Does the complex 3-sphere have a complex structure modulus?

Question

This question has a flavor which is more mathematical than physical, however it is about a mathematical physics article and I suspect my misunderstanding occurs because the precise mathematical definition of the concepts used is different than what I think. But then again, it might be some foolish mistake on my part.

I'm reading "Topological strings and their physical interpretation" by Cumrun Vafa and Andrew Neitzke. On p. 14 they introduce the "deformed conifold" (really complex 3-sphere) which is given by the equation

xy - zt = μ (equation 2.27 in the article)

in C^4 with coordinates (x, y, z, t) and μ a constant. According to the article, μ plays the role of a complex structure modulus, that is, varying μ we get diffeomorphic manifolds with different complex structure.

On p. 14 below equation (2.27) they write

This gives a Calabi-Yau 3-fold for any value μ ∈ C, so μ spans the 1-dimensional moduli space of complex structures

Also on p. 16 they write

In summary, we have two different non-compact Calabi-Yau geometries, as depicted in Figure 5: the deformed conifold, which has one complex modulus r and no Kahler moduli, and the resolved conifold, which has no complex moduli but one Kahler modulus t

Here μ was replaced by r since they change coordinates to rewrite (2.27) in the form

x_1^2 + x_2^2 + x_3^2 + x_4^2 = r (equation 2.30 in the article)

However, for any lambda non-zero, multiplication of the coordinates by lambda yields a biholomorphic mapping between the complex manifolds corresponding to μ and lambda^4 μ. Thus they are all isomorphic.

What am I missing here?

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Comments

Can you point us towards the section in the paper where that claim is made (it's a long paper)?

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Thx Joe, I added it to the text of the question

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Answer 1

I agree fully with Luboš's answer, but let me add a couple more remarks.

The notion of "non-compact Calabi-Yau" has been an extremely useful one in the topological string (and even in the physical string), even though the rules of the game are not always fully clear. One certainly doesn't want to consider e.g. the space obtained by deleting a single point from a compact Calabi-Yau as a "non-compact Calabi-Yau" in this sense. It is possible that every complete Ricci-flat Kahler manifold should be allowed.

Among the non-compact CY you almost certainly do want to allow are ones obtained by taking appropriate scaling limits of compact ones. Indeed, this is the original way non-compact CY entered the literature. In the rest of this answer I will try to say what "scaling limit" means; if what I say is correct, then it is surely in the literature somewhere, though I don't immediately know a reference.

You consider a 1-parameter family of Ricci-flat Kahler metrics $g_t$ on some compact $X$, where as $t \to 0$ the total volume of $X$ is diverging. You also take a family of open subsets $U_t$ of $X$, all equipped with diffeomorphisms to some fixed $U$. For any nonzero $t$, the induced metric $g_t$ on $U$ is incomplete. It may happen, though, that these metrics converge as $t \to 0$ to a complete (and still Ricci-flat Kahler) metric $g$ on $U$. If this happens, then $U$ is your non-compact CY.

The deformed conifold ought to be such a scaling limit. In this case the family $g_t$ would be obtained by varying the complex structure of $X$ in a particular way. Namely, let $L$ be some special Lagrangian 3-sphere in $X$, and suppose we have a family of complex structures on $X$, such that the period $Z_L = \int_L \Omega$ is going to zero as $t \to 0$ (here $\Omega$ denotes the holomorphic 3-form on $X$, normalized so that $\int_X \Omega \wedge \bar\Omega = 1$). Then let $g_t$ be a family of Kahler metrics compatible with those complex structures, such that the total volume of $L$ remains finite and nonzero as $t \to 0$. Finally we let $U_t$ be an appropriate tubular neighborhood of $L$ (contracting onto $L$ as $t \to 0$). To be honest, I think I have never seen the details of this worked out, and it might be tricky, since it involves some understanding of the Ricci-flat metric on $X$; but it may well have been done somewhere. Anyway, morally, the picture is that we are varying the moduli of $X$ in such a way that $L$ collapses to zero size, and "zooming in" on the behavior very near $L$.

Now the question arises: what should we consider to be the "moduli" of the non-compact $U$ obtained in this way? I would guess that the right answer is that they are all the moduli which come from moduli of $X$. In other words our parameter-space should include all of the Ricci-flat metrics on $U$ which are obtained from $X$ via such limits, up to isometries --- but here I mean isometries which also extend to 1-parameter families of isometries between the families of metrics $g_t$ on $X$.

In particular, in our example of $U = T^* S^3$, there is a 1-parameter family of Ricci-flat metrics, just obtained by overall rescaling. The members of this family are not isometric since they give different volumes to the special Lagrangian $S^3$. So there is at least a real modulus here. What I want to claim is that if you work "modulo isometries" in the above sense, you would find that this modulus is actually complexified. Morally, this complex modulus is keeping track of the ratio between $\int_L \Omega$ and some other period of $\Omega$, normalized by the appropriate power of the total volume so that it remains finite in the $t \to 0$ limit.

Incidentally: these days you often see people studying non-compact CY on their own, and then worrying later about whether they can actually be realized as "part" of a compact Calabi-Yau in some appropriate sense. My impression is that the question is not always straightforward (e.g. there was a lot of recent literature on such issues in the context of F-theory.)

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Answer 2

I believe that Andy and Cumrun didn't want to say that this manifold would have a complex structure modulus in isolation. However, as is clear from the "conifold" setup, the manifold given by $xy-zt=\mu$ is being incorporated into a larger manifold, so this equation only describes the vicinity of some region.

When you exploit the fixed asymptotic shape of the $xy-zt=\mu$ manifold which is $\mu$-independent and when you extend this deformed conifold geometry into a larger manifold, such as the quintic, the parameter $\mu$ connected with the neighborhood of the (deformed) singularity becomes a complex structure modulus labeling inequivalent complex structures of the whole (complicated) Calabi-Yau manifold.

Alternatively, you could count the single complex modulus even for this simple manifold itself but you would have to impose the constancy of the asymptotics i.e. ban your "scaling" transformations that were used to show the equivalence regardless of $\mu$.

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Answer 3

The explicit formula for the diffeomorphism between the affine quadric and the cotangent bundle $T^{*}S^3$ is known. It is given for example in: Hall and Mitchel's Coherent states on spheres (Equation 18). I'll write it here using the same notation of the question for completeness:

$\mathbf{z}(\mathbf{x}, \mathbf{p}) = \cosh(\frac{p}{\sqrt{\mu}})\mathbf{x}+i \frac{\sqrt{\mu}}{p}\sinh(\frac{p}{\sqrt{\mu}})\mathbf{p}$

where $(\mathbf{x},\mathbf{p})$ are the canonical coordinates of $T^{*}\mathbb{R}^4$, in which $T^{*}S^3$ is given by: $x^2 = \mu, \mathbf{x}.\mathbf{p}=0$.

The functions $\mathbf{z}(\mathbf{x}, \mathbf{p})$ (which define the induced complex structure on $T^{*}S^3$) depend explicitly on $\mu$.

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