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I am a mathematician who has become interested in some of the mathematics of string theory, of which I am largely ignorant, so please excuse any idiocies in what follows.

If $X$ is a Calabi-Yau $d$-fold (henceforth $CY_{d}$) then there is an associated sheaf of vertex algebras $\Omega^{ch}_{X}$, which is moreover endowed with an action of the $\mathcal{N}=2$ SUSY algebra at central charge $c=3d$. The reader can consult https://arxiv.org/abs/math/9803041 for a construction of this object. We can take cohomology of the above sheaf to produce a vertex algebra with an action of the $\mathcal{N}=2$, which we denote $H^{ch}(X)$.

Now my reading of the physics literature is that $H^{ch}(X)$ is expected (known?) to be invariant under a specific (outer) automorphism of the $\mathcal{N}=2$, known as the spectral flow, and denoted $\sigma$. This is an explicitly defined automorphism, one can find a definition in https://arxiv.org/pdf/1003.1555.pdf. Invariance here is presumably meant to mean that there is an isomorphism between $H^{ch}(X)$ and its $\sigma$-twist, ie the same underlying vector space with action of $x$ in the $\mathcal{N}=2$ defined by $\sigma(x)$. My first question is then, is this the correct notion of spectral flow invariance?

Now this seems to me to be quite a remarkable property, indeed even flowing the vacuum vector a number of times seems to produce some interesting universal (ie they exist naturally for each $X$) classes in $H^{ch}(X)$. These classes moreover have very particular properties with respect to the $\mathcal{N}=2$, specifically they are highest weight vectors for a spectral twist of a massless irrep. I can prove rigorously that these classes exist by hand (it is not quite trivial) but cannot prove that the representation $H^{ch}(X)$ is indeed invariant under spectral flow. Is a mathematically rigorous proof/ construction known in the literature?

Edit: I should stress that the isomorphism between $H^{ch}$ and its $\sigma$ twist is an isomorphism as $\mathcal{N}=2$ modules, as opposed to algebras. At least this is my reading of the physics.

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2 Answers 2

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I'll keep this answer here cause it has a couple of comments, but the $\sigma$ I describe here is not what's defined in the question, it is rather the automorphism responsible for the topological twist that I got mixed up $G^+ \leftrightarrow G^-$. I encourage you to vote it down cause it really has nothing to do with the spectral flow per-se.

I'll take a Dolbeaut resolution of your sheaf: for any smooth manifold, there exists a sheaf $\Omega^{ch,C_\infty}_X$ of vertex algebras. When $X$ is Calabi-Yau of real dimension $d$, this sheaf carries an action of two commuting $\mathcal{N}=2$ SUSY algebras of central charge $\frac{3}{2}d$. This vertex algebra admits an explicit order two automorphism which is simply determined by exchanging the odd generators $T_X \leftrightarrow T^*_X$ by using the Ricci flat metric on $X$ and it's inverse. Under this automorphism one of the two $\mathcal{N}=2$ structures is fixed, while this automorphism acts as the external automorphism $\sigma$ that you mention in your question.

The vertex superalgebra $H^{ch}(X)$ of your question is obtained from $\Omega^{ch,C_\infty}_X$ by first taking global sections, and then taking homology with respect to $Q_0^+$, here, $Q^+$ is one of the two odd generators of one of the two $\mathcal{N}=2$ algebras, and $Q^+_0$ is its zero mode. So if you take the cohomology with respect to the generator $Q^+$ corresponding to the invariant $\mathcal{N}=2$ you obtain $H^{ch}(X)$ with the remaining $\mathcal{N}=2$ structure, and the automorphism of the smooth CDR acts as $\sigma$ as you wanted.

The automorphism and its action on the two $\mathcal{N}=2$ is described in https://arxiv.org/abs/0806.1021

There are a few different ways that you can think of the relation between $\Omega^{ch}_X$ and $\Omega^{ch,C_\infty}_X$, the naive one: holomorphic sections are smooth sections, so there's a naive embedding $\Omega^{ch}_X \subset \Omega^{ch,C_\infty}_X$, or the BRST cohomology used above. This is discussed in an informal way in https://arxiv.org/abs/1702.02205 but I think it is a better explanation than the original (and slightly more general) result about generalized Calabi-Yau metric manifolds.

EDIT: I'll be happy to see if there's an algebraic proof of this avoiding the use of the Ricci flat metric as I did above. I can't see it right now, but that doesn't mean that it's not a triviality that my sleep deprived brain is missing.

EDIT2: note that I am not making any physical claim here. In particular that automorphism I mention above is precisely the right automorphism on the $\mathcal{N}=2$ subalgebra, but I do not claim this is the spectral flow of physicists. It was my understanding that on the Bosonic part for example the spectral flow would exchange $\beta \leftrightarrow \partial \gamma$, something that this automorphism is not doing: on flat space with generators $\beta, \gamma, b,c$ this automorphism is the identity on $\beta, \gamma$ and $b \leftrightarrow c$.

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  • $\begingroup$ Thank you for answering another of my questions Reimundo! I think, re ur edit 2, that this isn't the spectral flow that physicists envision. At least my reading of the physics is that they expect an infinite order such thing, and the classes formed by flowing the vacuum (nb it's an aut of modules for the N=2 not of VOAs with N=2 susy) already seem to be interesting. I've read that they expect the one unit flow of the vac to be the CY form for example. $\endgroup$
    – user108998
    Jul 27, 2020 at 21:47
  • $\begingroup$ You're absolutely right, but I don't know how to do this in the context of CDR. Note also that CDR on a CY manifold has an Odake subalgebra (adding the CY form to the N=2), but I never understood how to get this from what they call spectral flow. My collaborator Zabzine has thought about this though, perhaps you can contact him? $\endgroup$ Jul 27, 2020 at 22:02
  • $\begingroup$ Thanks for the suggestion, I'll ask Zabzine. Re the Odake subalgebra it's my opinion that one doesn't get the correct answer from it. Eg note that $\sigma$ invariance already implies the existence of a bunch of highest weight vecs for BPS reps, and I don't think the odake construction produces these (which at least I can write down explicitly by hand). Thanks again :) $\endgroup$
    – user108998
    Jul 27, 2020 at 22:09
  • $\begingroup$ I just updated this since I realized that I mixed up the automorphism you were referring to. This answer should really be voted down $\endgroup$ Jul 27, 2020 at 22:13
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    $\begingroup$ I had misinterpreted the automorphism in your question, I'm looking forward to see the classes that you can construct by flowing. In general constructing classes in $H^{ch}(X)$ is not trivial, please let me know when you post them $\endgroup$ Jul 27, 2020 at 22:48
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This answer addresses a comment of Reimundo above, it´s not a true answer to the question I asked in the OP.

We´ll construct a bunch of classes, $v_{s}(X)$, for $s\in \mathbb{Z}$, which we imagine to be the $s$-flow of the vacuum vector $\Omega\in H^{ch}(X)$. We note that $\Omega$ is of vanishing conformal weight and $U(1)$-charge so that $v_{s}(X)$ should have conformal weight $\frac{D}{2}s(s-1)$ and $U(1)$-charge $Ds$, where $D$ is the dimension of $X$. (This follows from the definition of the flow automorphism on the $\mathcal{N}=2$.) In fact the vector $v_{s}(X)$ should also have specific annihilation properties with respect to the odd currents $G^{+}$ and $G^{-}$, the classes we construct have the correct properties but we won´t make this explicit.

To construct the classes we work on the formal $D$-dimensional disc $\Delta^{D}$. To produce global classes we must ensure that the classes in $\mathbb{V}_{D}:=H^{ch}(\Delta^{D})$ that we produce are automorphism invariant, where automorphisms are understood as automorphisms of $\mathcal{N}=2$ vertex algebras. These are given precisely as automorphisms of $\Delta^{D}$ preserving the evident volume form, we denote the group of such automs $G^{CY}_{D}$.

We have generating fields $b^{i},c^{i},\beta^{i},\gamma^{i}$ for $i=1,...,D$, and $\mathbb{V}_{D}$ is spanned by monomials in $b^{i}_{j},\beta^{i}_{j},c^{i}_{1+j},\gamma^{i}_{1+j}$ as usual. The fields $b,c$ are bosonic and $\beta,\gamma$ fermionic. We identify power series in the variables $b^{i}_{0}$ with functions on the disc $\Delta^{D}$, so that we can think of $\beta^{i}_{0}$ as one forms on the $D$-disc etc.

Define vectors $v_{s}\in\mathbb{V}_{D}$ as follows, for $s\geq 0$ we set $$v_{s}:=\prod_{i=0}^{s-1}\prod_{j=1}^{d}\beta^{j}_{i},$$ and for $s<0$ we set $$v_{s}:=\prod_{i=1}^{-s}\prod_{j=1}^{D}\gamma^{j}_{i}.$$ We claim then that these vectors are preserved by the action of the group $G^{CY}_{D}$. Once this is proven it follows that there are corresponding classes $v_{s}(X)$ for each $X$ equipped with a CY-form. NB that $v_{1}(X)$ is the CY-form.

The proof essentially boils down to the observation that if $b^{j}\mapsto g^{j}(b^{1},...,b^{D})$ is a formal change of coordinates on a $D$-disc, then $\beta^{j}_{s}$ transforms under $g$ by $$\beta^{j}_{s}\mapsto\sum_{i}(\partial_{b^{i}}g^{j})\beta^{i}_{s}+(\beta_{<s}),$$ where the rightmost summand denotes a sum of monomials all containing variables $\beta^{i}_{l}$ for some $l<s$. We thus see that $\prod_{j=1}^{D}\beta^{j}_{s}$ transforms by the (by definition trivial) Jacobian plus a sum of monomials in $\beta$-variables of lesser conformal degree. Now the fact that the $\beta$ variables are fermionic implies that only the leading order terms survive, and we´re done.

Note now that the classes have the correct conformal weight and $U(1)$-charge. A slightly trickier argument shows that they have the correct annihilation properties with respect to the fermionic generating currents of the $\mathcal{N}=2$, again the point is that there are collisions between fermions forcing appropriate vanishing.

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  • $\begingroup$ Thanks, these work fine. I'd suggest though to call the $b,c$ ghosts Fermionic and the $\beta-\gamma$ ones Bosonic to avoid confusion with the standard literature (it's unfortunate that MSV used b for coordinates). $\endgroup$ Jul 28, 2020 at 20:18
  • $\begingroup$ Yes you're right, I don't think I'm going to be successful in my efforts to promote using Latin alph for Bosons and Greek for Fermions lol $\endgroup$
    – user108998
    Jul 28, 2020 at 20:38
  • $\begingroup$ Further it might be worth mentioning that the most basic case of the "find classes in H^{ch}(CY)" problem should prob thought of as this sort of universal problem, ie finding them formally locally and checking inv under CY auts, ie it's just computing inv of a huge group acting on a huge vec space. Ofc, u also have the N=2 acting compatible so that this cuts things down a fair bit. No idea what to even guess the space of invariants is mind u... $\endgroup$
    – user108998
    Jul 28, 2020 at 20:54
  • $\begingroup$ I thought about this (for any holonomy group, not just $SU(n)$) for a while and gave up. I think Bailin Song and Andrew Linshaw have too, they may have something to say about this. But I suppose it is a hard problem. The way I see this: we do not know what is the obstruction (if any) to constructing $N=1$, is it vanishing of the first Pontryagin class? $\endgroup$ Jul 28, 2020 at 22:17

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