What causes topology change in Calabi-Yau manifolds?

[dhillonv10]

Hi all,

I've been reading on phase changes that occur in manifolds such as flop transitions and conifold transitions for some time but I don't quite understand this one thing: flop transitions mathematically describe how one calabi-yau can change into another and most books mention that but what really causes this change? The book I am reading elegantly describes what happens to a calabi-yau manifold as it goes through but it doesn't say what causes it? I would love to know the name/description of the process that causes these phase changes, also any links to papers and such would be greatly appreciated. Thanks a lot :)

- Vikram

 

[mitchell porter]

Topology change is an extreme outcome of "geometry change". This goes back to Einstein's general relativity, the idea of gravity as curvature in space, and the equations describing how that curvature changes, in response to the presence of energy and because of its own intrinsic dynamics. Unless the Calabi-Yau metric is flat and stable everywhere, the curvatures of all the extra dimensions will be dynamical - increasing or decreasing. If one of the topology-defining curves in the manifold shrinks to zero, it's because the geometric and non-geometric forces are making it contract. It's not that different to the collapse of a star into a black hole, which is also a matter of dynamical geometry. And then the string-theory effects smooth out the transition to a new topology.

Also, because you have the quantum fluctuations or zero-point energy in quantum theory, there should be some possibility of a spontaneous-looking topology change at high energies / small scales. But again, that's because the quantum fluctuations temporarily concentrate the energy distribution in the necessary way.

All that is logically how things ought to be, but I'm not sure how close string theory comes to realizing this picture in mathematical detail. A lot of calculations are possible only because of special properties which allow you to skip difficult details. Calabi-Yau metrics are only known approximately, and there is a whole branch of string theory ("topological string theory") which is about calculations which are independent of the CY metric and only depend on its topology. But in the end, the "why" of topology change is broadly the same as the reason why black holes form or the universe expands - the dynamics of spatial curvature, as first figured out by Einstein.

edit: https://www.physicsforums.com/showthread.php?t=343045#post_message_2380389" on topological strings.

 

[dhillonv10]

If one of the topology-defining curves in the manifold shrinks to zero, it's because the geometric and non-geometric forces are making it contract. It's not that different to the collapse of a star into a black hole, which is also a matter of dynamical geometry. And then the string-theory effects smooth out the transition to a new topology.

Thanks for your insight mitchell, I am quite familiar with GR, but in this case if I am following right, dynamic geometry serves to be the cause of topology change or geometry change (in its mild form). So at a very fundamental level geometry is causing all these changes, right?

 

[dhillonv10]

mitchell, another question comes to mind if you think about stability, you answered my previous question about flop transitions so this ties in with it, if calabi-yau are stable and flop transitions preserve the observable physics as Brian Greene showed in his paper, can it be that flop transitions happen in nature all the time? This also relates to your idea of the spacetime fabric being dynamic, also how would one go about proving this rigorously?

 

[suprised]

I sense some confusion here and so let me try to clarify a bit.

You were asking what "causes" flop transitions; as if they would occur "by themselves" all the time. The confusion seems to originate in the word "transition".

Things are like follows. The topology and geometry that are considered are the ones of a Calabi-Yau compactification space. The size and shape of it can be changed by changing the "moduli", which are the size and shape parameters of the CY. These appear in the low-energy field theory as massless bosons. Changing the sizes and shapes corresponds to changing the vacuum expectation values of these moduli. One effect of this is that the masses of some other states of the theory change, like of states coming from branes wrapped around the cycles whose sizes change.

Depending on the values, some of these cycles can shink to zero size such that the CY becomes singular and the effective theory behaves badly. Sometimes one can formally analytically continue the size of the cycle past zero to negative values, however this strange geometry can then be reinterpreted in terms of a topologically different CY with positive cycle areas again. This is called a flop transition. What Brian was telling is that due to non-perturbative instanton effects, the physical effective theory actually perceives no singularity at this transition and the physical quantities stay smooth.

So all depends on the the vacuum expectation values of the massless moduli fields. If they stay constant (3+1D spacetime), then nothing changes ever. Now what determines them? This depends on the concrete physical string model that "uses" the CY. If you compactify type II strings, then the theory has N=2 supersymmetry, and this means that there is no potential for those fields; ie, any value is allowed and the value of these fields is undetermined. This is part of the landscape problem. So it is "up to you" to put in any value of these fields you like, and the physics depends on the values you choose.

The name "flop transition" refers to the hypothetical situation that you can smoothly change the values such that the CY runs through such an (apparent) singularity, and thus connect two topologically different CY's in a continuous manner. Despite the name "transition" this does not mean that this happens "by itself"; it is "you" who sits at the controls and changes the fields in a Gedanken experiment to "cause" these transitions.

In more realistic theories with less supersymmetry, there will be in general a potential for those moduli fields, which arises from all sorts of effects which are very hard to compute explicitly. The condition of being at the minimum of this potential implies that the possible values of these moduli are constrained or even completely fixed; this is called the moduli fixing problem. In such a situation there is no freedom to hypothetically change those values at will; rather they are determined by non-perturbative properties of the theory and only god knows what they actually are in a given model.

So don't be confused by the word "transition"; it mostly refers to a hypothetical Gedanken experiment. Only if one would consider a dynamical evolution of the moduli fields (potentially in the early universe), then one could run potentially through such a transition. But the tools for dealing with time-dependent backgrounds are limited and that's why this line of thoughts never has been developed AFAIK.

So this issue of flop transitions addresses an academic Gedanken exercise and its value was to demonstrate how non-perturbative string dynamics can blur out naive geometrical singularities. It does not have such an overall physical importance in itself (its significance may have been a bit over-emphasized in the book); it does not address the dynamics of 3+1D space time but rather of internal degrees of freedom.

 

[mitchell porter]

suprised is a working string theorist and I am not, so you should listen accordingly. But I'll still make a few more comments.

if calabi-yau are stable and flop transitions preserve the observable physics as Brian Greene showed in his paper, can it be that flop transitions happen in nature all the time?

Brian Greene used mirror symmetry to find an equivalent perspective where topology change doesn't happen. But in that mirror perspective, the geometry still changes, so the observable physics would still be affected.

Consider an ellipse. It has two parameters - the lengths of the two axes. These are its "moduli", and the moduli space for an ellipse is the set of all ellipse shapes. In an extreme case where one modulus goes to zero, it just flattens into a line.

Similarly, a CY has a lot of parameters which specify its size and shape, the moduli, and the moduli space for a CY consists of all the different shapes that the CY can end up in, and there are boundary cases (like the ellipse getting squashed) where the CY changes topology and becomes a different CY.

Like suprised says, those papers from 1993 were not looking at physical causes of topology change. They were just varying the moduli, like turning a dial, and seeing how the observable physics changed. One phenomenon was that you could take the moduli to values that made no sense geometrically - because they implied zero or negative lengths in the CY - but they still defined a working field theory. And in 1993 they showed that these cases do have a geometrical interpretation, in terms of another CY - you've moved into the moduli space of a CY with a different topology.

suprised makes an important technical point that all these CYs are thought to be stable; they shouldn't have any tendency to change. However, one consequence of the smooth transition between topologies, is that you can have a solution to string theory in which the topology of the extra dimensions varies in space. In effect, you'd have one CY everywhere on one side of the universe, the other CY everywhere on the other side of the universe, and a transition zone in the middle where you had CYs balanced on the transition point. (No one is proposing that the observable universe is like that, although there might be such boundaries out of sight, beyond our "Hubble volume".)

And then in some of Brian Greene's later papers he does develop models of topology change over time. http://arxiv.org/abs/1011.6588" , though not about topology change, can give you an idea of what's going on. It talks about the interplay between fluxes, cycles, and branes. The fluxes are lines of force in nongeometric fields (ultimately derived from strings) which encircle topological cycles in the CY. The amount of flux around a cycle constrains the size (e.g. radius) of the cycle, and the amount of flux can change by creation of a brane, but this costs energy. So those are the interacting ingredients in CY geometric change, and when the geometric change becomes large enough, it turns into topology change. Everyone assumes that the universe has settled into a highly stable state now, where these factors are all in a balance that's very hard to disturb. But we have to develop models where this is demonstrably the case, and where the observable physics matches the observed physics. And that's some of what string theorists are currently trying to do.