Can Membrane Field Theory Elucidate M-Theory?

[Julius H]

TL;DR Summary

Would a membrane field theory give any insight about non-perturbative M-theory?

Hey guys, I just wanted to know if you think that a membrane field theory could ellucidate the non-perturbative framework of M-theory?

Let me specify and explain what I mean by that: String field theory was intoduced to study the non-perturbative regime of string theory and some achievements in that direction were made. It captures the dynamics of the physical and topological string. However, the latter is much easier to compute.

So my question is how are the chances that if one constructs a membrane field theory from the relation between the A-model and B-model topological string and topological M-theory, would the field theory also capture the dynamcis of the physiclal M-theory? Classically, membrane fields would be paths in the compactified space of super CFT relating the A- and B-stratum on its boundary and functionals of the maps of a covariant Courant sigma model or something like it, thereby unifying complex and symplectic structure of the target.

Do you think this is an honest way to proceed towards non-perturbative M-theory? The partition function of its topological part would probably count instantons over Lagrangian cobordisms inside the target.

 

[mitchell porter]

Those are some deep ideas.

For now I suggest studying "Hypothesis H" by Schreiber et al - that's h for homotopy, I think - they want to get M-theory from a kind of super homotopy. See if it coincides with your own thoughts.

 

[Julius H]

Thanks for your answer,

I think I did read this before and some of it is familiar from the point of view I see the subject. However, I was able to construct the before mentioned membrane field theory and showed that it has the above mentioned properties and that it reduces to Seidberg-Witten theory in the strong coupling regime and to Donaldson-Witten theory in the weak coupling regime. Or more general to N=2 SUSY Yang-Mills in 4d. Although I was only able to show it for M=Y X R inside the target T*M,where Y is some three-fold.

Furthermore, I found out with the help of the membrane field action that topological M-theory relates A-model and B-model branes through the Penrose-Ward transform. The paper was accepted few days ago. I know that there are some relations between 5-brane compactification in M-theory and Seidberg-Witten theory.

Since I study string theory only since summer 2021, I am not so familiar with all the literature. That is why I asked if this membrane field action has relevance for non-perturbative M-theory. The lagrangian is basicely an abstractly defined curvature characteristic form of the underlying infinity-principal bundle constructed with infiniy-Chern-Weil theory from Wittens cubic OSFT.

 

[Demystifier]

Strings have a virtue that the Feynman diagrams are UV finite, due to the big 2-dimensional conformal invariance. Higher dimensional objects, that is (mem)branes, are not UV finite, because higher dimensional conformal groups are not so big. Hence one does not expect that second quantization of branes, that is brane field theory, can be UV finite. In other words, such a theory cannot be a theory of "everything".

 

[Julius H]

Strings have a virtue that the Feynman diagrams are UV finite, due to the big 2-dimensional conformal invariance. Higher dimensional objects, that is (mem)branes, are not UV finite, because higher dimensional conformal groups are not so big. Hence one does not expect that second quantization of branes, that is brane field theory, can be UV finite. In other words, such a theory cannot be a theory of "everything".

Thanks for the explanation. However, the membranes I talk about are one-parameter families of strings. In this sense the membrane field relates different string fields on different backgrounds with each other. They do this because they describe paths on the boundary of superconformal field theory space with N=2 and because of this one can show with their help how the A-model and B-model are related. This is due to the fact that the membrane coupling is the action of an additive semigroup, which controls the size of the two tori of the Narain lattice.

The tororidal CFTs degenerate into A-model and B-model in the weak and strong coupling limit respectively. So the membranes I talk about are deformations of the CFT on the worldsheet and this introduces an additional dimension such that we have something like a cobordism between two worldsheets. One the intertwining worldvolume lives a covariant O(d,d,Z)-invariant courant sigma model, which one can project onto a DFT. (O(d,d,Z) transformations are T-dualities and together with the strong/weak duality this points to a topological U-duality which is said to be an ingredient of M-theory). Such models are quantized with the BV-formalism and they satisfy the BV-master equation.

I can edit some speculation : Quantum CFTs correspond to semiclassical vacuum solutions of Einstein field equations on the target. What will a deformation of the CFT aka a membrane correspond too? Most likely the gradient flow of the Einstein-Hilbert action. Such gradient flow lines are instantons, which connect different vacua by semiclassical tunneling similar as in floer homology. In ay case we get something similar to the Ricci flow.

 

[mitchell porter]

I think I found your paper at ResearchGate... It's fascinating that you want to tie together so many famous theories and objects. It will take a while to understand it all.

I have another thought. At first the idea that 2-brane theory could be described by a path through the space of SCFTs seemed artificial. But then I remembered heterotic M-theory. If the E8xE8 string theory goes to strong coupling, the eleventh dimension appears and the strings indeed turn into cylindrical 2-branes with a string at either end. Could there be a relationship with your conception?

 

[Julius H]

What I can extract from the topological version of M-theory is that its physical cousin should be described by some sigma-model on a 2-brane, that degenerates into the five superstring theories if we play with its parameters. To put it in a similar geometric picture as for the topological theory, M-theory lies at the coincidence point in the space of SCFTs with given central charge, where the different versions of CFTs of the worldsheet of the superstring meet. This point should be acted on by an additive group of 6 parameters. Five of the directions to move into the regions of the respective superstring theories and one for the coupling in the direction of 11-dimensional supergravity. In this sense the M2-brane connects 5 worldsheets with each other. I cannot know obviously but this is exactly the way in which topological M-theory connects the TCFT of the A-model and B-model.

 

[Julius H]

I think I found your paper at ResearchGate... It's fascinating that you want to tie together so many famous theories and objects. It will take a while to understand it all.

I have another thought. At first the idea that 2-brane theory could be described by a path through the space of SCFTs seemed artificial. But then I remembered heterotic M-theory. If the E8xE8 string theory goes to strong coupling, the eleventh dimension appears and the strings indeed turn into cylindrical 2-branes with a string at either end. Could there be a relationship with your conception?

I have further evidence that the action gives access to the non-perturbative regime of M-theory: The finite non-perturbative regime of the membrane field action is equal to (2,0) 6d superconformal field theory on M cross C, where M is a four-manifold and C is a, possibly punctured, complex curve. In the perturbative regime the membrane model is the cohomological Yang-Mills theory, which computes the Donaldson polynomials on M. What naturally arises is the AGT correspondence.

In conclusion:

membrane coupling greater than 1:
topological M-theory on the eight-manifold X=M cross T*C = partially twisted (2,0) 6d superconformal field theory on M cross C

Perturbative limit:
C shrinks to a point (geometric transition)
topological M-theory on the eight-manifold T*M = N=2 gauge theory on M

AGT correspondence
N=2 gauge theory on M=CFT on C

In the same way in that one can show with string field theory that A-model and B-model reduce to Chern-Simons and holomorphic Chern simons-theory, respectively, one can show the prescribed correspondence of topolgical M-theory with the membrane field theory. The equality is on the level of partition functions.

In the limit when the coupling aproaches infinity or zero, topological M-theory degernates into B-model and A-model respectively.

In my opinion this is beautiful.

 

[mitchell porter]

Looking at your work again and trying to grasp it. In some ways it's like asking what are the relations between topological string theory and the (2,0) theory - since it's the latter which I know can give rise to many N=2 theories of interest. As for topological string theory, I confess that I have never clarified for myself, exactly how it's related to physical string theory. It's as if we're considering physical strings on M4 x CY6, and then somehow only considering the CY6.

I also see (at a few points in your ResearchGate paper) you're interested in relations between general relativity and Yang-Mills. The two major cases might be, the gauge theory for gravity on 3d AdS (as described by Witten), and then attempts to do the same thing in 4d (using Palatini action, Ashtekar variables, etc). The 4d case has been at the heart of the controversy between string theory and loop quantum gravity, although perhaps twistors offer a third way to develop it. This makes me wonder if some part of your work consists of logically combining these gauge/gravity correspondences with more conventional stringy results, and therefore I also wonder if the technical criticisms of those correspondences, will also affect the validity of your claims.

 

[Julius H]

Losely speaking, a topological twist has the effect that it exchanges some of the worldsheet supersymmetry with topological symmetry, which can be much easier investigated. Building a BRST charge from these symmetries, we find in the two-dimensional case that in its cohomology are either Kähler deformations, or deformations of the conplex structure of the Kähler target. So there are two possible ways to twist the N=2 superconformal algebra, resulting in TQFTs with critical dimension 6. Another cool effect of the twisting is that the theories are semi-classical exact and that they are related by topological T-Duality (mirror symmetry). As in the phyisical case there is also a theory connecting both models.
We need to think of such a geometric transtition as a continuous physical process. This can be described with the help of a Narain lattice, which is the composition of a lattice and its dual. The authomorphisms of this lattice are elements of O(d,d,Z) and they lift to the symmetries of the CFT.
Additionally, A-model and B-model are also S-dual. This means that while making a geometric transition we vary the coupling continiously from the A-model geometry to the B- model geometry. For example we go from the deformed conifold over the singular to the resolved conifold. This is, the theory has four sectors. Let's start in the limit of zero coupling. Here the geometry is the deformed conifold which is isomomorphic to the cotangent bundle of some three-manifold T*Y. With Wittens string field theory on calculates that the A-model reduces to Chern -Simons theory because the higher modes of the string decouple and only zero modes contribute. Therefore the string field is just a U(N) Lie algebra-valued gauge potential. The group comes from the N branes we have to wrap around the zero locus were the open A-strings end. Now we increase the coupling to 1 and move towards the singular geometry. The three-manifold decribes a path and shrinks, the CFT is getting deformed and the geometry looks like T*M for M a four-manifold. Hence we need something else to describe the theory on the worldsheet times R in this regime, as well as the worldvolume the three-branes describe while transitioning. Here, with the help of the membrane field theory one shows that topological M-theory becomes twisted N=2 super Yang-Mills on Y×R in the finite perturbative regime and the membrane field is an instanton. This theory is a four-dimensional twisted cohomological theory, which computes the Donaldson polynomials. Moving further into the non-perturbative regime, a two-sphere grows instead and the geometry is M×T*S^2. We have 5-branes wrapped around M×S^2, which is the twistorspace of M. The worldvolume theory is partially twisted U(N) (0,2) superconformal field theory in 6d and one can show that this is topological M-theory on M×T*S^2 and the membrane field is a torsion-free coherent sheaf over the twistor space. This fits also well because it is known that the 4d N=2 theory is the infrared limit of the 6d theory. Topological M theory is a way to understand why. If the coupling approaches infinity the boundary looks like the resolved conifold were the membrane field takes values in the string field, which is a section of holomorphic bundle. Thus the B-model reduces to holomorphic Chern-Simons as it should be.
The whole transition is a membrane field and the dynamics are described by the membrane field action. Very cool. The claim is now that this can be generalized to arbitrary 4 and 6 manifolds. One can also show that the 6d theory is the holomorphic dual to M theory on AdS_7×S^4.
In this way we have a theory whose UV regime is a quantum theory of non-abelian gerbes. Furthermore, since the topological string is part of the physical one, both are described by the string field. We should expect the same from the topological membrane.

With regard to the second part of your post, it is known that the gradient flow lines of the CS functional are instantons. I have shown, that in the case of FLRW metrics with semi-definite Ricci curvature, the EH action has the same structure as topological Yang-Mills on Y×R. Its vacuum looks like the background of N infinite D3-branes. This made me wonder if one could apply the open/closed string duality in this case. Indeed, in the category of A branes the morphisms are strings stretched between them, mathematically described with symplectic Floer homology. Using the Atiyah-Floer conjecture I can instead consider instantons. So the situation in general relativety is mirrored by the equivalence of the wrapped Fukaya category and the (1,infinity)-category with morphisms as Lagrangian cobordisms. Understanding membrane fields as instantons over M or semi-stable coherent sheaves over the twistorspace opens a deep connection between GR and topologcial M-theory. In this szenario our expanding spacetime is an instanton as a tunneling process of closed strings between two 3-branes . It also follows that GR beyond the Planck length is somehow a 6d superconformal theory.