,.

(38)

We want to consider “spherically” symmetric connections on the bundles that define the exotic 7-spheres, $M^{7}_{k}$ , simplified by the Kaluza-Klein reduction. These are then $\mathfrak{so}(4)-$ connections on a bundle corresponding to the manifolds defined by ${\rm SO}(5)-$ invariant gauge potentials (the spherical symmetry) whose components are identified with those of the spin connections of $\mathbb{S}^{4}$ . Spherically symmetric solutions of the Yang-Mills equations (instantons) allow us to solve for the spectrum of the Dirac operator. For general $h$ and $l$ there are no spherically symmetric instantons, i.e. solutions of the Yang-Mills equations. However, if one can find the appropriate embeddings, then the Dynkin indices of the embeddings Francesco et al. (1997), will be related to the topological invariants, $h,l$ of the connection and one can consider spherically symmetric instantons.

A clear example of this situation is given by Wilczek Wilczek (1976). Here he considers a spherically symmetric instanton in an ${\rm SU}(3)$ gauge theory, but one that has topological charge 4. The instanton corresponds to in fact, an instanton in the $3\times 3$ spin 1 representation of ${\rm SU}(2)$ embedded into ${\rm SU}(3)$ . However, the spherical symmetry (and the fact that the configuration is a solution) can be destroyed if one spatially separates the instanton into four charge 1 instantons corresponding to different embeddings of the fundamental representation of ${\rm SU}(2)$ into ${\rm SU}(3)$ . By local topologically trivial gauge transformations, these embeddings can then be gauge transformed into configurations corresponding to one specific embedding, say the standard embedding which corresponds to the $SU(2)$ subgroup of ${\rm SU}(3)$ sitting in the upper left $2\times 2$ bloc of the fundamental $3\times 3$ representation of ${\rm SU}(3)$ . Then the instantons can be brought together, giving rise to a charge 4 configuration in the standard embedding of ${\rm SU}(2)$ into ${\rm SU}(3)$ . Of course, this construction does not give a solution to the Yang-Mills equations, However, it is clear that the configuration will not be spherically symmetric, and it is also well known that a solution to the Yang-Mills equations with charge 4 in the fundamental representation of ${\rm SU}(2)$ exists and can be described by the ADHM construction Atiyah et al. (1978), and it is not spherically symmetric.

Correspondingly, we imagine we have a fundamental bundle of ${\rm SO}(4)$ instantons with charge $2h$ and $-2l$ in the left and right sector respectively. These are not spherically symmetric in principle, however, if an appropriate representation of the gauge group is chosen, then we can have a spherically symmetric configuration with the same given topological charges. Depending on the embedding of the representation of ${\rm SO}(4)$ that we pick, we can get charge $2h$ or $-2l$ instantons with spherical symmetry. We refer to Schellekens (1985) and Schellekens (1984) for more details. These embedded representations of $\mathfrak{so}(4)$ will be denoted by $\mathcal{R}_{h,l}$ which would not necessarily be an irreducible representation. The irreducible representations of ${\rm SO}(4)$ are labelled by two half-integers, and $r,s$ with representation noted as $(r,s)_{4}$

We can now construct spherically symmetric ${\rm SO}(4)-$ multi-instantons ${\color[rgb]{0,0,0}A}$ on $\mathbb{S}^{4}$ with topological invariants $2h-2l$ (instanton number) and $h+l$ (Euler number) as follows. We consider the following $\mathfrak{so}(4)-$ valued singular $1-$ form locally defined on $\mathbb{S}^{4}$ :

\displaystyle A\equiv\sum_{r=1}^{5}A_{r}\,dz_{r}:=\sum_{m=1}^{4}\sum_{n=1}^{4}-\frac{1}{1+z_{5}}\eta^{i}_{mn}T^{[{\color[rgb]{0,0,0}h}]}_{i}z_{n}\,dz_{m}\,+\,\sum_{m=1}^{4}\sum_{n=1}^{4}-\frac{1}{1+z_{5}}\eta^{i}_{mn}T^{[{\color[rgb]{0,0,0}l}]}_{i}z_{n}\,dz_{m}\,,

where $\Big{\{}T^{[h]}_{i}\,,\,i=1,2,3\Big{\}}$ and $\Big{\{}T^{[l]}_{i}\,,\,i=1,2,3\Big{\}}$ are generators of the two $\mathfrak{su}(2)$ factors in $\mathfrak{so}(4)=\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ which corespond to the representations of ${\rm SO}(4)$ under which the fermions that will satisfy th Dirac equation transform. The left chirality spinors transform independently of the right chirality spinors, the corresponding gauge fields are self-dual and anti-self-dual, respectively. We label the representations by $h$ and $l$ , however, the representations of the left and right factors of ${\rm SU}(2)$ have the first Casimir (Dynkin index) given by $2h$ and $-2l$ respectively. Additionally, the fermions carry intrinsic spin $\pm(1/2)$ . We take:

\displaystyle[T^{[h]}_{i},T^{[h]}_{j}]=\epsilon_{ijk}T^{[h]}_{k}\qquad,\qquad[T^{[l]}_{i},T^{[l]}_{j}]=\epsilon_{ijk}T^{[l]}_{k}\qquad,\qquad[T^{[h]}_{i},T^{[l]}_{j}]=0\,.

By definition, they have the properties

\displaystyle\mbox{Tr}\Big{(}T^{[h]}_{i}T^{[h]}_{j}\Big{)}=-{\color[rgb]{0,0,0}h}\,\delta_{i\,j}\qquad,\qquad\mbox{Tr}\Big{(}T^{[l]}_{i}T^{[l]}_{j}\Big{)}={\color[rgb]{0,0,0}l}\,\delta_{i\,j}\,,

where we take $h>0$ and $l<0$ and which are the Dynkin indices of the embeddings of higher representations of $\mathfrak{so}(4)=\mathfrak{su}(2)\oplus\mathfrak{su}(2)$ which determine $\mathcal{R}_{h,l}$ .

For the specific case $h=2$ , $l=-1$ we can take

{T_{i}^{[2]}}_{jk}=-\epsilon_{ijk}

(39)

which satisfy

Tr\left(T_{i}^{[2]}T_{j}^{[2]}\right)=h\delta_{ij}=2\delta_{ij}

(40)

for the left component of $\mathfrak{so}(4)$ . This representation of $\mathfrak{so}(4)$ embeds smoothly into the fundamental representation of $\mathfrak{so}(5)$ . For $l=-1$ we can take

T_{i}^{[-1]}=-i\frac{\sigma^{i}}{2}\oplus-i\frac{\tau^{i}}{2}

(41)

where $\sigma^{i}$ and $\tau^{i}$ are independent Pauli matrices, which satisfy

Tr\left(T_{i}^{[-1]}T_{j}^{[-1]}\right)=-l\delta_{ij}=\delta_{ij}

(42)

for the right component of $\mathfrak{so}(4)$ . This representation is unitarily equivalent to the right isoclinic factor of the fundmental representation of ${\rm SO}(4)$ that was described above, Eqn.(26). This representation embeds smoothly into the dimension 4 spinor representation of $\mathfrak{so}(5)$ . The manifold with $h=2$ , $l=-1$ satisfies $h+l=1$ but $h-l=3\neq\pm 1\,\,{\rm modulo}\,\,7$ and hence describes an exotic sphere.

IV.3 Spectrum of the Dirac operator (squared)

We now compute the spectra of the squared Dirac operator on $\mathbb{S}^{4}$ in the gauge fields that we have constructed for all values of $h$ and $l$ . This spectrum constrains the mass/energy spectrum for fermions on $\mathbb{S}^{4}$ after Kaluza-Klein reduction. We will also show how the choice of the smooth structure on $7-$ spheres affects the energy/mass spectrum for fermions on compactified space-time $\mathbb{R}^{4}\cup\{\infty\}\cong\mathbb{S}^{4}$ .

We consider the standard Riemannian metric on $\mathbb{S}^{4}$ . After Kaluza-Klein reduction, the Einstein-Yang-Mills-Dirac action on the compactified space-time $\mathbb{S}^{4}$ is given by :

\displaystyle\mathcal{S}_{\mathcal{E}-\mathcal{YM}-\mathcal{D}}=\int_{\mathbb{S}^{4}}\Big{(}R_{\mathbb{S}^{4}}+R_{\mathbb{S}^{3}}+\frac{1}{2}\mathscr{L}_{\mathcal{Y}\mathcal{M}}[A]+\overline{\psi}(i\mathcal{D}_{A})\psi\Big{)}\,\mathrm{d}vol_{\mathbb{S}^{4}}\,.

The Dirac operator on $\mathbb{S}^{4}$ in a ${\rm SO}(4)-$ gauge field background $A=A_{r}dz_{r}$ is given (using our conventions for the indices) by

\displaystyle\mathcal{D}_{A}

\displaystyle=\gamma^{l}e^{r}_{l}\Big{(}\partial_{r}+\frac{1}{4}\omega_{mn,r}\gamma^{mn}+iA_{r}\Big{)}\,.

(43)

Here $\{e^{m}\equiv e^{m}_{r}dz_{r}\,,\,\,m=1,\dots,4\}$ form the standard orthonormal coframe for $\mathbb{S}^{4}$ and the components of the spin connection $1-$ form of $\mathbb{S}^{4}$ are given by Eqn.(38),

\displaystyle\omega_{mn}=\frac{1}{1+z_{5}}(z_{m}\mathrm{d}z_{n}-z_{n}\mathrm{d}z_{m})\,

and $\gamma^{mn}:=\frac{1}{2}[\gamma^{m},\gamma^{n}]$ , with $\gamma^{m}$ the usual Dirac gamma matrices satisfying $\{\gamma^{m},\gamma^{n}\}=2\delta^{mn}$ . Then, the Dirac equation for $\psi$ is

\displaystyle i\gamma^{l}e^{r}_{l}\Big{(}\partial_{r}+\frac{1}{4}\omega_{mn,r}\gamma^{mn}+iA_{r}\Big{)}\psi=0\,.

(44)

We will aim to find the spectrum of the Dirac operator $\mathcal{D}_{A}$ . However, exploiting the assumed spherical symmetry of the gauge field, Dolan Dolan (2003) has found general formulas for the spectrum of the square of the Dirac operator on a homogeneous space. The square of an eigenvalue $\lambda$

i\cal D_{A}\psi=\lambda\psi

(45)

of the Dirac operator will of course be an eigenvalue, $\lambda^{2}$ , of the square of the Dirac operator $(i\mathcal{D}_{A})^{2}$ , however, the converse, that $\pm\sqrt{\lambda^{2}}$ will correspond to eigenvalues of the Dirac operator, does not necessarily follow.

Dolan’s results are obtained as follows, we note that his work, as he himself notes, leans heavily on previous work of Salam-Strathdee Salam and Strathdee (1982) and was well understood in the mathematics literature Kobayashi and Nomizu (1963). Recording the more general case, let $G/H$ be a Riemannian homogeneous coset space, with $G$ and $H$ compact Lie groups and $G$ simple, such that its isometry group is $G$ and its holonomy group is $H$ . Let $t_{M},M=1\dots,\dim G$ be the anti-hermitean generators of $G$ , with $[t_{M},t_{N}]=C_{MN}^{P}t_{P}$ , and $t_{a},a=1,\dots,\dim H$ will denote the generators of $H$ . Let $A$ be a $G-$ symmetric gauge potential on $G/H$ and (using our conventions for the indices)

\mathcal{D}_{A}:=\gamma^{\alpha}e^{\mu}_{\alpha}\Big{(}\partial_{\mu}+\frac{1}{4}\omega_{\delta\beta,\mu}\gamma^{\delta\beta}+iA_{\mu}\Big{)}

is the Dirac operator on $G/H$ , where $\{e^{\alpha}_{\mu}dx^{\mu}\,,\,\alpha=1,\dots,\dim(G/H)\}$ form an orthonormal coframe for $G/H$ . Here $\alpha,\beta=1,\dots,\dim G/H$ are orthonormal indices and $\mu,\nu=1,\dots,\dim G/H$ are coordinate indices. The orthonormal 1-forms can be taken as the Maurer-Cartan 1-forms on the whole of $G$

g^{-1}dg=e^{A}t_{A}

(46)

such that

de^{A}=\frac{1}{2}C^{A}_{\,\,\,BC}e^{B}\wedge e^{C}.

(47)

The set of 1-forms separate into a subset $e^{\alpha}$ for a $G-$ invariant metric on $G/H$ and the remaining $e^{a}$ can be expanded as $e^{a}=\Pi^{a}_{\,\,\,\alpha}e^{\alpha}$ on the manifold $G/H$ . The ensuing spin connection is obtained from

de^{\alpha}+\omega^{\alpha}_{\,\,\,\beta}\wedge e^{\beta}=0

(48)

yielding the curvature 2-form

R^{\alpha}_{\,\,\,\,\beta}=\frac{1}{2}R^{\alpha}_{\,\,\,\,\beta\gamma\delta}e^{\gamma}\wedge e^{\delta}=\frac{1}{2}C^{\alpha}_{\,\,\,\,\beta a}C^{a}_{\,\,\,\,\gamma\delta}e^{\gamma}\wedge e^{\delta}.

(49)

We can calculate $(i\mathcal{D}_{A})^{2}$ to find

(i\mathcal{D}_{A})^{2}=-D_{\alpha}D^{\alpha}+\frac{R}{4}\mathbb{I}+\frac{i}{2}F_{\alpha\beta}\gamma^{\alpha\beta}

(50)

where $R$ is the Ricci scalar and $\Delta=-D_{\alpha}D^{\alpha}$ is the $G$ symmetric Dirac Lapacian acting on spinors including the spin connection and the gauge connection defined on $G/H$ . For the specific, spherically symmetric gauge fields, all three terms on the RHS of Eqn.(50) are mutually commuting and therefore can be simultaneously diagonalized. One can compute and find

[D_{\alpha},D_{\beta}]=iF^{a}_{\alpha\beta}t_{a}+\frac{1}{4}R_{\alpha\beta\gamma\delta}\gamma^{\gamma\delta}

(51)

where $t_{a}$ are the generators of the chosen representation of $H$ .

The notion of spherical symmetry means that we choose a metric and connection that are $G$ invariant. In our case, $G={\rm SO}(5)$ and $H={\rm SO}(4)$ giving $G/H={\rm SO}(5)/{\rm SO}(4)=\mathbb{S}^{4}$ as the base and the fibre $H={\rm SO}(4)\simeq{\rm SU}(2\times{\rm SU}(2)/\mathbb{Z}_{2}$ is 6-dimensional. The gauge field being spherically symmetric means that a Killing vector $K$ , generates via the Lie derivative just a gauge transform, $F$ is invariant up to a gauge transformation

{\cal L}_{K}F=g^{-1}Fg.

(52)

Such an invariance is obtained by taking the gauge connection to be equal to the spin-connection, which is possible as the gauge group is the holonomy group $H$ . The gauge field strength

F^{a}=\frac{1}{2}F^{a}_{\,\,\,\,\alpha\beta}e^{\alpha}\wedge e^{\beta}=\frac{1}{2}C^{a}_{\,\,\,\,\alpha\beta}e^{\alpha}\wedge e^{\beta}.

(53)

The Riemann tensor is covariantly conserved hence so is the field strength

D_{\alpha}F^{a}_{\,\,\,\,\beta\gamma}=0

(54)

and with this choice for the gauge field, it is easy to verify

[D_{\alpha},D_{\beta}]=C^{a}_{\alpha\beta}\left(\mathbb{I}\otimes t_{a}-\frac{1}{4}C_{a\gamma\delta}\gamma^{\gamma\delta}\otimes\mathbb{I}\right).

(55)

However, interestingly, $T_{a}=-\frac{1}{4}C_{a\gamma\delta}\gamma^{\gamma\delta}$ give a representation of the holonomy gauge group $H$

[T_{a},T_{b}]=C^{c}_{\,\,\,\,ab}T_{c}

(56)

which then implies the commutator

[D_{\alpha},D_{\beta}]=C^{a}_{\,\,\,\alpha\beta}D_{a}

(57)

where

D_{a}=\mathbb{I}\otimes t_{a}+T_{a}\otimes\mathbb{I}.

(58)

Then we can write the Dirac Laplacian as

\Delta=-D_{\alpha}D_{\alpha}=-D_{A}D_{A}+D_{a}D_{a}

(59)

but these are just the quadratic Casimirs of $G$ and $H$ respectively. These Casimirs simply depend on the representation of the groups that is being considered. Therefore we can write

\Delta=C_{2}(G,\cdot)-C_{2}(H,D_{a}).

(60)

where the $C_{2}(G,\cdot)$ indicates any representation of $G$ that contains the representation $D_{a}$ of $H$ . As we scan over all such representations, we get all the possible eigenvalues of the Dirac Laplacian. This is completely analogous to the action of the spherical Laplacian on the spherical harmonics, the result there is $l(l+1)$ for the eigenvalue of the spherical Laplacian, depending on which spherical harmonic is considered. The eigenvalue is obtained from pure group theory, there is actually no necessity to solve for the eigenfunctions of the partial differential operator given by the Laplacian! Therefore, in total we have

\displaystyle(i\mathcal{D}_{A})^{2}=C_{2}^{G}-C_{2}^{H}+\frac{1}{8}R_{G/H}\,.

In our case, we consider a symmetric homogeneous space ${\rm SO}(5)/{\rm SO}(4)\cong\mathbb{S}^{4}$ of unit radius (endowed with its standard ${\rm SO}(5)-$ invariant Riemannian metric) with holonomy group ${\rm SO}(4)$ the scalar curvature is

R_{\mathbb{S}^{4}}=12\,

giving a contribution of $\frac{3}{2}$ as a cosmological constant. The irreducible representations $(p,q)_{5}$ of $\mathfrak{so}(5)$ have quadratic Casimirs (eigenvalues)

C_{2}^{{\rm SO}(5)}((p,q)_{5})=\frac{p^{2}+q^{2}}{2}+2p+q\,.

Hence, the full spectrum of the squared Dirac operator $\mathcal{D}_{A}^{2}$ on $\mathbb{S}^{4}$ in any of our symmetric gauge field backgrounds constructed before will have the form

\displaystyle E^{[h,l]}_{p,q}={\frac{p^{2}+q^{2}}{2}+2p+q-C_{2}^{{\rm SO}(4)}(\mathcal{R}_{h,l})+\frac{3}{2}}\,,

where the quadratic Casimir operator $C_{2}^{{\rm SO}(4)}(\mathcal{R}_{h,l})$ also denotes its eigenvalues in the representation $\mathcal{R}_{h,l}$ . Here, there is the constraint that $p\geq q$ and that the irreducible representations $(p,q)_{5}$ of $\mathfrak{so}(5)$ used to compute the spectrum should contain the (embedded) representation $\mathcal{R}_{h,l}$ of $\mathfrak{so}(4)$ . Additionally, the total eignevalue will have independent contributions from the left and right sectors.

It was shown by Yang Yang (1978), in his prescient study of ${\rm SU}(2)$ monopoles on ${\rm S}^{4}$ , that the representations of ${\rm SO}(5)$ which contain a given representation $I$ of the ${\rm SU}(2)$ , satisfy

p-q=2I

(61)

where $I$ is the total “isospin” of the fermion, comprising of the combination of the gauge “isospin” $J$ and the intrinsic “isospin” of the fermion, $1/2$ , Yang (1978). Thus $I=J\pm(1/2)$ , $p=q+2I$ and $C_{2}^{{\rm SO}(4)}(\mathcal{R}_{J,0})=J(J+1)$ . This gives for the case of Eqn.(39)

E^{[J,0]}_{q+2I,q}={q^{2}}+q(2I+3)+2I^{2}+4I-J(J+1)

(62)

There will be an independent contribution for the left-handed spinors and the right-handed spinors, transforming according to representation labeled by $h$ and $l$ respectively. The relationship between $J$ and $h$ or $l$ can be slightly complicated. In our example, $h=2$ corresponds to an irreducible representation $(1,0)_{4}$ of ${\rm SO}(4)$ while for $l=-1$ , the representation corresponds to the reducible representation $(0,1/2)_{4}\oplus(0,1/2)_{4}$ . Thus the complete eigenvalue will have a representation in $(p,q)_{5}$ of ${\rm SO}(5)$ for the left-handed sector in which is embedded the representation labelled by $h$ of ${\rm SO}(4)$ and a representation in $(p^{\prime},q^{\prime})_{5}$ of ${\rm SO}(5)$ for the right-handed sector in which is embedded the representation labelled by $l$ of ${\rm SO}(4)$ . Thus the full spectrum of eigenvalues will be

\lambda^{2}(q,I,q^{\prime}.I^{\prime})=E^{[h,0]}_{q+2I,q}+E^{[0,l]}_{q^{\prime}+2I^{\prime},q^{\prime}}+3/2

(63)

Example 1 : If we start the Kaluza-Klein reduction process with the standard sphere $\mathbb{S}^{7}\cong M^{7}_{1}$ , where $h=1$ and $l=0$ , i.e. $h+l=1$ and $k=h-l=1$ . Then we find that the spectrum for the Dirac operator will be (in the representation $h=1$ , whch corresponds to $(\frac{1}{2},0)_{4}=\frac{1}{2}$ of $SU(2)_{L}$ and the representation for $l=0$ , which corresponds to $(0,0)_{4}$ or the trivial representation of ${\rm SU}2)_{R}$ ). The left and right handed spinors then will be independently appended by a representation of ${\rm SO}(5)$ which permits the embedding of the given representation of ${\rm SO}(4)$ . As the right handed spnor is trivial, we will have simply $I^{\prime}=\pm(1/2)$ and then we get

	$\displaystyle E^{[1,0]}_{q+2I,q}+E^{[0,0]}_{q^{\prime}+2I^{\prime},q^{\prime}}+3/2={q^{2}}+q(2((1/2)\pm(1/2))+3)+2((1/2)\pm(1/2))^{2}+4((1/2)\pm(1/2))-(1/2)((1/2)+1)$
	$\displaystyle+{q^{\prime 2}}+q^{\prime}(2(\pm(1/2))+3)+2(\pm(1/2))^{2}+4(\pm(1/2))-(0)(0+1)+3/2$
			(64)

In this case, the reduced/effective 4D theory is just the standard Einstein-Yang-Mills theory on $\mathbb{S}^{4}$ with ${\rm SU}(2)$ Yang-Mills gauge group and our ${\rm SO}(4)-$ multi-instanton reduces to the BPST $1-$ instanton. The Milnor’s bundle is just the standard quaternionic Hopf fibration.

Example 2 : If we start the Kaluza-Klein process with an exotic $7-$ sphere $M^{7}_{3}$ , where $h=2$ and $l=-1$ , i.e. $h+l=1$ and $k=h-l=3$ . The extra term corresponding to the eigenvalues of $C_{2}^{{\rm SO}(4)}(\mathcal{R}_{2,-1})$ will depend on the integers $h=2$ and $l=-1$ . Clearly the spectrum will not be the same as for the theory on the standard sphere. In this case, the isopin for the left-handed sector will have $J=1$ so that $I=1\pm(1/2)$ while for the right handed sector, the isospin of the direct sum of two spin one-half representations will act in concert and be $J=1/2$ giving $I^{\prime}=(1/2)\pm(1/2)$ . Then the eigenvalues of the Dirac operator (squared) will be

	$\displaystyle E^{[2,0]}_{q+2I,q}+E^{[0,-1]}_{q^{\prime}+2I^{\prime},q^{\prime}}+3/2={q^{2}}+q(2(1\pm(1/2))+3)+2(1\pm(1/2))^{2}+4(1\pm(1/2))-1(1+1)$
	$\displaystyle+{q^{\prime 2}}+q^{\prime}(2((1/2)\pm(1/2))+3)+2((1/2)\pm(1/2))^{2}+4((1/2)\pm(1/2))-(1/2)((1/2)+1)+3/2.$		(65)

V Conclusions and Future Work

Thus, we see directly how different choices of smooth structures on the $7-$ sphere affect the energy/mass spectrum for fermions. The spectrum is affected because of global topological reasons. Diffeomorphically inequivalent topological 7-spheres, exist because the map between these manifolds, for certain values of $h$ and $l$ cannot be made everywhere smooth. The problem with differentiability occurs at at least one point Milnor (1956). Our results show that the spectrum of the Dirac operator, when considering the Dirac operator on an exotic 7-spheres, will be modified relative to the spectrum of the Dirac operator on a standard 7-sphere.

These results will have other applications in quantum mechanics, condensed matter physics and Kaluza-Klein supergravity. In condensed matter physics, the ground state degeneracies of the quantum Hall effect in higher dimensions are related to the Atiyah-Singer index theorem for spinors in gauge fields backgrounds. Observable effects of inequivalent differential structures would have unimagined physical consquences, Here we only look at physics on 7-spheres, which does not correspond to any given physical system. However, effective theories in condensed matter physics can easily correspoind to effective, higher dimensional physics. The example of the analysis of higher dimensional quantum Hall effect Zhang and Hu (2001) is interesting.

Indeed, the square of the Dirac operator in a gauge field background on a curved manifold $M$ is represented by the relativistic Hamiltonian for a spinor particle moving on $M$ in that background and consists of a kinetic term, a Zeeman term and a curvature term. For spinors on the coset space ${\rm SO}(5)/{\rm SO}(4)\cong\mathbb{S}^{4}$ and gauge group ${\rm SO}(4)$ moving in a homogeneous background gauge field identified with the spin connection in our (embedded) representation $\mathcal{R}_{hl}$ of $\mathfrak{so}(4)$ , the Hamiltonian (square of the Dirac operator) can be diagonalised for spherically symmetric gauge fields. In such a background, zero-modes of the Dirac operator exist, the ground states are the ones in which the Zeeman energy exactly cancels the kinetic energy and the degeneracy is the number of zero-modes, which is equal to the index of the Dirac operator in the background gauge field for $M=\mathbb{S}^{4}$ . But, as given by the Atiyah-Singer index theorem, the net number of zero-modes on $\mathbb{S}^{4}$ in a gauge field background is equal the second Chern number of the gauge field, which in our case is equal to $2(h-l)$ and related to the Milnor’s invariant of exotic $7-$ spheres.

VI ACKNOWLEDGEMENTS

We thank NSERC, Canada for financial support and Richard MacKenzie, Ben Webster(Perimeter) and Benedict Williams (UBC) for useful discussions. We thank the University of Auckland, Department of Physics, Auckland, New Zealand, where this work was finally completed and written up.

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