This brief instroduction is based on David Tong’s TASI Lectures on Solitons Lecture 1:Instantons.

We’ll first talk about the instantons arise in SU(N) Yang-Mills theory and then explain the connection between them and supersymmetry. By the end we’ll try to explain how string theory jumps in this whole business.

Instantons are nothing but a special kind of solution for the pure SU(N) Yang-Mills theory with action S=\frac{1}{2 e^2}\int d^4x Tr F_{\mu\nu}F^{\mu\nu}. Motivated by semi-classical evaluation of path integral, we search for finite action solutions to the Euclidean equations of motion, \mathcal{D}_{\mu} F^{\mu\nu}=0. In order to have a finite action, we need field potential A_{\mu} to be pure gauge at the boundary \partial R^4=S^3, i.e. A_{\mu}=ig^{-1}\partial_{\mu}g.

Then the action will be given roughly by a surface integral which depends on the third fundamental group of SU(N), given by \Pi_3(SU(N))=k, k is usually called the charge of the instanton. For the original action to be captured by this number k, we need to have self-dual or anti self-dual field strength.

A specific solution of k=1 for SU(2) group can be given by A_{mu}=\frac{\rho^2(x-X)_{\nu}}{(x-X)^2((x-X)^2+\rho^2)}\bar{\eta_{\mu\nu}}^i(g\sigma^i g^{-1}) where Xs are coordinate parameter \rho is scale parameter and together with the three generator of the group, we have 8 parameters called collective coordinates. \eta is just some matrix to interwine the group index i with the space index \mu.

For a given instanton charge k and a given group SU(2), an interesting question is how many independent solutions we have. The number is usually counted by given a solution A_{\mu}and we try to find how many infinitisiaml perturbation of this solution \delta_\alpha A_\mu, known as zero modes, \alpha is the index for this solution space, usually called moduli space.

When we consider a Yang-Mills theory with an instanton background instead of a pure Yang-Mills theory. We’d like to know if we still have non-trivial solutions, and especially if these solutions will give rise to even more collective coordinates. This is where fermion zero modes and supersymmetry comes in. For \mathcal{N}=2 or 4 supersymmetry in D=4, it’s better to promote the instanton to be a string in 6 dimentiona or a 5 brane in 10 dimensions respectively. The details of how to solve the equation will be beyond the scope of this introduction, and We’ll refer the reader to the original lecture notes by David Tong.

These are some notes on arXiv:hep-th/0701050 by Denef, Douglas and Kachru.

Flux compactification is an ominous term that often scares people away. Here are some notes I come across can give a simple idea how to do moduli stabilization using flux compactification in 6 dimensions. We choose 6 dimensions because it equals  4(Minkowski spacetime we live in) plus 2(like a torus, the simplest low dimensional Calabi-Yau we can find).

The idea is that we start with Einstein-Hilbert action L=\int d^6 x \sqrt{-g} M_6^4 {\cal_R} and dimensional reduce it to a 4d spacetime.

A compact 2d internal space can be characterized by its genus. (g is the number of holes in the “donut”, g=0 is sphere, g=1 is torus…). An ansatz for the 6d metric can be taken to be ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}+R^2\tilde{g_{mn}}dy^m dy^n, where R^2 is the volume of the 2d manifold M_g. Then the action can be written as M_6^4R^2\int d^4x \sqrt{-g}((\int d^2y\sqrt{\tilde{g}}{\cal_R}_2)+R^2{\cal_R}_4)+....

We realize that \int d^2y\sqrt{\tilde{g}}{\cal_R}_2 is a topology constant which equals \chi(M_g)=2-2g and rescale to Einstein frame(where we have canonical Einstein-Hilbert action \int \sqrt{-g}{\cal_R}):g\rightarrow h=R^2 g, we find the 4d lagrangian to be M_4^2\int d^4 x\sqrt{-h}({\cal_R}_h-V(R) where M_4^2=M_6^4 R^2 and V(R)\sim (2 g-2)\frac{1}{R^4} . Apparently this one term is not enough to help us to stabilize the volume modulus R(x).

Let’s add new ingrediants: suppose there is n units of magnetic flux on the 2d internal space M_g,i.e. \int_{M_g} F=n then the term in 6d action \int d^6 x \sqrt{-g}|F|^2 can give a term proportional to \frac{1}{R^2} given by \int_{M_g}|F|^2=R^2\times (\frac{n}{R^2})^2, after rescale to Einstein frame, we obtain V(R)\sim (2 g-2)\frac{1}{R^4}+\frac{n^2}{R^6}. Now if g=0, it is easy to see that the two terms can compete with each other and stabilize R.

If further more we add m O planes(an ingrediant has negative tension), we have one more term in the potential -m\frac{1}{R^4}, which can stabilize the modulus even when we have g=1, i.e. a torus.

In 10 dimensional string theory, we adopt the same idea: inclusion of fluxes and branes and planes will give us a potential that eventually can stabilize all the moduli.

Let us continue the summary following Denef’s notes. arXiv:0803.1194 [hep-th]

All the discussion so far is based on a simple analysis with fluxes turned off. (Flux has the same definition as in Gauss’s Law in E & M but with more generalized potential field, e.g. n dimensional antisymmetric tensor).

When we turn the flux on, they give a nice tree level potential for the moduli field, so we can currently evade from the non-computable corrections problem. However, it’s not nice enough, since for a Ricci-flat (remember we start with a CY) internal space, we can always rescale the metric to make the potential depend on r^{-a} (r characterizes the rescale of the metric and a is some positive number that we don’t care much now). So the flux potential looks like r^{-a}\int_X F^2(X is the internal manifold and F is the field strength of the potential field), which means it’s positive definite and we have a runaway solution at r\rightarrow \infty.

The idea is that adding flux potential is going to deform our original Ricci-flat internal space to be nonflat anymore. The Einstein-Hilbert action scales differently from the flux potential and can compete with it(if it has the correct sign) to generate a minimum for the moduli at some finite field range. The problem with this is that for p\geq 2 form fluxes we’ll end up in an Ads space at the KK scale, which means we’ll need to find quantum corrections to lift the Ads to ds space and the quantum corrections are (again!) of the order of the potential we constructed with flux and geometry! Something interesting happens when we allow F_1\neq 0 or F_0\neg 0. I’ll refer the reader back to the Denef’s notes.

Now we’ll have to add a lot of other stuff (like D-branes, O-planes) in order to generate some vacua with general fluxes, so we’ll have a lot of terms in the potential, and intuitively we’ll get a minimum much easier. For example, for type IIB, we can use orientifold, O3 and O7 planes, D3 and D7 branes and NSNS and RR 3-form fluxes to stabilize all the complex structure (shape) moduli, and use non-perturbative effects to stabilize the Kahler(size) moduli. In conclusion, we might be able to produce some reasonably controlled vacua, but it’s not the simple smooth exact classical solutions we were hoping for anymore.

Hence, we are living in a dirty world.

This is a summary of Denef’s argument about the moduli in string theory in his paper/lecture notes arXiv:0803.1194[hep-th]. First I’ll give the list of moduli and then I’ll argue why they are always a problem.

List of moduli:

1 We always have the overall volume modulus, since g_{\mu \nu}\rightarrow r g_{\mu \nu} is a symmetry of vacuum Einstein equations.

2 The dilaton e^{\phi}, for all string theories. This is the string coupling g_s.

3 Axions. These comes from p-form potentials C_p integrated over nontrivial p-cycles \Sigma_p.

4 Metric moduli: complex structure(shape) and Kahler(size) moduli.

5 Brane deformation and/or vector bundle moduli. (Who wants to explain this please do so.)

There are thousands of these moduli and they are no good. Even if these light scalars only couple with matter with gravitational strength, they’ll induce a “long range fifth force” which is not observed. Also, it’s difficult to allow light scalars while maintaining the predictions of standard cosmology.

Dine and Seiberg have a famous argument about the universal problem of theoretical physics,

When the corrections are computable, they are not important; when they are important, they are not computable.

In terms of moduli stabilization, an example is given as follows.

Suppose we have the property that when the corresponding modulus \rho\rightarrow \infty, our string theory is weakly coupled. Then the quantum corrections will generate a potential for \rho, which will satisfy lim_{\rho \rightarrow \infty} V(\rho)=0. Now we have two choices for this potential. If it’s positive, then we’ll have our modulus runaway to infinity. If it’s negative, then our modulus will be trapped at strong coupling where we can’t trust our calculation. We can add higher order corrections to deform the potential so that we can stabilize the modulus at a finite nonzero regime. But it’s usually the case that the corrections we add are so important that we can’t ignore the even higher order corrections anymore, i.e. we are in the strong coupling regime again. Thus Dine and Seiberg concluded: we might as live in a strong coupling string vacuum now.