Spectral Flow Invariance for Calabi-Yau Sigma Models

Question

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.

Answer 1

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 $b \leftrightarrow c$.

Answer 2

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.