Meaning of "Bundle Connection" in String Theory?

["pi"mp]

Hi,
So in an informal sense, we hear about string theory requiring small, curled up dimensions locally at every point of spacetime. In my very, very limited knowledge of geometry, I would like to think of this as a fibre bundle structure over each point of Minkowski space. However, analogous to gauge theories, don't we need to specify the bundle connection A_{\mu}(x)? I'm not sure if that's the right terminology, but I'm trying to refer to the difference between cylinder vs. Mobius band as line bundles, for example.

So what is the physical significance of this bundle connection, if any? Thanks!

 

[fzero]

You want to think of the higher-dimensional space time $S$ as a bundle $S\rightarrow M_4$ over Minkowski space $M_4$. The compact dimensions would be viewed the fibers. The reason that you don't see people discussing a connection here is that, almost all of the time, one considers a genuine product space $S= M_4\times X$. So this is a trivial fiber bundle and there is always a choice of coordinates such that the connection in question vanishes.

The main reason that one only considers the product structure is the observed Lorentz invariance of the universe. The parameters describing the shape and size of $X$ (called moduli) are directly related to the parameters like the gauge couplings and masses of the low-energy observable physics. Astrophysical measurements tell us that quantities like the fine structure constant don't seem to vary much, if at all, over the observable universe. Significant changes to $X$ between points in 4-space would tend to lead to different physical constants in different parts of the universe.

That said, there is a certain way in which string theory takes into account that the moduli of $X$ can change. There is a low-energy effective description of string physics into quantum fields analogous to the photon, electron, etc. In that language the moduli are promoted to scalar fields on Minkowski space, which we can represent by $\Phi_i$. Within QFT, there is a notion of splitting a field into a classical value $\phi_i$ and a purely quantum fluctuation $\varphi_i$:

$$\Phi_i = \phi_i + \varphi_i.$$

The classical geometry only depends on the $\phi_i$ which appear to be the same everywhere in order to agree with observation. The ability to study the $\varphi_i$ as well in string and the associated quantum field theory is a way in which string theory encodes a type of "quantum geometry" that doesn't necessarily have a simple explanation in terms of classical geometric concepts.

 

["pi"mp]

Thanks for your reply. Let me see if I'm appreciating this properly. You say the size, shape, (and orientation?) of the fibres determine observed physical constants in our base space universe. A non-trivial connection would mean these constants change at different points in the universe. Therefore, we claim the connection is trivial.

I am confused by you mentioning moduli. I've only heard of this referred to as a "space of possibilites" where each point in the moduli space represents a different possible geometry. So if we have a Calabi-Yau bundle over Minkowski space, in what sense is the Calabi-Yau manifold a "moduli" ?

 

[fzero]

Thanks for your reply. Let me see if I'm appreciating this properly. You say the size, shape, (and orientation?) of the fibres determine observed physical constants in our base space universe. A non-trivial connection would mean these constants change at different points in the universe. Therefore, we claim the connection is trivial.

I am confused by you mentioning moduli. I've only heard of this referred to as a "space of possibilites" where each point in the moduli space represents a different possible geometry. So if we have a Calabi-Yau bundle over Minkowski space, in what sense is the Calabi-Yau manifold a "moduli" ?

By moduli I mean the specific parameters that distinguish two manifolds that are topologically the same. For a torus (a 2d CY!) we have three real numbers. In the picture where we think of it as a solid formed by rotating a circle, we have two radii and an angle. The unit torus has two parameters: the ratio of the radii and the angle. If we draw the unit torus in the complex plane (by identifying edges), these parameters are naturally assembled into the complex structure modulus $\tau$.

The torus is a still simple enough to consider fibering over a line. This is directly analogous to the QFT description. Say we just consider $S\rightarrow \mathbb{R}$ to be the total space where we have torus fibers over the real line and we let the complex structure be a function $\tau(t)$. String theory gives a natural dynamics for $\tau$ that looks symbolically like

$$ \Delta \tau = J ,$$

where $\Delta$ is the Laplacian on the base and $J$ could be source terms or additional interactions between fields.

 

["pi"mp]

So the function \tau(t) gives us the different moduli of the tori attached above each point? If I'm understanding this correctly, we have a trivial connection but thanks to this function, the tori can have different sizes. Is that right?

So a non-trivial connection corresponds to observed parameters varying throughout the universe, but what does it mean physically to have tori with different moduli in the fibre? Even if the connection is trivial.

 

[fzero]

So the function \tau(t) gives us the different moduli of the tori attached above each point? If I'm understanding this correctly, we have a trivial connection but thanks to this function, the tori can have different sizes. Is that right?

Yes, $\tau(t)$ tells us the value of the modulus for a given point $t$ on the base. For a generic choice of function $\tau(t)$, the connection will be nontrivial. The modulus $\tau$ that I'm referring to controls the shape rather than the size (area) of the torus, but as a math problem one could also allow the size to change.

So a non-trivial connection corresponds to observed parameters varying throughout the universe, but what does it mean physically to have tori with different moduli in the fibre? Even if the connection is trivial.

As a physical problem, our modulus $\tau$, or more generally $\Phi_i$ will take the classical value $\phi_i$ everywhere in Minkowski space in order to maintain Lorentz invariance. $\phi_i$ can be thought of as an equilibrium solution to the equation of motion or as the vacuum expectation value of the quantum field $\Phi_i$. Since we are dealing with a quantum theory, small, localized variations due to excitations $\varphi_i$ are expected. These excitations cost energy of order the Planck scale, so they wouldn't happen often enough to have an observable effect.