John Huerta visited here for about a week earlier this month, and gave a couple of talks. The one I want to write about here was a guest lecture in the topics course Susama Agarwala and I were teaching this past semester. The course was about topics in category theory of interest to geometry, and in the case of this lecture, “geometry” means supergeometry. It follows the approach I mentioned in the previous post about looking at sheaves as a kind of generalized space. The talk was an introduction to a program of seeing supermanifolds as a kind of sheaf on the site of “super-points”. This approach was first proposed by Albert Schwartz, though see, for instance, this review by Christophe Sachse for more about this approach, and this paper (comparing the situation for real and complex (super)manifolds) for more recent work.

It’s amazing how many geometrical techniques can be applied in quite general algebras once they’re formulated correctly. It’s perhaps less amazing for supermanifolds, in which commutativity fails in about the mildest possible way.  Essentially, the algebras in question split into bosonic and fermionic parts. Everything in the bosonic part commutes with everything, and the fermionic part commutes “up to a negative sign” within itself.

### Supermanifolds

Supermanifolds are geometric objects, which were introduced as a setting on which “supersymmetric” quantum field theories could be defined. Whether or not “real” physics has this symmetry (the evidence is still pending, though ), these are quite nicely behaved theories. (Throwing in extra symmetry assumptions tends to make things nicer, and supersymmetry is in some sense the maximum extra symmetry we might reasonably hope for in a QFT).

Roughly, the idea is that supermanifolds are spaces like manifolds, but with some non-commuting coordinates. Supermanifolds are therefore in some sense “noncommutative spaces”. Noncommutative algebraic or differential geometry start with various dualities to the effect that some category of spaces is equivalent to the opposite of a corresponding category of algebras – for instance, a manifold $M$ corresponds to the $C^{\infty}$ algebra $C^{\infty}(M,\mathbb{R})$. So a generalized category of “spaces” can be found by dropping the “commutative” requirement from that statement. The category $\mathbf{SMan}$ of supermanifolds only weakens the condition slightly: the algebras are $\mathbb{Z}_2$-graded, and are “supercommutative”, i.e. commute up to a sign which depends on the grading.

Now, the conventional definition of supermanifolds, as with schemes, is to say that they are spaces equipped with a “structure sheaf” which defines an appropriate class of functions. For ordinary (real) manifolds, this would be the sheaf assigning to an open set $U$ the ring $C^{\infty}(U,\mathbb{R})$ of all the smooth real-valued functions. The existence of an atlas of charts for the manifold amounts to saying that the structure sheaf locally looks like $C^{\infty}(V,\mathbb{R})$ for some open set $V \subset \mathbb{R}^p$. (For fixed dimension $p$).

For supermanifolds, the condition on the local rings says that, for fixed dimension $(p \bar q )$, a $p|q$-dimensional supermanifold has structure sheaf in which $they look like $\mathcal{O}(\mathcal{U}) \cong C^{\infty}(V,\mathbb{R}) \otimes \Lambda_q$ In this, $V$ is as above, and the notation $\Lambda_q = \Lambda ( \theta_1, \dots , \theta_q )$ refers to the exterior algebra, which we can think of as polynomials in the $\theta_i$, with the wedge product, which satisfies $\theta_i \wedge \theta_j = - \theta_j \wedge \theta_i$. The idea is that one is supposed to think of this as the algebra of smooth functions on a space with $p$ ordinary dimensions, and $q$ “anti-commuting” dimensions with coordinates $\theta_i$. The commuting variables, say $x_1,\dots,x_p$, are called “bosonic” or “even”, and the anticommuting ones are “fermionic” or “odd”. (The term “fermionic” is related to the fact that, in quantum mechanics, when building a Hilbert space for a bunch of identical fermions, one takes the antisymmetric part of the tensor product of their individual Hilbert spaces, so that, for instance, $v_1 \otimes v_2 = - v_2 \otimes v_1$). The structure sheaf picture can therefore be thought of as giving an atlas of charts, so that the neighborhoods locally look like “super-domains”, the super-geometry equivalent of open sets $V \subset \mathbb{R}^p$. In fact, there’s a long-known theorem of Batchelor which says that any real supermanifold is given exactly by the algebra of “global sections”, which looks like $\mathcal{O}(M) = C^{\infty}(M_{red},\mathbb{R}) \otimes \Lambda_q$. That is, sections in the local rings (“functions on” open neighborhoods of $M$) always glue together to give a section in $\mathcal{O}(M)$. Another way to put this is that every supermanifold can be seen as just bundle of exterior algebras. That is, a bundle over a base manifold $M_{red}$, whose fibres are the “super-points” $\mathbb{R}^{0|q}$ corresponding to $\Lambda_q$. The base space $M_{red}$ is called the “reduced” manifold. Any such bundle gives back a supermanifold, where the algebras in the structure sheaf are the algebras of sections of the bundle. One shouldn’t be too complacent about saying they are exactly the same, though: this correspondence isn’t functorial. That is, the maps between supermanifolds are not just bundle maps. (Also, Batchelor’s theorem works only for real, not for complex, supermanifolds, where only the local neighborhoods necessarily look like such bundles). Why, by the way, say that $\mathbb{R}^{0|q}$ is a super “point”, when $\mathbb{R}^{p|0}$ is a whole vector space? Since the fermionic variables are anticommuting, no term can have more than one of each $\theta_i$, so this is a finite-dimensional algebra. This is unlike $C{\infty}(V,\mathbb{R})$, which suggests that the noncommutative directions are quite different. Any element of $\Lambda_q$ is nilpotent, so if we think of a Taylor series for some function – a power series in the $(x_1,\dots,x_p,\theta_1,\dots,\theta_q)$ – we see note that no term has a coefficient for $\theta_i$ greater than 1, or of degree higher than $q$ in all the $\theta_i$ – so imagines that only infinitesimal behaviour in these directions exists at all. Thus, a supermanifold $M$ is like an ordinary $p$-dimensional manifold $M_{red}$, built from the ordinary domains $V$, equipped with a bundle whose fibres are a sort of “infinitesimal fuzz” about each point of the “even part” of the supermanifold, described by the $\Lambda_q$. But this intuition is a bit vague. We can sharpen it a bit using the functor of points approach… ### Supermanifolds as Manifold-Valued Sheaves As with schemes, there is also a point of view that sees supermanifolds as “ordinary” manifolds, constructed in the topos of sheaves over a certain site. The basic insight behind the picture of these spaces, as in the previous post, is based on the fact that the Yoneda lemma lets us think of sheaves as describing all the “probes” of a generalized space (actually an algebra in this case). The “probes” are the objects of a certain category, and are called “superpoints“. This category is just $\mathbf{Spt} = \mathbf{Gr}^{op}$, the opposite of the category of Grassman algebras (i.e. exterior algebras) – that is, polynomial algebras in noncommuting variables, like $\Lambda(\theta_1,\dots,\theta_q)$. These objects naturally come with a $\mathbb{Z}_2$-grading, which are spanned, respectively, by the monomials with even and odd degree: $\Lambda_q =$latex \mathbf{SMan}$ (\Lambda_q)_0 \oplus (\Lambda_q)_1$$(\Lambda_q)_0 = span( 1, \theta_i \theta_j, \theta_{i_1}\dots\theta{i_4}, \dots )$ and $(\Lambda_q)_1 = span( \theta_i, \theta_i \theta_j \theta_k, \theta_{i_1}\dots\theta_{i_5},\dots )$ This is a $\mathbb{Z}_2$-grading since the even ones commute with anything, and the odd ones anti-commute with each other. So if $f_i$ and $f_j$ are homogeneous (live entirely in one grade or the other), then $f_i f_j = (-1)^{deg(i)deg(j)} f_j f_i$. The $\Lambda_q$ should be thought of as the $(0|q)$-dimensional supermanifold: it looks like a point, with a $q$-dimensional fermionic tangent space (the “infinitesimal fuzz” noted above) attached. The morphisms in $\mathbf{Spt}$ from $\Lambda_q$ to$llatex \Lambda_r\$ are just the grade-preserving algebra homomorphisms from $\Lambda_r$ to $\Lambda_q$. There are quite a few of these: these objects are not terminal objects like the actual point. But this makes them good probes. Thi gets to be a site with the trivial topology, so that all presheaves are sheaves.

Then, as usual, a presheaf $M$ on this category is to be understood as giving, for each object $A=\Lambda_q$, the collection of maps from $\Lambda_q$ to a space $M$. The case $q=0$ gives the set of points of $M$, and the various other algebras $A$ give sets of “$A$-points”. This term is based on the analogy that a point of a topological space (or indeed element of a set) is just the same as a map from the terminal object $1$, the one point space (or one element set). Then an “$A$-point” of a space $X$ is just a map from another object $A$. If $A$ is not terminal, this is close to the notion of a “subspace” (though a subspace, strictly, would be a monomorphism from $A$). These are maps from $A$ in $\mathbf{Spt} = \mathbf{Gr}^{op}$, or as algebra maps, $M_A$ consists of all the maps $\mathcal{O}(M) \rightarrow A$.

What’s more, since this is a functor, we have to have a system of maps between the $M_A$. For any algebra maps $A \rightarrow A'$, we should get corresponding maps $M_{A'} \rightarrow M_A$. These are really algebra maps $\Lambda_q \rightarrow \Lambda_{q'}$, of which there are plenty, all determined by the images of the generators $\theta_1, \dots, \theta_q$.

Now, really, a sheaf on $\mathbf{Spt}$ is actually just what we might call a “super-set”, with sets $M_A$ for each $A \in \mathbf{Spt}$. To make super-manifolds, one wants to say they are “manifold-valued sheaves”. Since manifolds themselves don’t form a topos, one needs to be a bit careful about defining the extra structure which makes a set a manifold.

Thus, a supermanifold $M$ is a manifold constructed in the topos $Sh(\mathbf{Spt})$. That is, $M$ must also be equipped with a topology and a collection of charts defining the manifold structure. These are all construed internally using objects and morphisms in the category of sheaves, where charts are based on super-domains, namely those algebras which look like $C^{\infty}(V) \otimes \Lambda_q$, for $V$ an open subset of $\mathbb{R}^p$.

The reduced manifold $M_{red}$ which appears in Batchelor’s theorem is the manifold of ordinary points $M_{\mathbb{R}}$. That is, it is all the $\mathbb{R}$-points, where $\mathbb{R}$ is playing the role of functions on the zero-dimensional domain with just one point. All the extra structure in an atlas of charts for all of $M$ to make it a supermanifold amounts to putting the structure of ordinary manifolds on the $M_A$ – but in compatible ways.

(Alternatively, we could have described $\mathbf{SMan}$ as sheaves in $Sh(\mathbf{SDom})$, where $\mathbf{SDom}$ is a site of “superdomains”, and put all the structure defining a manifold into $\mathbf{SDom}$. But working over super-points is preferable for the moment, since it makes it clear that manifolds and supermanifolds are just manifestations of the same basic definition, but realized in two different toposes.)

The fact that the manifold structure on the $M_A$ must be put on them compatibly means there is a relatively nice way to picture all these spaces.

### Values of the Functor of Points as Bundles

The main idea which I find helps to understand the functor of points is that, for every superpoint $\mathbb{R}^{0|n}$ (i.e. for every Grassman algebra $A=\Lambda_n$), one gets a manifold $M_A$. (Note the convention that $q$ is the odd dimension of $M$, and $n$ is the odd dimension of the probe superpoint).

Just as every supermanifold is a bundle of superpoints, every manifold $M_A$ is a perfectly conventional vector bundle over the conventional manifold $M_{red}$ of ordinary points. So for each $A$, we get a bundle, $M_A \rightarrow M_{red}$.

Now this manifold, $M_{red}$, consists exactly of all the “points” of $M$ – this tells us immediately that $\mathbf{SMan}$ is not a category of concrete sheaves (in the sense I explained in the previous post). Put another way, it’s not a concrete category – that would mean that there is an underlying set functor, which gives a set for each object, and that morphisms are determined by what they do to underlying sets. Non-concrete categories are, by nature, trickier to understand.

However, the functor of points gives a way to turn the non-concrete $M$ into a tower of concrete manifolds $M_A$, and the morphisms between various $M$ amount to compatible towers of maps between the various $M_A$ for each $A$. The fact that the compatibility is controlled by algebra maps $\Lambda_q \rightarrow \Lambda_{q'}$ explains why this is the same as maps between these bundles of superpoints.

Specifically, then, we have

$M_A = \{ \mathcal{O}(M) \rightarrow A \}$

This splits into maps of the even parts, and of the odd parts, where the grassman algebra $A = \Lambda_n$ has even and odd parts: $A = A_0 \oplus A_1$, as above. Similarly, $\mathcal{O}(M)$ splits into odd and even parts, and since the functions on $M_{red}$ are entirely even, this is:

$( \mathcal{O}(M))_0 = C^{\infty}(M_{red}) \otimes ( \Lambda_q)_0$

and

$( \mathcal{O}(M))_1 = C^{\infty}(M_{red}) \otimes (\Lambda_q)_1)$

Now, the duality of “hom” and tensor means that $Hom(\mathcal{O}(M),A) \cong \mathcal{O}(M) \otimes A$, and algebra maps preserve the grading. So we just have tensor products of these with the even and odd parts, respectively, of the probe superpoint. Since the even part $A_0$ includes the multiples of the constants, part of this just gives a copy of $U$ itself. The remaining part of $A_0$ is nilpotent (since it’s made of even-degree polynomials in the nilpotent $\theta_i$, so what we end up with, looking at the bundle over an open neighborhood $U \subset M_{red}$, is:

$U_A = U \times ( (\Lambda_q)_0 \otimes A^{nil}_0) \times ((\Lambda_q)_1 \otimes A_1)$

The projection map $U_A \rightarrow U$ is the obvious projection onto the first factor. These assemble into a bundle over $M_{red}$.

We should think of these bundles as “shifting up” the nilpotent part of $M$ (which are invisible at the level of ordinary points in $M_{red}$) by the algebra $A$. Writing them this way makes it clear that this is functorial in the superpoints $A = \Lambda_n$: given choices $n$ and $n'$, and any morphism between the corresponding $A$ and $A'$, it’s easy to see how we get maps between these bundles.

Now, maps between supermanifolds are the same thing as natural transformations between the functors of points. These include maps of the base manifolds, along with maps between the total spaces of all these bundles. More, this tower of maps must commute with all those bundle maps coming from algebra maps $A \rightarrow A'$. (In particular, since $A = \Lambda_0$, the ordinary point, is one of these, they have to commute with the projection to $M_{red}$.) These conditions may be quite restrictive, but it leaves us with, at least, a quite concrete image of what maps of supermanifolds

### Super-Poincaré Group

One of the main settings where super-geometry appears is in so-called “supersymmetric” field theories, which is a concept that makes sense when fields live on supermanifolds. Supersymmetry, and symmetries associated to super-Lie groups, is exactly the kind of thing that John has worked on. A super-Lie group, of course, is a supermanifold that has the structure of a group (i.e. it’s a Lie group in the topos of presheaves over the site of super-points – so the discussion above means it can be thought of as a big tower of Lie groups, all bundles over a Lie group $G_{red}$).

In fact, John has mostly worked with super-Lie algebras (and the connection between these and division algebras, though that’s another story). These are $\mathbb{Z}_2$-graded algebras with a Lie bracket whose commutation properties are the graded version of those for an ordinary Lie algebra. But part of the value of the framework above is that we can simply borrow results from Lie theory for manifolds, import it into the new topos $PSh(\mathbf{Spt})$, and know at once that super-Lie algebras integrate up to super-Lie groups in just the same way that happens in the old topos (of sets).

Supersymmetry refers to a particular example, namely the “super-Poincaré group”. Just as the Poincaré group is the symmetry group of Minkowski space, a 4-manifold with a certain metric on it, the super-Poincaré group has the same relation to a certain supermanifold. (There are actually a few different versions, depending on the odd dimension.) The algebra is generated by infinitesimal translations and boosts, plus some “translations” in fermionic directions, which generate the odd part of the algebra.

Now, symmetry in a quantum theory means that this algebra (or, on integration, the corresponding group) acts on the Hilbert space $\mathcal{H}$ of possible states of the theory: that is, the space of states is actually a representation of this algebra. In fact, to make sense of this, we need a super-Hilbert space (i.e. a graded one). The even generators of the algebra then produce grade-preserving self-maps of $\mathcal{H}$, and the odd generators produce grade-reversing ones. (This fact that there are symmetries which flip the “bosonic” and “fermionic” parts of the total $\mathcal{H}$ is why supersymmetric theories have “superpartners” for each particle, with the opposite parity, since particles are labelled by irreducible representations of the Poincaré group and the gauge group).

To date, so far as I know, there’s no conclusive empirical evidence that real quantum field theories actually exhibit supersymmetry, such as detecting actual super-partners for known particles. Even if not, however, it still has some use as a way of developing toy models of quite complicated theories which are more tractable than one might expect, precisely because they have lots of symmetry. It’s somewhat like how it’s much easier to study computationally difficult theories like gravity by assuming, for instance, spherical symmetry as an extra assumption. In any case, from a mathematician’s point of view, this sort of symmetry is just a particularly simple case of symmetries for theories which live on noncommutative backgrounds, which is quite an interesting topic in its own right. As usual, physics generates lots of math which remains both true and interesting whether or not it applies in the way it was originally suggested.

In any case, what the functor-of-points viewpoint suggests is that ordinary and super- symmetries are just two special cases of “symmetries of a field theory” in two different toposes. Understanding these and other examples from this point of view seems to give a different understanding of what “symmetry”, one of the most fundamental yet slippery concepts in mathematics and science, actually means.