By definition, a super vector space is a direct sum of two ordinary vector spaces (over the same base field, of course), graded mod 2, with “even” and “odd” elements according as these belong to one or the other of the direct summands. The authors of the book under review start off their first chapter with the following evocative and tantalizing remarks: “The theory of manifolds and algebraic geometry are ultimately based on linear algebra. Similarly the theory of supermanifolds needs super linear algebra, which is a linear algebra in which vector spaces are replaced by vector spaces with a **Z**/2**Z**-grading, namely, super vector spaces.” So it is that this first chapter takes the reader through super Lie algebras (coincidentally something my graduate school office mate, a ring theorist, sprang on me back in the 1980s; something than which few more exotic things could be imagined, I thought at that time of innocence…), universal enveloping superalgebras, and even Hopf superalgebras.

After this it’s off to the races (well, they’re already underway, of course, but we’re going to go a lot faster): algebraic geometry, or rather the theory of sheaves is discussed in the next chapter, and then we get to supergeometry. We encounter a superspace (p.46) as a super ringed space, i.e., a topological space endowed with a sheaf of super rings, with the property that at each point of the space, the stalk of the structure sheaf is a local super ring; happily, “as in the ordinary setting, a commutative super ring is local if it has a unique maximal ideal.” On the heels of this definition comes the application of these notions to manifolds, which is to say that a couple of pages later we get to supermanifolds quickly followed by superschemes.

The third chapter ends with a discussion of the functor of points, actually the second one in the book: on p.38 it’s presented in the usual algebraic geometric setting with the following pithy introduction: “When we are dealing with classical manifolds and algebraic varieties, we can altogether avoid the use of their functor of points… However, if we go to the generality of schemes, the extra structure overshadows the topological points and leaves out crucial details so that we have little information without the full knowledge of the sheaf [belonging to the scheme: see e.g. Grothendieck-Dieudonné (or p.36 of the present book)] … The same happens for supergeometric objects … [Indeed] the structure sheaf of a supermanifold cannot be reconstructed from its underlying topological space …”

Thus, already around p.50 or so we are in the thick of it: yes, it’s linear algebra, and, yes, it’s algebraic geometry, and even differential geometry, but it’s supered-up, to coin a phrase, and this means that there’s a lot more to it than introducing **Z**/2**Z**-gradings. And this is of course a very good thing on a number of counts: we certainly do have something new and different, even if it’s built up from classical notions (plus Grothendieck, so that the word “classical” is immediately compromised).

Anyway, at this stage we are hitting (super) geometry hard: differentiable supermanifolds in the fourth chapter, next; then the Frobenius comes in two chapters later, after which it’s Lie theory time: super Lie groups and their actions. The book’s last three chapters go on to deal with, respectively, homogeneous spaces (super Minkowski spacetime!), supervarieties and superschemes (with an appearance by the Grassmannian superscheme), and algebraic supergroups (e.g. “Lie(G) for a supergroup scheme G; and linear representations are discussed on p.223 ff.). Finally, there are three appendices, devoted respectively to Lie superalgebras, categories, and Fréchet superspaces.

So much for sketching what the book’s about. What’s the rationale for it? Well, first off, as they authors put it early on, “[s]upersymmetry … is the machinery mathematicians and physicists have developed to treat two types of elementary particles, bosons and fermions, on the same footing. Supergeometry is the basis for supersymmetry …” And this entails some really beautiful stuff: “Einstein’s special relativity requires that physical theories must be invariant under the action of the Poincaré group. Since observable operators (e.g. Hamiltonians) must commute with this action, the classification [of elementary particles] corresponds to finding unitary representations of the Poincaré group. A ‘super’ representation gives a ‘multiplet’ of ordinary particles which include both fermions and bosons.” Very, very pretty!

Of course, there’s a lot more: the authors note that whereas supersymmetry was discovered by the physicists in the ’70s, featuring, e.g., Abdus Salam, to identify some one whose name is a household word, “the mathematical treatment of it began much later and grew out of the works notably of Berezin [whose name is attached to the super-counterpart to the determinant, the “Berezinian” — see p.12], Kostant, Leites, Manin, Bernstein, Freed, Deligne, Morgan, [and] Varadarajan.” So we’re dealing with a great deal of interesting material of significance well beyond the confines of physics.

And who should read the book? Well, in the mathematical sense, everybody! This is truly wonderful stuff, and the authors have taken pains to make it all quite accessible (and I think they succeed beautifully). The formal prerequisites are as follows: “Our work is primarily directed to second or third year graduate students who have taken a one year graduate course in algebra and a beginning course in Lie groups and Lie algebras [and I’d add some algebraic geometry: sheaves, schemes, &c.] … Our book can very well be used [for] a one-semester course or a participating seminar on supersymmetry, directed to second and third year graduate students.” And indeed it should be.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.