Modular invariance in string theory

[petergreat]

Is it proved that the bosonic string and superstring partition functions are modular-invariant for arbitrarily high loop order? If not, how many loops have been analyzed?

 

[crackjack]

I may be wrong but, as far as I understand, modular invariance is the mapping class symmetry of one-loop diagrams only. Under some conditions (what are they?), they can be extended to symmetries of higher-genus Riemann surfaces.

Modular invariance is one necessary (but not always sufficient) anomaly cancellation constraint. Tadpole cancellations (which includes K-theory constraints) are stronger than modular invariance.

I am sure there can be a better answer.

 

[suprised]

The analog of the modular group of the torus is the Siegel modular group Sp(2g,Z), where g is the genus of the Riemann surface.

For bosonic strings it it relatively easy to prove the the genus-g amplitude is modular invariant.

I am not completely up to date for superstrings, but I believe that this problem boils down, as always, to how to properly define a measure of the supermoduli space. Moreover, if I recall correctly, one has to put picture changing operators at certain locations and then the task is to show that the given choice (which naively breaks modular invariance) drops/cancels out in the end. As said, I am not sure about the current status, but at any rate this is the flavor of the typical problems. I believe things were sorted out to some genus like g=2 or 3.

 

[tom.stoer]

... how to properly define a measure of the supermoduli space. ... I believe things were sorted out to some genus like g=2 or 3.

What about the measure at all? If remember some papers where a measure for g=3 and 4 was constructed, but I am not 100% sure about that. Is it true that beyond g=4 the measure is in general not known? Is this perturbative approach still considered to be useful or required - or outdated?

 

[suprised]

Here is a review summarizing some research post d'Hoker-Phong:

http://arxiv.org/pdf/0804.3167

It appears that old problems (concering the measure) were overcome, but not rigorously.

 

[petergreat]

I have a (somewhat) related question. Is 4D general relativity invariant under large diffeomorphisms, in a hypothetical universe with a non-trivial 4D topology?

 

[tom.stoer]

I am not sure if I understand. Do you mean transformations that change topology? Or do you mean ordinary diffeomorphisms in different topological sectors?

 

[petergreat]

I am not sure if I understand. Do you mean transformations that change topology? Or do you mean ordinary diffeomorphisms in different topological sectors?

I mean diffeomorphisms which cannot be smoothly connected to identity, similar to the modular group of the 2D torus. After all, the symmetry of GR is diffeomorphism invariance. I think "large diffeomorphisms" exist when the topology have a non-trivial fundamental group, that's why R^4 is not of interest here.

 

[tom.stoer]

Diffeomorphisms by construction respect the topology, so within each "superselection sector" with fixed topology nothing is wrong with diffeomorphisms; different "superselection sectors" i.e.different topologies cannot be related by diffeomorphisms, but that's not a problem.

Think about universes R*R³, R*S³ and R*T³; (where R means the time direction); these topologies are unrelated and GR does not establish any geometrical or dynamical relationship.

 

[tom.stoer]

Just an idea: are you talking about Dehn twists on the 2-torus?

 

[petergreat]

Just an idea: are you talking about Dehn twists on the 2-torus?

Exactly. This is a topology-preserving transformation, so your earlier objection that "different topologies don't mix up" does not apply here. I suppose similar transformations also exist in 4D, but I may be wrong.

 

[tom.stoer]

The Dehn twist does NOT change the topology of the torus, it only twists the geometry. Neigboured points are transformed into neigboured points and closed curves are transformed into closed curves, so it's topology-preserving.

But I see what you are talking about. Such transformations could indeed be called "large" and I guess they cannot be generated dynamically as they belong to different "superselection sectors", just like solitons in ordinary field theory.

There is one construction I am aware of where such "gluings" or "twists" are used. This is the Poincare dodecahedral space which is generated from a dodecaeder where opposite faces are twisted and glued together. This (closed, finite) space allows for a negative curvature and is discussed as a model for our universe. It seems that it reproduces some structures known from the cosmic mivrowave background.

Of course you could do the same thing with a 3-torus but it seems that this is not relevant for cosmology.

 

[tom.stoer]

Wait.

There is another puzzle, namely Dehn twists which are not diffeomorphisms (using arbitrary angles, not only N*360° rotations). But even if these twists are not homeomorphisms they map tori to tori.

I think we better discuss these questions here: https://www.physicsforums.com/showthread.php?t=477213

 

[crackjack]

The analog of the modular group of the torus is the Siegel modular group Sp(2g,Z), where g is the genus of the Riemann surface.
...

Thanks @surprised, for that clarification.

As a side note, there is also the pure-spinor formulation that has gone a little further than RNS formalism in loop and n-point amplitudes.

I have a (somewhat) related question. Is 4D general relativity invariant under large diffeomorphisms, in a hypothetical universe with a non-trivial 4D topology?

The modular symmetry (that includes large-diffeomorphisms) of string theory comes up as a result of the conformal Killing symmetry of the 2D sigma model. In 4D Minkowski spacetime, the global Killing symmetries are just the Poincare symmetries. If the geometry is non-trivial, and if the Killing equation admits global solutions which are also large-diffeomorphisms (etc), I am not sure why we should not include the new symmetries.

 

[crackjack]

The modular symmetry (that includes large-diffeomorphisms) of string theory comes up as a result of the conformal Killing symmetry of the 2D sigma model. In 4D Minkowski spacetime, the global Killing symmetries are just the Poincare symmetries. If the geometry is non-trivial, and if the Killing equation admits global solutions which are also large-diffeomorphisms (etc), I am not sure why we should not include the new symmetries.

Reading my post now, I think it is complete nonsense! :-S Global/large & Killing fields should have never appeared together.