Timeline for Why "holomorphic" vertex algebra?
Current License: CC BY-SA 4.0
18 events
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| Apr 20, 2021 at 10:55 | comment | added | André Henriques | @TheoJohnson-Freyd To answer your questions: yes for (i) <=> (ii). We also have (ii) => (iii). I'm not sure about (iii) => (ii). Certainly, the tensor category Bimod(A) has a good chance of remembering A up to Morita equivalence. For example, the absorbing object of Bimod(A) has endomorphism algebra given (stably) by $A\otimes A^{op}$. So Bimod(A) remembers to Mortia equivalence class of $A\otimes A^{op}$. | |
| Apr 19, 2021 at 23:06 | comment | added | Theo Johnson-Freyd | In the “irrational radius circle” case: the irreps of each copy of Heis are labeled by $\mathbb{R}$. Each copy has a dense subset $S_L, S_R \subset \mathbb{R}$, namely the projection of the corresponding Narain lattice to the x and y axes. This subset spans a subcategory of the representations (I resist calling it a “dense” subcategory). The full CFT does include an isomorphism between these subcategories. | |
| Apr 19, 2021 at 23:00 | comment | added | Theo Johnson-Freyd | @AndréHenriques In particular, it seems that I understand (the formal aspects of) Schur-Weyl duality less well than I thought I did. Let me ask you a more down to earth question. Suppose I have two von Neumann algebras A and B. Which of the following data are equivalent? (i) A and B are each other’s commutators inside a Type I factor. (ii) A and B are (oppositely?) Morita equivalent. (iii) Bimod(A) and Bimod(B) are (antimonoidally?) equivalent. | |
| Apr 19, 2021 at 22:54 | comment | added | Theo Johnson-Freyd | @AndréHenriques I thought, but your comment makes me realize that I am mistaken, that the radius was encoded in the identification of modules. | |
| Apr 19, 2021 at 21:10 | comment | added | André Henriques | @TheoJohnson-Freyd you write: "A good working definition of "2D CFT" is "pair of [unitary] vertex operator algebras with an identification between their categories of representations"". How does that account for CFTs such as free boson compactified on a circle of irrational square radius? Such theories has Heis as both the chiral and antichiral vertex algebras. | |
| Apr 19, 2021 at 3:24 | comment | added | David Ben-Zvi | Thanks, this is all extremely helpful. I like the ``erasing dR H^*" -- maybe we should be thinking about something like the classifying space of a symplectic groupoid as the corrected moduli space of leaves in a Poisson manifold. Though I am partial to having both states and operators (or if you'd like, having access to eg correlation functions). | |
| Apr 19, 2021 at 2:44 | comment | added | Theo Johnson-Freyd | Analogously, if you have any old Poisson manifold, then I want you to think of it as a family of symplectic manifolds parameterized by the “space” of symplectic leaves. Actually, in derived geometry, I don’t know myself what the correct definition of that space is. But you, @DavidBen-Zvi, might. This is a good thing to do since it’s easy enough to contemplate parameters in which at special values the number of degrees of freedom drops, and Poisson geometry organizes that. It also organizes “dualities” where seemingly different parameters give the same system: these are the large dense leaves. | |
| Apr 19, 2021 at 2:34 | comment | added | Theo Johnson-Freyd | If you have any old vN algebra, then it isn’t a single QM system. What it is is a family of QM systems parameterized by Spec(centre). Well, that’s true non stackily. I really want you to take the monoidal category of bimodules, and think of it as sheaves on some sort of “E2 stack”, and then that stack is the parameter space. The coarse moduli space is Spec(centre), if you can define things correctly (I have not thought through the details), and the stack should be a scheme exactly then the fibres of the bundle of algebras (ie set the central character) are all type I. | |
| Apr 19, 2021 at 2:27 | comment | added | Theo Johnson-Freyd | The reason I bring up Morita: a factor is Type I (and this is the definition in the super case) when it is Morita invertible. See, bosonically without symmetry, Type I factors are Morita trivial — they are B(H) — but they are not so in families, whereas Morita invertibility is an “in families” notion. | |
| Apr 19, 2021 at 2:23 | comment | added | Theo Johnson-Freyd | He hoped that these would all be B(H). But he was disappointed, discovering some of the Type II factors. Iirc, he did not find the hyper finite Type III factor. What von Neumann lacked was a principle to exclude the Type II factors, so he was sad. The problem, I think, was that he didn’t know Morita theory. I’d have to check the dates, but I think von Neumann was working in the 30s and Morita in the 40s? Someone will hopefully correct me. | |
| Apr 19, 2021 at 2:21 | comment | added | Theo Johnson-Freyd | He also, I think, knew that commutative C* algebras have, and are determined by, their spectrums, and that for a commutative vN, the spectrum is a probability space. This is very nice for axiomatizing QM. And he knew the Heisenberg uncertainty principle, and should have know the Noether theorem except she proved it in her Habilitation, if my memory is correct, and then didn’t publish it. Anyway, my history might be off. The point is that von Neumann was reasonably led to the notion of “vN factor”, ie simple vN algebra. | |
| Apr 19, 2021 at 2:17 | comment | added | Theo Johnson-Freyd | It helps to remember why von Neumann invented his algebras. The Hilbert space is obviously not physical. States aren’t vectors or even lines: only pure states are. What is obviously physical are the operators, and it’s reasonable to think that they form a C* algebra. von Neumann knew that if he could show that the algebra of operators was abstractly isomorphic to B(H), then the Hilbert space would be uniquely determined. And so he gave a good argument that the algebra should be weak-closed, ie von Neumann. | |
| Apr 19, 2021 at 2:12 | comment | added | Theo Johnson-Freyd | Quantumly, what I think the true definition of QFT should be is ... well, it should have the following form. There should be some combination of factorization algebra, higher categories of extended operators, and functional analysis, at the end of which you should know what is a “von Neumann n-algebra”. (Compare: a multifusion category is a finite semisimple 2-algebra.) Then a qft should be a von Neumann n-algebra, plus some Poincare symmetry data (Hamiltonian operator, etc), plus that it is a type I factor. | |
| Apr 19, 2021 at 2:09 | comment | added | Theo Johnson-Freyd | ... has cohomology even if you have a dense leaf. Well, it always has cohomology equal to the de Rham cohomology — I really mean to subtract that off, and work with the sheaf Casimirs/Constants. (And if we’re hanging out late in the evening at the conference hotel bar, then I might tell you that I want to do something to collapse the de Rham stack to a point. But I would be wrong, because this is only supposed to be (semi)classical, and in particular perturbative, so I shouldn’t care about “large” automorphisms.) | |
| Apr 19, 2021 at 2:04 | comment | added | Theo Johnson-Freyd | @DavidBen-Zvi I should emphasize the caveat that my opinions on the matter continue to evolve. So I might disagree with myself in a few years. With that said, yes, classically, I would use a word like “degenerate” for non symplectic phase spaces. Here is a test: a Noether theorem should hold. Classically, Noether’s theorem is the statement that all symplectic vector fields are Hamiltonian. Maybe this succeeds for Poisson-with-dense-leaf if you insist on smooth vector fields, but i think it fails derivedly? Equivalently, there should be a sheaf of Casimir functions, and I think this sheaf ... | |
| Apr 17, 2021 at 23:19 | comment | added | David Ben-Zvi | That's a very insightful response! One question: when you talk about triviality of centers of operators in QFT, that seems to me a version of saying your classical phase spaces are all [shifted] symplectic, rather than Poisson -- is this restriction necessary? or would you use some words like "degenerate" to apply to "Poisson" QFTs? (TBH I'm confused if this is the analog of symplectic or of Poisson with dense symplectic leaf, so no Casimirs) | |
| Apr 17, 2021 at 19:19 | history | edited | Theo Johnson-Freyd | CC BY-SA 4.0 |
corrected dimension versus codimension on one statement
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| Apr 14, 2021 at 16:46 | history | answered | Theo Johnson-Freyd | CC BY-SA 4.0 |