# Exotic spheres in string/M-theory

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An exotic sphere of dimension $n$ is a smooth manifold of dimension $n$ which is homeomorphic to the standard $n$-dimensional sphere but not diffeomorphic to it. In what follows, I always assume that $n$ is greater than $5$. One can show that the set of differentiable structures on a topological sphere of dimension $n$ is a finite group, isomorphic to the group of connected components of the group of diffeomorphims of the standard $(n-1)$-dimensional sphere (one constructs something homeomorphic to the  $n$-sphere by gluing two $n$-balls by a diffeormorphism of the $(n-1)$-sphere boundary. One obtains an exotic sphere if this diffeomorphism is not continously connected to the identity).

The possibility of an interest of exotic spheres in physics was first suggested by Witten in the paper "Global gravitational anomalies" (http://projecteuclid.org/euclid.cmp/1103943444 ). The main subject of the paper are the global gravitational anomalies of the $10$ dimensional superstring theories. In this context, the appearance of exotic $11$-dimensional spheres is quite natural. We want to study the behavior of the theory under a diffeomorphism of the $10$-dimensional sphere not continously connected to the identity. To do that, we consider a $11$-dimensional manifold with a metric that interpolates between the $10$-dimensional metric and its image under the diffeomorphism i.e. we have essentially inserted an $11$-dimensional exotic sphere. To compute the anomaly, we have to compute the variation of the spectrum of the $10$-dimensional Dirac (and Rarita-Schwinger) operators under this interpolation, which is related to the index of some $12$-dimensional Dirac (and Rarita-Schwinger) operators. This relation between anomalies in dimension $d$ and indices in dimension $d+2$ is quite standard and is true for local/global gauge and gravitational anomalies. But in this construction, the dimensions $d+1$ and $d+2$ have no physical meaning, they are just a mathematical trick to compute a physical quantity, the anomaly, in dimension $d$. The fact that the computation of the global gravitational anomaly for $10$-dimensional superstring theories uses exotic $11$-spheres and $12$-dimensional Dirac operators has a priori nothing to do with the existence of well-defined quantum theories in $11$ and $12$ dimensions.

So we can ask for a more physical meaning of exotic spheres. A proposal in this direction is made in the Witten's paper: exotic $n$-spheres should be thought as gravitational instantons in $n$-dimensional gravity and should give rise to solitons in $(n+1)$-dimensional gravity. For example, Witten asks: is it possible to find the $10$-dimensional solitons corresponding to the $7$ exotic $9$-spheres in the $10$-dimensional string theories ? One could ask the analogue for M-theory: is it possible to find the $11$-dimensional solitons corresponding to the $5$ exotic $10$-spheres in the $11$-dimensional M-theory ? Has any progress be made on these questions since Witten's paper?

In fact, I am more interested by instantons than by solitons. Is there any indication of the presence of the gravitational instantons corresponding to the $991$ exotic $11$-dimensional spheres in the $11$-dimensional M-theory? If I assume that they exist, I could ask: if I put a gravitational instanton between the Horava-Witten walls (take an Euclidean time) defining the strong coupling limit of the $E_8 \times E_8$ heterotic superstring theory (http://arxiv.org/abs/hep-th/9510209 ) and if I take the walls very close from each other, what becomes the gravitational instanton from the $E_8 \times E_8$ heterotic superstring point of view ?

This question is motivated by the fact that there is a hidden $E_8$ in the exotic $11$-spheres. More precisely, the group of differentiable structures on the $11$-sphere is cyclic of order $992$ (the standard one + $991$ exotic). The cyclic generator $M$ can be constructed as follows. Consider the $12$-dimensional singular manifold $X$ defined by the equation $x_0^3+x_1^5+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=0$ in $\mathbb{C}^7$. Its has a singularity at the origin $O$.  Take a small $13$-dimensional sphere around $O$ in $\mathbb{C}^7$ and intersect it with $X$: this gives a smooth $11$-dimensional manifold which is our $M$. Now $X$ is related to $E_8$ because the intersection matrix of the vanishing cycles of the singularity of $X$ at $O$ is the Cartan matrix of $E_8$ (the analogue result for the complex surface singularity $x_0^3+x_1^5+x_2^2$ is well-known and plays an important role in the Heterotic on $T^4$/ type II on $K3$ duality: this complex surface singularity can locally appear in a K3 surface and this corresponds to an enhancement to $E_8$ of the gauge group of the theory). I ask myself if there is a relation between this $12$-dimensional $E_8$ and the $10$-dimensional $E_8 \times E_8$ of the heterotic superstring theory.

recategorized Aug 28, 2014

Let $r$ denote the radius of the 13-dim sphere and $M_r$ denote the 11-dim manifold that you get by intersecting the sphere with $X$. Is $M_r$ Einstein? What is known about $M_r$ in the limit $r\rightarrow 0$? Does it decompose into an interval times a 10-d manifold? Does Milnor have something to say on these issues?

@suresh1: It is known that $M_r$  admits Sasakian-Einstein metrics : http://arxiv.org/abs/math/0311293 ;

I don't know what can be said about the behaviour in the limit $r \rightarrow 0$.

One could ask the analogue for F-theory: is it possible to find the 12-dimensional solitons corresponding to the 991 exotic 11-spheres in the 12-dimensional F-theory?

There are not exotic 12-dimensional spheres in the 12-dimensional F-theory.

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