# Is there an analogue of the moduli space of the torus in semi-Riemannian signature?

I'm starting to study Riemann surfaces and already met the fact that Riemann surfaces have both a complex structure and a conformal structure that are, in fact, very closely related.

If we consider a Riemann surface one can classify the different conformal structures and if I correctly understand the space whose points label these different conformal structures is the so-called Riemann moduli space.

For the torus the moduli space is $${\cal M}={\cal H}/{\rm PSL(2,\mathbb{Z}})$$

where $${\cal H}$$ is the upper half plane in $$\mathbb{C}$$ and $${\rm PSL}(2,\mathbb{Z})={\rm SL}(2,\mathbb{Z})/\mathbb{Z}_2$$. So the distinct conformal structures of the torus are identified by a complex number $$\tau$$ with $${\rm Re}(\tau)\leq 1/2$$, $${\rm Im}(\tau)>0$$ and $$|\tau|>1$$.

Now, all of this is in the context of Riemann surfaces which are equipped with a conformal structure and hence with an equivalence class of Riemannian metrics.

But I would say nothing stops us from picking the manifold $$\mathbb{T}^2=S^1\times S^1$$ and endowing it with a Lorentzian metric, like $$g=-d\phi\otimes d\phi+d\psi\otimes d\psi,$$

where $$\phi$$ and $$\psi$$ are angle coordinate functions on $$S^1$$ and $$(\phi,\psi)$$ is the product chart.

Question: Is there an analogue of the moduli space and of the modular parameter for such a Lorentzian torus?

My intuition says that there should be because we can still consider two Lorentzian metrics conformally equivalent if they are a Weyl rescaling of one another, and we can still talk about a conformal structure as an equivalence class of such metrics.

• It is, most likely, going to be non-Hausdorff. Commented Feb 9, 2021 at 20:50
• This is a very intricate question. The surprising fact about the moduli space here is not that it exists, but that it is finite dimensional and admits a simple explicit description. In the Riemannian case, the relation to complex analysis plays an important role for this, and this relation is not there in Lorentzian signature. The replacement for it would be the "product structure" on $\mathbb R^2=\mathbb R\times\mathbb R$, which corresponds to the two distinguished isotropic lines for the Lorentzian metric that you have in each tangent space. But I don't know where this leads. Commented Feb 10, 2021 at 10:59

This is a corrected version of my earlier answer.

Many things are different in the pseudo-Riemannian setting. First of all, while every pseudo-Riemannian metric on a surface locally conformally-flat, there are pseudo-Riemannian metrics on 2-dimensional tori which are not globally conformally-flat (unlike in the Riemannian case):

Sánchez, Miguel, Structure of Lorentzian tori with a Killing vector field, Trans. Am. Math. Soc. 349, No. 3, 1063-1080 (1997). ZBL0873.53046.

I am unaware of any classification of pseudo-conformal structures on tori in full generality. Hence, I will consider only conformal classes of flat pseudo-Riemannian tori: Quotients $$T_{\Lambda}$$ of $${\mathbb R}^{1,1}$$ by lattices, i.e. subgroups $$\Lambda$$ of $${\mathbb R}^2$$ generated by two translations $$T_u, T_v$$ along two linearly independent vectors $$u, v$$. The choice of the generators $$T_u, T_v$$ of the lattice means that we are actually working with "marked tori;" the space of marked globally conformally flat Lorentzian tori is the Lorentzian analogue $${\mathcal T}_{1,1}$$ of the classical Teichmuller space $${\mathcal T}$$ of marked conformal structures on 2-dimensional tori. (The space $${\mathcal T}$$ is identified with the upper half-plane $$U$$, which is a model of the hyperbolic plane; the modular group $$SL(2, {\mathbb Z})$$ acts on $$U$$ via linear-fractional transformations and the moduli space $${\mathcal M}$$ of conformal structures on the torus is the quotient of $$U$$ by the action of $$SL(2, {\mathbb Z})$$. Below, I will explain, to which extent the same holds for globally conformally flat Lorentzian tori. First of all, I will consider oriented tori and I will assume that conformal maps are orientation-preserving (as it is done in the classical setting).
This orientation will translate into the assumption that the basis $$\{u, v\}$$ in $${\mathbb R}^2$$ has the standard orientation.

One can show that two marked pseudo-Riemannian tori $$T_{\Lambda}$$, $$T_{\Lambda'}$$ (with the given generators $$u, v$$ for $$\Lambda$$ and $$u', v'$$ for $$\Lambda'$$) are pseudo-conformally equivalent if and only if there exists a pseudo-conformal transformation $$A$$ of $${\mathbb R}^{1,1}$$ such that $$Au= u', Av=v'$$ where $$A$$ is an (invertible) linear transformation preserving the given Lorentzian inner product up to a scale factor. It will be convenient for my answer to assume that this inner product is given by the Gram matrix $$\left[\begin{array}{cc} 0 & 1\\ 1 & 0 \end{array}\right],$$ i.e. is $$dx dy$$ in terms of differentials. Then the matrix $$A$$ has to be diagonal with the diagonal entries of the same sign. If I were to write $$u, v$$ as column vectors in a matrix $$B$$, then the above condition reads: $$AB=B'.$$ I can also consider the effect of this multiplication on the rows of $$B$$ and will find that each row gets rescaled. The matrix $$B$$, of course, contains all the information about the vectors $$u, v$$. The fact that $$u, v$$ form a positively oriented basis means simply that $$\det(B)>0$$. It is convenient (at first) to allow more matrices $$B$$ than that and assume only that both rows $$b_1, b_2$$ of $$B$$ are nonzero. Thus, $${\mathcal T}_{1,1}$$ will be contained in a space $$X$$ obtained from pairs of nonzero vectors $$b_1, b_2\in {\mathbb R}^2$$ we would be identifying pairs if they differ by rescaling $$b_1$$ by a positive factor and $$b_2$$ by a (possibly different) positive factor. This rescaling is done independently for $$b_1$$ and $$b_2$$. The next observation is that the space of nonzero vectors $$b\in {\mathbb R}^2$$ up to such rescaling is simply the unit circle, since you can just normalize the vector $$b$$ (divide it by its Euclidean length) to make it a unit vector. Thus, our space $$X$$ is nothing but the 2-dimensional torus $$T^2=S^1\times S^2$$ (with points $$(z_1, z_2)$$, where $$z_1, z_2$$ are complex numbers of unit absolute value). Now, the torus $$T^2$$ contains the subset $$D$$ which corresponds to pairs of vectors $$u, v$$ which do not form a basis because they are linearly dependent; in other words, $$z_1=\pm z_2$$, hence, $$D$$ consists of two disjoint circles. Removing these two circles, results in two open annuli, one, $$A_+$$, for the positively oriented bases and the other one for negatively oriented ones. (In fact, we need to do one more identification, since we can multiply both $$u, v$$ by $$-1$$; this sends $$(z_1, z_2)$$ to $$(-z_1, -z_2)$$. This quotient of $$T^2$$ is still the 2-dimensional torus.)

Thus, the annulus $$A_+$$ is our Teichmuller space $${\mathcal T}_{1,1}$$. The trouble comes from an attempt to form the moduli space $${\mathcal M}_{1,1}$$ which is the quotient of $${\mathcal T}_{1,1}$$ by the action of $$SL(2, {\mathbb Z})$$. This amounts to "forgetting the basis." Unlike the classical case, this action of $$SL(2, {\mathbb Z})$$, however, is quite bad: Almost every orbit is dense. Hence, taking the naive quotient we get a highly non-Hausdorff space: almost every point is "infinitesimally close" to any other point.

Algebraic geometers teach us how to work around this problem (the keyword is "stacks"), but I will stop here. The bottom line: Yes, there is a moduli space, just it does not look like a surface (or, for this matter, anything you have seen before in your math or physics classes).

Two more references:

1. The most comprehensive treatment of pseudo-Riemannian surfaces:

Weinstein, Tilla, An introduction to Lorentz surfaces, De Gruyter Expositions in Mathematics. 22. Berlin: de Gruyter. xiv, 213 p. (1996). ZBL0881.53001.

1. This one should be highly relevant, but I do not have access to it:

Smyth, Robert; Weinstein, Tilla, How many Lorentz surfaces are there?, Gindikin, Simon (ed.), Topics in geometry. In memory of Joseph D’Atri. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20, 315-330 (1996). ZBL0860.53039.

• Thans for such a great answer ! This is all very interesting. I've met this Lorentzian torus in a Physics problem and the question of understanding the change in its conformal structure to an inequivalent one seems to be important. It also seems very interesting from the mathematical point of view. Could you suggest some references where I can learn more about this subject ? Thanks again, you gave a very nice answer !
– Gold
Commented Feb 11, 2021 at 23:40
• @user1620696 There is a book by Tilla Weinstein on this subject, although she does not talk about tori per se. I will try to find some other references for you as well. Commented Feb 12, 2021 at 1:12