Do negative dimensions make sense?

Question

Some time ago I read in a popular physics book that in M-theory, there are some "things" which can be said to have dimension $-1$.

Probably, the author was vastly exaggerating, but this left me wondering:

Are there mathematical theories which contain a notion that can be regarded as some sort of generalization of the classical notion of dimensionality and which allows negative values?

For example the Hausdorff dimension can assume, as far as I know, only nonnegative real values. I think that stable homotopy groups can be defined for arbitrary integer dimensions, but this doesn't really count, since one is not dealing with negative dimensional objects per se.

Answer 1

One way to define the dimension of a finite-dimensional vector space naturally extends to the definition of the Euler characteristic of a bounded chain complex of finite-dimensional vector spaces, and these can be negative. Somewhat relatedly, there is a notion of super vector space which also have a notion of dimension which can be negative. These explain, in some sense, the formula

$$\left( {n \choose d} \right) = (-1)^d {-n \choose d}$$

if you think of ${-n \choose d}$ as the dimension of the exterior power $\Lambda^d(V)$ where $\dim V = -n$. See this blog post for details.

Thinking in terms of negative dimensions also suggests some interesting dualities between Lie groups; for example, I think an inner product on a negative-dimensional vector space is a symplectic form, so in some sense the orthogonal groups of negative-dimensional vector spaces are the symplectic groups, or something like that. See, for example, this paper.

Answer 2

It is not a general notion of dimensionality, but homotopy theory and higher category theory (very closely related subjects) have found it useful to consider, for example, the $-1$-sphere: $$S^{-1}=\{x\in\mathbb{R}^0:\|x\|=1\}=\varnothing.$$ The nLab articles on negative thinking, the periodic table, and the sphere are relevant (I link to the Google cache because I can't get the pages themselves to work for me for some reason).

Answer 3

Linear group representations are vector spaces equipped with certain symmetries. More specifically a rep is a space $V$ and group $G$ with homomorphism $G\to{\rm GL}(V)$, or equivalently a $K[G]$-module where $K$ is the underlying scalar field. There is a trace map ${\rm tr}:{\rm End}(V)\to K$ which sends any linear transformation $A:V\to V$ to its trace ${\rm tr}(A)\in K$. If we postcompose $G\to{\rm GL}(V)$ with this trace map, we get a character $\chi_V:G\to K$ (when $G$ is abelian, it's a morphism $G\to K^\times$).

In particular, the dimension can be recovered via the relation $\chi(1)=\dim_KV$. Thus, in general, group characters function as a sort of "twisted" dimension value, as $\chi_V(g)$ for various $V$ and $g$ can take on negative or even nonreal values (when $K\subseteq\bf C$). Character theory is intimately related to abstract harmonic analysis / Fourier theory and roots of unity, which is why the adjective "twisted" is sensible in this context. In other places as well, when character-like values (themselves viewed as "information" or "data" about other things) are placed as new coefficients on already understood series, the series are then called "twisted."

Another representation-theoretic way of creating nonnatural number dimensions is through the creation of "virtual" objects (in the same way that going from the semiring of positive naturals with addition and multiplication to the full ring of integers involves creating negatives as "virtual" numbers designed to cancel against the "tangible" numbers). There are two major operations on representations, direct sum $V\oplus W$ and tensor product $V\otimes W$. As it happens, tensor products distribute through direct sums in the same way that multiplication distributes through addition.

Thus, our representations form a semiring under $\oplus$ and $\otimes$ operations; adjoin formal additive inverses of the representations and we have ourselves a full ring (called the representation ring, as well as other names). These give rise to "virtual" characters, which can take negative values even when applied to $1\in G$, which is to say the dimension itself can be interpreted as negative.