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A Short Guide to Supergeometry

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A Short Guide to Supergeometry A Short Guide to Supergeometry Nadia Ott July 2019 Contents 1 Notes on the Notes1 2 Introduction2 3 Superalgebras3 3.1 Supervector Spaces..........................3 3.2 Superalgebras.............................3 3.3 Supermodules.............................5 4 Supermanifolds5 4.1 Super Ringed Spaces.........................6 4.2 Split and Projected Super Ringed Spaces.............6 4.3 Superschemes.............................8 4.4 Complex Supermanifolds.......................9 4.5 Smooth Supermanifolds....................... 11 4.6 Something Weird: A complex supersymmetric (CS) manifold.. 11 5 Vector Bundles 12 5.1 Tangent Bundle............................ 12 6 Super Riemann Surfaces 13 6.1 The supermoduli space Mg and its Compactification....... 16 6.2 A look at genus zero 1j1 supercurves................ 18 1 Notes on the Notes These notes were prepared to give a survey of results from supergeometry aimed at mathematicians who are interested but not already familiar with the field. Personally, I find that conversations with other mathematicians have been the most conducive to learning the basics of a new field. Therefore, I have chosen to present much of the material in the style of a very casual conversation. 1 2 Introduction Since the late 1980s, it has been known to physicists| and mathematicians!| that a superstring scattering amplitude may be formulated as a Berezin integrals over the supermoduli space of super Riemann surfaces. The story behind this jargon heavy first sentence boils down to the following collection of facts: Remark. In the following list I use the words super Riemann surface, superstring, superstring scattering amplitude, etc. Its not important to know the definitions of these words, they are introduced only so that you may gain a sense of why (super)string theorists|and super-mathematicians!| care about these objects. 1. The worldsheets propagated by superstrings are super Riemann surfaces. Click here for some pictures illustrating this in the case of ordinary strings. 2. A superstring scattering amplitude refers to the "probability" that a super- string hanging out in space will propagate a certain type of super Riemann surface. By "a certain type" I am referring to a super Riemann surface of a pre-imposed genus and number of punctures. It is natural for physicists to want to compute this probability but to do this |in the sense of ac- tually getting a value out of it| one needs a good understanding of the geometry of the moduli spaces of super Riemann surfaces. This is where mathematicians come in... 3. First of all, to compute an integral over a supermoduli space one first needs to define what exactly to take an integral of. Ordinarily, this is done by integrating a global section of the determinant bundle, however, super Riemann surfaces don't have a natural top form. Felix Berezin, resolved this complication by constructing a new type of determinant called the Berezinian, or the super-determinant. This new formulation of a top form makes it possible integrate over superspaces via Berezinian integration. Click here for more information on how to integrate over a supermanifold. 4. As is true in the ordinary case, the moduli spaces of super Riemann sur- faces are not compact. However, a good way to determine integrability of the the thing (super Mumford form) we are trying to integrate to look at the Deligne-Mumford compactification of the moduli spaces. The Deligne- Mumford compactification of a supermoduli space is slightly more compli- cated than the compactification in the ordinary setting. This is because we allow for two types of degnerations to occur, the first type, which is called a Neveu-Schwarz degeneration, is due to the appearance of nodes while the second type, called Ramond degeneration, is due to singularities in the "super structure" placed on a super Riemann surface. 5. As you might be already able to tell, the Neveu-Schwartz type of degener- ations are analogous to the punctures one encounters in ordinary moduli theory. The Ramond punctures however are fairly exotic and have no ordinary counterpart. 2 Remarks on Notation: The lower-case k will always refer to an alge- braically closed field of characteristic zero. Actually, its best just to pretend k = C. 3 Superalgebras Basic Idea: Put a Z2 grading on it and require morphisms to preserve it. 3.1 Supervector Spaces Definition 3.1. A super vector space V is a vector space with a Z2-grading. Morphisms of supervector spaces are Z2-grading preserving linear maps. 1. Can you give an example of a supervector space of dimension mjn ? Take any ordinary vector space W of dimension m + n and choose a basis e1; ··· ; em+n of W . Take the vector space spanned by e1; ··· ; em and de- note it by W0. Similarly, take the vector space spanned by em+1; ··· ; em+n and denote it by W1. Then W0 ⊕ ΠW1 is a supervector space! 2. What does a morphism of T : kmjn ! kpjq look like as a matrix? A B T = : C D where A and D are matrices of dimension r×p and s×q, respectively. The blocks B and C are matrices of dimension r × q and s × p, respectively. Important Remark: Notice that in order for T to preserve grading B and C must be comprised of "odd" elements. 1 However, T is matrix over the ground field k which has no "odd" elements and therefore B, C are zero matrices. This is exactly why the matrix representation of a grading preserving morphism is usually not given until one has defined the notion of a supermodule over a superalgebra. 3.2 Superalgebras Definition 3.2. A superalgebra A over a field k is a supervector space equipped with a grading-preserving multiplication map A ⊗ A ! A. A su- peralgebra is always assumed to be k-associative and unital. Morphisms of superalgebras are grading-preserving k-algebra maps. 1. What about commutativity? Let a; b 2 A be homogeneous elements. Ordinarily, we say that a; b com- mute iff ab = ba. However, this is not what we want in superalgebra. In- stead, we say that a; b supercommute iff ab = (−1)jajjbjba, that is, if they 1 The notion of odd elements will make more sense once we give the definition of a super- algebra. 3 obey the rule of signs. Similarly, we say that a; b skew-supercommute iff ab = −(−1)jajjbjba. We say that A is a supercommutative algebra if A is a superalgebra such that for all a; b 2 A, ab = (−1)jajjbjba. 2. Can you give an example of a supercommutative algebra? The classical exterior algebra on a vector space V =∼ km+n is a familiar example of an object which obeys the rule of signs. We can decompose M 2p M 2q+1 ^•(V ) := ( ^ V ) ⊕ ( ^ V ) p q The even 2p-tensors and odd 2q + 1-tensors obey the rule of signs. Thus, the exterior algebra naturally has the structure of a supercommutative • algebra. We refer will refer to ^ V as the Z2-graded exterior algebra of V . 3. Wait, doesn't a superalgebra have something to do with Grassmann vari- ables? Of course, if θ1; ··· ; θn are Grassmann variables, then the polynomial algebra k[θ1; ··· ; θn] is an example of a superalgebra over k. I gave the example in 2 first only because it will become important in the discussion of split supermanifolds and because it makes sense for those of use who haven't seen Grassmann variables before. 4. If A is an ordinary commutative algebra over k, then we know that A =∼ k[x1; ··· ; xn]=I for some n 2 N0 and some ideal of relations I on the generators xi. Is something similar true for supercommutative algebras? Yes. Let θi be Grassman variables. Any supercommutative algebra A over k is a quotient of k[x1; ··· ; xm] ⊗k k[θ1; ··· ; θn], for some m; n 2 N0 2 and homogeneous ideal Q. The decomposition of A into its Z2-graded components is given by, X A0 = ff0 + fI θI jf0; fI 2 k[x1; ··· ; xm];I 2 fi1; ··· ; irgg (1) I;ev X A1 = f fJ θJ jfJ 2 k[x1; ··· ; xm];J 2 fj1; ··· ; jsgg: (2) J;odd This implies that A1 is an A0-module generated by fθig for all 1 ≤ i ≤ n and that, thereby, the superalgebra A has the structure of an A0-module as well. Of course, the same is true for any quotient of A, and thus for any supercommutative algebra. 5. So what are the odd and even elements of A? The elements belong to A0 are even and those belonging to A1 are odd. 2 An ideal Q of a supercommutative algebra over k is homogeneous if Q = (Q \ A0) ⊕ (Q \ A1). 4 Every supercommutative algebra A over k comes with an ideal J ⊂ A generated by A1. we mod out A by J we are left with an ordinary commutative ∼ algebra. Usually, this is denoted by A=J = Ared. As an example, take the • superalgebra A = k[x] ⊗ ^ θ1; θ2. Then J = hθ1; θ2i and A=J = k[x]. This gives us a nice way of recovering the ordinary commutative algebra underlying a superalgebra! 3.3 Supermodules Definition 3.3. Let A be a supercommutative algebra over k.A free super- module R over A of rank pjq is a module R such that R =∼ A⊕p ⊕ (ΠA)⊕q. Morphisms of free supermodules over A are grading-preserving A "linear" maps. 1. What do morphisms of supermodules over A look like? A morphism of free A-modules T : Apjq ! Arjs has matrix representation A B T = : C D where A and D are matrices of dimension r × p and s × q, respectively, comprised of elements belonging to A0.
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