Without being informed before it actually happened, I found myself sitting with K. Costello, M. Douglas, J. Morgan, D. Sullivan, A.J. Tolland, B. Vallette etc. in the common room of the Math department at Stony Brook. It was supposed to be one of a series of dialogues between physicists and mathematicians. During the conversation Douglas proposed a conjecture of Kontsevich-Soibelman:
A CFT satisfies gapped positive energy condition if all the eigenvalues of the Hamiltonian () are non-negative and or for some . The conjecture is that the space of all CFTs with a fixed central charge satisfying gaped positive energy condition is precompact.
The conjecture is an analogue of a theorem in Riemannian geometry (I don’t know if the following statement of the theorem is correct. I just copy it from Douglas’ handwriting on blackboard. The precise statement is not very important to us.) :
The space of Riemannian manfolds with the diameter of smaller than and satisfying , is compact.
Kevin Costello was asked if it is true in TCFT context. Kevin replied no. It quickly became clear that it is perhaps a good sign because the states in a TCFT are all vacuum states. Maybe it can be viewed as an evidence of the necessity of the gapped positive energy condition.
One might feel weird when he/she see this conjecture for the first time. The conjecture is not so well-defined. We don’t even know what the definition of CFT is, needless to mention the topology on the space of CFTs. However, this conjecture make perfect sense to me!
After Michael Douglas wrote down the conjecture, Kevin asked immediately how they come up with such conjecture. Michael answered “well, you had better ask themselves.” Although this conjecture make a lot of sense to me immediately, only when I was preparing my talk which was delivered today, I found more to say about this conjecture. So I added it to my today’s talk as the last part.
After nearly an hour introduction to my own proposal that a CFT should be viewed as a stringy algebraic geometry or a 2-spectral geometry, which should recover Riemannian geometry in certain classical limit, I put down the following derivation of K.-S. conjecture:
a 2-spectral geometry = a CFT
energy is gaped.
By adding the natural positive energy condition and ignoring ??, we arrive at the K.S. conjecture.
It is already quite interesting. But one should not stop there. It is natural to ask if we can do better?
If you are familiar with the classification of open-closed rational CFTs, you definitely can say more. Although the classification result is only available for the rational cases, it indeed suggests a lot for the irrational case as well. For example, it is reasonable to believe that the closed algebra in an irrational CFT should also be a commutative symmetric Frobenius algebra in a braided tensor category. So let be a VOA generated by its Virasoro element with central charge . It is natural to expect that the category of -modules, satisfying gapped positive energy condition plus some other natural conditions, gives a braided Frobenius tensor category as some kind of a categorification of a commutative Frobenius algebra. Then we can reformulate K.-S. conjecture and propose the following three potentially different conjectures.
Conjecture 1: The space of isomorphic classes of commutative symmetric Frobenius algebras in + some conditions (modular invariance condition) is pre-compact.
Conjecture 2: The space of the Morita classes of simple symmetric Frobenius algebras in is pre-compact.
Conjecture 3: The space of the equivalent classes of indecomposable module categories over is pre-compact.
Look, these conjectures are still not well-defined because we don’t know what topology to choose if there is any interesting topology at all. But let us leave such problem aside for now. Conjecture 3 reminds me of a conjecture by Ostrik:
Conjecture (Ostrik): For a given rigid monoidal category with finitely many irreducible objects there exists only finitely many inequivalent indecomposable -modules.
Motivated by Ostrik’s conjecture, we would like to give a continuous version of it.
1. A topological Frobenius tensor category is a Frobenius tensor category endowed with a proper topology.
2. is called compact if the space of 0-dimensional objects (whatever it means, D0-branes?) is compact.
Then we can propose the following conjecture:
If a topological Frobenius tensor category is compact, so is the space of equivalent classes of indecomposable -modules.
I don’t know how to make the space of equivalent classes of indecomposable -modules into a topological space. But I hope that readers will find this endeavor interesting.
I don’t remember how the dialogue was ended. I forgot to ask them if such a dialogue is a regular event at Simons Center. But my impression is that it is certainly not the first one nor the last one. I expect that it will happen very often in the future. I think that Stony Brook is a very good place to study mathematical physics.