Manifolds with negative dimension – Definition, References

Question

Does the concept of differential manifold with negative dimension make sense, in differential geometry?
If yes, how is it defined? Do you have any reference to recommend?

My problem was born in Einstein warped-product manifolds, i.e. to admit a type of metric, the fiber-manifold must have a negative dimension, but I do not know if that manifold makes sense in differential geometry, General Relativity or String Theory…

Thank you for any help

Answer 1

Smooth manifolds of negative dimension are defined in derived geometry.

Recall that if A→M and B→M are two transversal submanifolds of codimension a and b respectively, then their intersection C is again a submanifold, of codimension a+b.

Derived geometry explains how to remove the transversality condition and make sense out of a nontransversal intersection C as a derived smooth manifold of codimension a+b. In particular, dim C = dim M - a - b, and the latter number can be negative.

See Spivak, Derived Smooth Manifolds. Simplicial approach to derived differential manifolds by Borisov and Noel simplifies Spivak's foundations considerably, and Taroyan's Equivalent models of derived stacks simplifies them even further.

Answer 2

One place where negative dimensional manifolds appear naturally is complex cobordism $U^*$. Intuitively, elements of the abelian group $U^n(X)$ are represented by families of $(-n)$-dimensional manifolds varying over $X$.

Let us only define $U^*(X)$ for $X$ a finite-dimensional manifold, without boundary but not necessarily compact. Suppose $X$ is one such, of dimension $d$. Then, the abelian group $U^n(X)$ is zero for $n>d$ and can be nonzero for all $n\leqslant d$, also negative. It is generated by pairs $(\iota:M\hookrightarrow E,c)$ where $M$ is a $d-n$-manifold, $E$ is either the total space of a complex vector bundle over $X$ (for $n$ even) or $\mathbb R$ times such total space (for $n$ odd), and $c$ is a complex structure on the normal bundle of $\iota$ which is an embedding. Relations identify pairs that are cobordant. For details, see

Quillen, D., Elementary proofs of some results of cobordism theory using Steenrod operations, Adv. Math. 7, 29-56 (1971). ZBL0214.50502.

For an element of $U^{-n}(X)$ represented by $(\iota,c)$ as above the composite of the projection of the bundle with $\iota$ gives a map $M\to X$ from a $d+n$-manifold $M$ to the $d$-manifold $X$ which can be thought of as a family of $n$-manifolds varying over $X$. This is intuitively understandable for $n\geqslant0$, but also keeps making sense for $n\leqslant0$ all the way until $n=-d$.

Answer 3

Yeah, so, there is a rather fringe (albeit compelling) construction involving “negative virtual dimensions,” called a PNDP manifold, or partially negative-dimensional product manifold. The idea is very simple: start with an Einstein manifold $\tilde{B}$, take its product with a fiber bundle $F$, and project to the PNDP, which we will denote by $\bar{B}$.

We have the formula: $$\dim(\bar{B}) = \dim(\widetilde{B}) + \dim(F) = \dim(\tilde{B}) - \operatorname{rank}(E)$$ where $E$ is the obstruction bundle of F. The obstruction bundle is a rather delicate object to define, but the interested reader may consult Fukaya, Oh, Ohta, and Ono's lengthy treatise “Technical details on Kuranishi structure and virtual fundamental chain” for some background information.

These manifolds were discussed for the first time by Pigazzini and his collaborators in their paper On PNDP Manifold, and eventually recast via a physical formulation, where they were interpreted as PNDP branes in an attempt to explain gravity.

Essentially, there are two types of “embeddings” one can perform inside the negative dimensions to create a “type II PNDP,” which is a Lagrangian submanifold of dimension (d,-d). The dimensions cancel out with the negative virtual dimensions, and so it becomes pointlike. The first embedding results from taking the inclusion $I \times I \times S^1 \hookrightarrow \smash{\bar{B}}^{(d,-d)}$; i.e., embedding a cylinder into the point. This results in an unorientable point, which is interpreted by the author's as a gravitationally active closed string. The second, and perhaps more interesting route, is to embed a Möbius strip inside the virtual sector of the PNDP, which should, in practice, result in an orientable point. This is remarkable, because orientable is a property only thought to be possessed by manifolds with positive dimension.

In short, negative dimensional manifolds do exist, and they are a niche but ongoing area of investigation, which lead to rich dualities and symmetries. There is still much to explore here, so let's get to work!

Edit: I forgot to add, if you desuspend a point, you get $$\Sigma^{-1}(\ast) = \{\}$$ which is just the empty set; ergo, the empty set is a $-1$-dimensional manifold.