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 1|1 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 superstring 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 actually 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 surfaces 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 degenerations 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 m|n ? 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 denote 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 : km|n → kp|q 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 superalgebra 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 ∈ 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)|a||b|ba, that is, if they

1 The notion of odd elements will make more sense once we give the deﬁnition of a superalgebra.

3 obey the rule of signs. Similarly, we say that a, b skew-supercommute iﬀ ab = −(−1)|a||b|ba. We say that A is a supercommutative algebra if A is a superalgebra such that for all a, b ∈ A, ab = (−1)|a||b|ba. 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 variables?

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 ∈ 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 ∈ N0 2 and homogeneous ideal Q. The decomposition of A into its Z2-graded components is given by, X A0 = {f0 + fI θI |f0, fI ∈ k[x1, ··· , xm],I ∈ {i1, ··· , ir}} (1) I,ev X A1 = { fJ θJ |fJ ∈ k[x1, ··· , xm],J ∈ {j1, ··· , js}}. (2) J,odd

This implies that A1 is an A0-module generated by {θi} 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 supermodule R over A of rank p|q 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 : Ap|q → Ar|s 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. The blocks B and C are matrices of dimension r ×q and s×p, respectively, comprised of elements belonging to A1.

4 Supermanifolds

Basic Idea: A supermanifold, is a locally super ringed space which is locally isomorphic to a certain type of local model. There are three categories of supermanifolds we are interested in: Smooth supermanifolds, complex analytic supermanifolds, and superschemes. Of course, we distinguish these based on their respective local models.

Before we begin, let us talk for a second about ordinary ringed spaces since this will help us better understand the upcoming deﬁnition of a super ringed space. Essentially, a ringed space is a topological space which locally supports the notion of a ”ring of functions”. More explicitly, a ringed space is a pair (M, O) consisting of M a topological space and O a sheaf of algebras such that Ox at each point x ∈ M is a local ring. A good example of a ringed space is aﬃne space An for which M is an n-dimensional vector space and O is a sheaf of functions on M which we can choose, depending on what we want, to be either algebraic, smooth, analytic, or whatever.

5 4.1 Super Ringed Spaces

Basic Idea: Z2-grade the structure sheaf and require morphisms to preserve this. A super ringed space is a topological space which locally supports the notion of a supercommutative algebra of ”functions”. More explicitly, a super ringed spaces is a pair (M, O) where M is a topological space and O is a sheaf of supercommutative algebras.

Deﬁnition 4.1. A super ringed space is a pair X = (|X|, OX ) consisting of a topological space |X| and a sheaf OX of supercommutative algebras over k.A morphism of super ringed spaces f : X → Y is a pair (|f|, f #) where # |f| : |X| → |Y | is continuous map of topological spaces and f : OY → f∗OX is a morphism of sheaves of supercommutative algebras. If for all p ∈ |X|, the stalk OX,p at p is a local ring, we say that X is a locally super ringed space. A morphism of locally super ringed spaces is a morphism of super ringed spaces # for which the induced grading-preserving morphism fp : OY,f(p) → OX,p of #−1 stalks preserves maximal ideals, i.e fp (mp) = mf(p).

4.2 Split and Projected Super Ringed Spaces Basis Idea: Some super ringed spaces are harder to work with than others. The easiest ones to work with are those whose structure sheaves are constructed from vector bundles. These ”simple” types of super ringed space will ultimately be used as the local models from which we build supermanifolds.

First oﬀ, let us discuss how to recover an ordinary ringed space from a super ringed space (M, O). Let J be the sheaf of ideals in OM generated by OM,1. Here OM,1 denotes the odd component of the structure sheaf OM = OM,0 ⊕ OM,1. We deﬁne the reduction of (M, OM ) to be the ordinary ringed space Mred = (M, OM /J ). We often denote OM /J as OMred . We will mainly be interested in those super ringed spaces (M, OM ) where Mred is either a smooth manifold, a complex analytic manifold, or a scheme. This just means that the topological space M has either the standard topology or the Zariski topology and that OMred is a sheaf of functions in either the analytic, smooth, or algebraic category. It turns out we can always build a super ringed space from an ordinary space Y and a vector bundle V on Y .

1. Let Y be a smooth manifold. Then the total space of a vector bundle V on Y is a space locally isomorphic to Rm+n where m is the dimension of Y and n is the rank of V . Of course, we can give V local coordinates by pulling back a choice of coordinates on Rm+n. These coordinates will glue along the intersections according to the respective gluing functions on Y and V . To give V the structure of a ringed space we simply just pullback smooth functions on Rm+n and make sure everything works out on the intersections.

6 2. Let Y be a complex analytic manifold. Then the total space of a vector bundle V on Y is a space locally isomorphic to Cm+n where m is the dimension of Y and n is the rank of V . Of course, we can give V local coordinates by pulling back a choice of coordinates on Cm+n. These coordinates will glue along the intersections according to the respective gluing functions on Y and V To give V the structure of a ringed space we simply just pullback holomorphic function on Cm+n. 3. Now let Y be a scheme. Recall that given any locally free sheaf E on Y , we can construct a corresponding vector bundle on Y as follows: Let S(E) denote the symmetric algebra on E. Then Spec(S(E)) → Y is a vector bundle on Y . Conversely, any vector bundle on Y arises from such a construction as follows: Let V → Y be a rank n vector bundle and let J (V/Y ) denote its sheaf of sections. Then J (V/Y ) is a rank n locally free sheaf of OY -modules. Then it can be shown that V = Spec(S(J (V/Y )∗)) → Y . If Y is a scheme over A, then we can use the structure morphism Y → A to give the total space Spec(S(E)) the structure of a scheme over A. The most useful example of this is when A = C and Y is of finite type over A. Suppose Y is integral, seperated, and of finite type over C. 3. Then the total space of the vector bundle V = Spec(S(E)) is locally isomorphic to m+n AC . These of course glue together according to the respective gluing on Y and E. Its obvious that V is a ringed space. Anyway, let us get back to describing how we can use a vector bundle V over Y to give us a super ringed space. The description I give in the case of Y a complex analytic manifold extends easily to the cases where Y is either a real manifold or a scheme. 1. Let Y be a complex analytic manifold and let V be a vector bundle on Y . Then locally the total space of V is isomorphic to Cm+n. Let m+n (x1, ··· , xmθ1, ··· , θn) denote a choice of coordinates on C . Then θ1, ··· , θn are duals of the generators of V . Now, suppose that we instead declared the generators of V to be odd, then, locally we would have that the total space of V is isomorphic to a space we denote as Cm|n. We would denote declaration of taking the generators of V to be odd by ΠV . Then the functions on Cm|n are holomorphic functions in (x, θ). Now since the m|n θ are declared to be odd, this given C a sheaf of Z2-graded algebras, i.e a sheaf of supercommutative algebras. In particular, the total space of ΠV is the super ringed space (Y, ∧•V ∗), where we give ∧•V ∗ the structure V• ∗ V2n ∗ of a sheaf of supercommutative algebras designating ( V )0 = V V• ∗ V2n + 1 ∗ and( V )1 = V . Using the above construction we can now define a very important class of super ringed spaces called split super ringed spaces. We say that a super ringed

3All of that is just to say that Y can be covered by aﬃne varieties whose intersections are also aﬃne varieties

7 ∼ space (Y, OY ) is split if there exist a vector bundle V on Y such that OY = V• V ∗.

4.3 Superschemes Definition 4.2. An affine superscheme is locally super ringed space Spec A = ∼ ∼ (Spec(A0),A ), where A is a supercommutative algebra, and A denotes the 4 sheaf of A0-modules associated to the A0-module A. Here we use the notation Spec(A) to denote the set of prime ideals in A with the Zariski topology. A superscheme is a locally super ringed space locally isomorphic to an affine superscheme. An affine supervariety is a affine superscheme of finite type over k such that its reduction is integral, separated, and of finite type over k. Notice that the topological space of an affine superscheme is equivalent to the topological space of its ordinary counterpart in the sense that SpecA = SpecAred.

1. Is there a super version of aﬃne n-space?

(a) Or another way, let E be a locally free sheaf of rank n 5 on Am and • m let Spec(∧ (E)) → Ak be the corresponding parity reversed vector bundle. The total space of this vector bundle is the super ringed m|n space Ak (b) Obviously, any aﬃne supervariety is split by the construction above.

2. Do superschemes have local models? In the sense that, does there exist a cover by open subsets on which they are isomorphic to the total space of a trival vector bundle on the reduction?

If we limit ourselves to superschemes X of finite type over k such that Xred is integral, separated, and of finite type over k, then X may be covered m|n by affine superschemes isomorphic to Ak . From above we see that these are spaces built from a trivial vector bundle on the reduction. On an arbitrary affine superscheme X we may have nontrivial vector bundles and thus it is not clear what would serve to build a local model...

3. What about a super version of Proj?

4 Recall that A1 generates A as an A0-module. 5Here E is of course trivial.

8 This is also fairly straightforward. A projective super scheme over k is a locally super ringed space Proj A = (Proj(A),A∼), where A is a bi- graded supercommutative algebra over k, and A∼ is the sheaf of modules associated to the graded A0-module A. Be careful to keep track of the bigrading going on here! 4. What is the super version of projective n-space?

If A = k[x1, ··· , xm|θ1, . . . , θn], we deﬁne projective super space of m|n m|n dimension m|n over k, Pk , to be Proj A. The reduction of Pk is the m ordinary projective m-space Pk . 5. Are all aﬃne superschemes split? Need to think this out in detail.

4.4 Complex Supermanifolds Basic Idea: A cx. analytic supermanifold is a super ringed space locally isomorphic to the super ringed space Cp|q. Let’s begin by desribing Cp|q...

Cp|q is a super ringed space with topological space isomorphic to Cp and the • following sheaf of supercommutative algebras: O : U → Hol(U) ⊗ ∧ θ1, ··· , θq. Here U is an open subset of Cp|q which is equivalent to an open subset of Cp. Why ? This is because as topological spaces Cp|q and Cp are the same by deﬁnition.

p|q 1. What is (C )red ? p|q p p|q p (C )red = C . Make sure not to confuse (C )red = C the complex manifold Cp with the topological space |Cp|q| = Cp. We red index is always meant to refer to the underlying manifold, not the underlying topological space.

2. Where do the Grassmann variables θi come from in the structure sheaf definition? First, let me answer this with something that might at first seem unrelated. Let V be a rank q vector bundle on the underlying manifold Cp. Then we p|q • ∗ • ∗ can define C = Spec(∧ V ), where ∧ V is a Z2-graded sheaf of OCp - algebras. We use the definition of Spec as given on pg.128 in Hartshorne.

Now let me give an actual useful answer. Let η1, ··· , ηq be generators of the above vector bundle V . Then for each U ⊂ Cp|q, we have that θ1 = η1|U , ··· , θq = ηq|U . 3. What is a function (section of the structure sheaf) on Cp|q invertible? A function on Cp|q is of the form

i j I f : f0 + f1θ θ + ··· fI θ .

We say f is invertible if f0 is.

9 4. Is Cp|q an example of a split supermanifold? Of course! Recall, that we introduced splitness because this is a property we wanted our local models to have. Anyway, its obvious that Cp|q is split from the discussion of the Grassmann variables

p|q 5. What about coordinates? C has coordinates (x1, ··· , xp, η1, ··· , ηq) p where xi are a choice of coordinates on the manifold C and the ηi are a choice of generators of a trivial rank q vector bundle on Cp. Deﬁnition 4.3. A complex analytic supermanifold is a super ringed space locally isomorphic to Cp|q. Let us discuss for a moment the rigorous deﬁniton of a split supermanifold. To any p|q supermanifod, we can associate the following sheaf called the associated graded,

2 q Gr(OX ) := OX /J ⊕ J /J ⊕ · · · J . Obviously the sequence stops at q because J q+1 = 0. ∼ Deﬁnition 4.4. We say that a supermanifold X is split iﬀ OX = Gr(OX ) as a sheaf of supercommutative algebras.

1. I thought splitness had something to do with a vector bundle on Xred?I don’t see this in the above deﬁnition. • ∗ ∗ 2 ∗ q ∗ If V is a vector bundle on Xred, then ∧ V = OXred ⊕∧V ⊕∧ V ⊕∧ V . Now let J /J 2 =∼ V ∗.

2. If M is a cx. analytic supermanifold, then what is Mred?

Mred is an ordinary cx. analytic manifold. 3. What is the gluing construction for complex supermanifolds? See page 2. in this paper https://arxiv.org/pdf/1209.2199.pdf for a full discussion. 4. Are all p|q dimensional cx. analytic supermanifolds split?

No. Why? Take {Ui} to be a trivializing open cover of M. This means ∼ p|q that each Ui = C . Then

• OM : Ui 7→ Hol(Ui) ⊗ ∧ θ1, ··· , θq

p where the θi are generators of a rank q trivial vector bundle on C . And,

• OM : Uj 7→ Hol(Uj) ⊗ ∧ ϕ1, ··· , ϕq

p where the ϕi are the generators of a rank q vector bundle on C . However, there is no reason that the θi and ϕi are local generators of some vector bundle on Mred. Notice, however, that every cs. supermanifold is locally split because Cp|q is split. Click here for a construction of a non-split supermanifold. Its actually not so easy to come up with one.

10 5. Is anything weaker true? Namely, is OM at least always a module over

OMred ?

Also, no. We call supermanifolds for which OM is a OMred -module projected. 6. What is the relationship between split and projected supermanifolds? Every split supermanifold is projected. But not every projected supermanifold is split.

Theorem 4.5. Any cx. 1|1 dimensional supermanifold is split.

2 2 Proof. Its pretty easy to see that J = OM,1 ⊕ OM,1. Since OM,1 = 0 (since locally it is generated by a single Grassmann variable θ), we have that OM = OM,0 ⊕J . Again, since locally OM,1 is generated by a single Grassmann variable ∼ θ, we also have that OM,0 = OMred . Thus, O = OMred ⊕ J = Gr(OM ).

4.5 Smooth Supermanifolds Basic Idea: A smooth supermanifold is a super ringed space locally isomorphic to the local model Rp|q.

Deﬁnition 4.6. A smooth supermanifold is a super ringed space locally isomorphic to the super ringed space Rp|q. Equivalently, Let V be a rank q trivial vector bundle on the smooth manifold Rp, then Rp|q is the total space of the parity reversed vector bundle ΠV . So let us deﬁne this local model Rp|q....Rp|q is a super ringed space with topological space isomorphic to Rp and the following sheaf of supercommutative ∞ • algebras O : U → C (U) ⊗ ∧ θ1, ··· , θq.

p|q 1. What is (R )red? The smooth manifold Rp. 2. Exercise: Show that all smooth supermanifolds are split.

4.6 Something Weird: A complex supersymmetric (CS) manifold Basic Idea: These are kinda weird super ringed spaces locally isomorphic to the weird local models, Rp|∗q. It turns out that these are exactly the type of supermanifolds that we will eventually want to integrate over.

Deﬁnition 4.7. As cs manifold of dimension p|q is a topological space X with a sheaf of C-algebras such that locally (X, O) is isomorphic to the super ringed p|∗q p ∞ space R := (R ,C [θ1, ··· , θq] ⊗ C)

11 A cs manifold of dimension p|0 can be identiﬁed with a ordinary smooth manifold of dimension p. A cs manifold of dimension 0|q is the same thing as a complex analytic supermanifold of dimension 0|q.

5 Vector Bundles

Basic Idea: Just sheafify the definition of a supermodule over a superalgebra. Definition 5.1. A sheaf of modules on a supermanifold X is a sheaf F such that, for any open subset U of X, F(U) is a OX (U)-module. The sheaf F is a locally free sheaf of rank p|q, or equivalently an vector bundle of rank p|q, if there exists an open cover {Uα} of X such that ∼ ⊕p ⊕q F|Uα = (OX |Uα ) ⊕ Π(OX |Uα )

Deﬁnition 5.2. A line bundle, or invertible sheaf, on a superscheme X is a rank 1|0 locally free sheaf of OX -supermodules.

1. What about invertible sheaves of rank 0|1? Why not call those line bundles as well? First of all, we will want a that a super version of the Picard group is 1 ∗ given by H (X, OM,0). This only works out if we deﬁne line bundles to be of rank 1|0.

As in the classical case, line bundles of rank 1|0 on a supermanifold M are ∼ 1 ∗ classiﬁed by the Picard group Pic0(M) = H (M, OM,0). The restriction to ∗ OM,0 is a simple consequence of the condition that transition functions must be grading-preserving.

Important: Let F be a sheaf of supermodules over a supermanifold X, then F ⊗ OX /J , denoted as Fred, is a sheaf of OXred -modules. If F is locally free of rank m|n, then Fred is locally free of rank m|n over OXred , i.e reducing does not eliminate the Z2-grading!!

5.1 Tangent Bundle Basic Idea: No huge surprises if you write out everything using local coordinates.

Let X be a p|q dimensional supermanifold. Then locally X has coordinates 6 (x1, ··· , xp, θ1, ··· , θq). Section of the tangent sheaf TX of X are locally of the form, i j i j f(x , θ )∂xi + g(x , θ )∂θj .

6These are just the coordinates on the local models Cp|q.

12 1 Sections of the cotangent sheaf ΩX of X are locally of the form,

i j i j f(x , θ )dxi + g(x , θ )dθj .

Both the tangent and cotangent sheaf are locally free of rank p|q.

The reduction of the tangent sheaf, denoted as (TX )red, is an OXred -module but it is NOT equivalent to the tangent sheaf of the reduction. Instead, we have that 2 ∗ (TX )red = TXred ⊕ (J /J ) (3)

6 Super Riemann Surfaces

Basic Idea: A SRS is a cx. 1|1 supermanifold on which we place an additional structure. This additional structure is sometimes called a superconformal structure.

Deﬁnition 6.1. A super Riemann surface (SRS), or SUSY curve, is a 1|1 dimensional complex supermanifold Σ with the additional data of a ”superconformal structure”. A superconformal structure is a rank 0|1 subsheaf D ⊆ TΣ ⊗2 [] such that D −→ TΣ/D is an isomorphism, i.e D is maximally nonintegrable. Here [ , ] denotes the super Lie bracket. We will also require the underlying topological space of Σ to be compact and connected.

1. What is Σred?

Σred is a compact, connected, Riemann surface. The genus of Σ is the genus of Σred. 2. What exactly is the signiﬁcance of D ? I am honestly not entirely sure right now what its signiﬁcance is in terms of superstring theory. But here are some possible ideas to work with:

(a) As far as I can tell, one of the reasons physicists started studying string theory is because (apparently) replacing a particle-point of view (as represented by Feynman diagrams) with a string-point of view resulted in ”softer divergences” of the Feynman path integrals. (b) Turns out that even when we replaced particles with strings we still had problems with integration, i.e integrating over the moduli space of Riemann surfaces (as opposed to integrating over the set of all world lines as is done in the particle point of view) will still give us inﬁnite values. (c) One way to try to get rid of these singularities is to add extra structure to the worldsheets propagated by bosonic strings and, apparently, the correct extra structure to add was a square root of the

13 7 derivative ∂z. Notice that such a square root does not exist ordinarily. Therefore, the natural thing to do was add an extra nilpotent coordinate to a Riemann surface. This extra nilpotent coordinate then allows for a square root of the derivative. If none of this talk about the square root of the derivative makes sense, don’t worry! I will explain this in detail in the upcoming discussion. (d) Now think of a super Riemann surface as a (worldsheet of a bosonic string) + (extra nilpotent coordinate) + (square root of derivative).

Before we move on in our discussion of SRS, let us talk about why the above deﬁnition of a SRS can become confusing.

1. What we deﬁned above is a SRS over a point, i.e over Spec C. However, if you see the word super Riemann surface in the literature (Witten’s papers) what it (he is actually referring to is a family of super Riemann surfaces. (Deﬁned below)

2. It turns out that a SRS over a point is the same thing as a spin curve and apparently we already know too much about those. Don’t worry, we will discuss this family vs. over a point situation in detail! 3. In the upcoming deﬁnition of a family of SRS pay attention to the fact that we have switched over to the language of algebraic geometry. This means that instead of describing a SRS as a 1|1 cx supermanifold we instead describe it in terms of superschemes, supercurves, etc.

Deﬁnition 6.2. A family of SUSY curves is a smooth, proper morphism π : X → S of superschemes of relative dimension 1|1, with a choice of distribution D ⊆ TX/S of rank 0|1 such that [ , ]: D ⊗ D → TX/S/D is an isomorphism of sheaves of OX -modules. From the deﬁnition of a family of SRS we have the following short exact sequence: ⊗2 0 → D → TX/S → D → 0 (4) Make sure to remember this because we will use it over and over again.

Remark: In the literature you might come across the word supercurve. A supercurve is a smooth superscheme X of ﬁnite-type over k of dimension 1|N. Its genus g is equal to the genus of its reduction. As in the classical case, the phrase a family of supercurves over S is used to denote a smooth, proper morphism π : X → S of superschemes over k of relative dimension 1|N. If we want to move to the language of algebraic geometry, we replace the phrase ” a 1|1 cx. supermanifold” with ”a 1|1 supercurve”.

7You will see in just a minute that locally D is generated by a vector ﬁeld D such that 2 ∂ D = ∂z

14 1. Ok. Let us go back and talk a bit about the distribution D. Can you give an example? Sure, lets just work over a point for now and ignore the compactness requirement so that we can consider the supermanifold C1|1. Let (z, θ) be a choice of global coordinates on C1|1 and deﬁne D to be the subsheaf of T generated by the section D = ∂θ + θ∂z. It turns out 1/2[D,D] = ∂z. In 2 other words, D = ∂z is a square root of the ordinary derivative ∂z!! Moreover, it turns out that given any family of SRS Σ and any point x ∈ Σ there exist a coordinate path U(z, θ) around x such that D|U is generated by the section ∂θ + θ∂z. Click here for a proof of this statement. 2. So why exactly is a SRS over a point not interesting?

Take the SES in4 and tensor each term with OX /J , the result is the

following SES of OXred -modules,

∗ ⊗2 0 → Dred → TXred ⊕ (J ) → Dred → 0.

⊗2 Now Dred is a rank 0|1 OXred -module, Dred is a rank 1|0 OXred -module, ∗ J is of rank 0|1 and TXred is of rank 1|0. Therefore, since we need to preserve grading, we have that,

∼ ∗ ∼ ⊗2 Dred = J , TXred = Dred.

Now, recall, that in the first definition of a SRS we defined it as a cx. 1|1 dimensional supermanifold. Now we saw earlier that any cx analytic 1|1 dimensional supermanifold is split. We also know that OX,1 = J . Also, recall that for any 1|1-dimensional split supermanifold, J /J 2 = V ∗ where ∗ 2 ∗ V is a line bundle on Σred. Since J = 0,we must have that J = V . Now we can use the above two isomorphism to conlude that J =∼ K1/2, i.e J is a square root of the canonical sheaf. A line bundle which is a square root of the canonical sheaf is called a spin structure. Still, why is this not considered interesting? Well, now we actually know everything about 8 the SRS X. Namely it is the total space of a spin structure on Xred. In other words, a SRS over a point is equivalent to a spin curve in the sense that the data defining each of them is the same. What do you mean by the data defining them? Both spin curves and SRS are defined by the data of a Riemann surface and a spin structure on that Riemann surface.

(a) So a SRS over a point is a spin curve. Is the converse of this true? Namely, does every spin curve give rise to a SRS? Yes. Let X be an ordinary Riemann surface and let J (aptly named) be a spin structure on X. Now let Σ = (X, ∧•J ∗), this is obv. a 1|1 cx. supermanifold which we can locally give coordinates (z, θ). Now we somehow need to specify D in a unique way. The fact that

8**super space

15 this is possible comes from what we discussed in question1, namely that locally (i.e on an open subset isomorphic to C1|1) we can always ﬁnd a coordinate chart in which D is generated by ∂θ + θ∂z and that, moreover, any supeconformal structure on C1|1 is isomorphic (SUSY isomorphic) to this one. Therefore, take our Σ we just construction and deﬁne D as to be locally generated by the section ∂θ + θ∂z. Ok. But how do we know for sure that non isomorphic spin structures won’t give rise to isomorphic SRS? Let J and J 0 be spin structures on X. Then Σ = (X, ∧•J ∗) and Σ0 = (X, ∧•(J 0)∗) aren’t even isomorphic as cx. supermanifolds simply because their respective structure sheaves are constructed from non-isomorphic line bundles on X.

3. So why is a family of SRS so much more interesting? Let Σ → S be a family of SRS. It turns out that the family is actually only interesting if S is an aﬃne superscheme whose coordinate ring contains at least on Grassmann variable. If S is the spectrum of an ordinary commutative algebra, then the family Σ → S is again boring. 4. Let S be an ordinary aﬃne scheme, why is a family Σ → S still considered boring? Well, the family is still split, i.e it is equivalent to the data of a family of Riemann surfaces over S and a spin structure on that family. The proof of this statement is basically equivalent to the proof regarding the splitness of a SRS over a point. It turns out that a family Σ → S is possibly non- split only if S is the spectrum of a superalgebra containing at least one Grassmann variable.

6.1 The supermoduli space Mg and its Compactiﬁcation

Throughout this section Mg will denote the supermoduli space of genus g super

Riemann surfaces and Mg,nNS ,nR will denote the supermoduli space of genus g super Riemann surfaces with nNS NS punctures and nR Ramond punctures. The moduli space Mg of ordinary genus g Riemann surfaces is not a manifold but an orbifold. This is because a genus g Riemann surface may have automorphisms. The supermoduli space Mg is an orbifold for the same reason- ing.

1. What is the dimension of Mg? 3g − 3|2g − 2.

2. Is Mg compact? Deﬁnitely not. There are actually two types of degenerations that we need to allow in order to compactify Mg. The ﬁrst type of degeneration you may already be familiar with while the second is a bit more exotic.

16 Type 1: Let Σ be a super Riemann surface. The ﬁrst type of degeneration, called a Neveu-Schwarz degeneration, occurs when a node appears in the underlying Riemann surface Σred. We distinguish such nodes by whether they are separating (b) or non-separating (c).

- (a) How do punctured super Riemann surfaces come about from this type of degeneration? A naked super Riemann surface Σ with a node of type (c) is of course singular. However, if we simply unglue the two branches of Σ which meet at the node, we get a Riemann surface Σ0 of genus g − 1 with 2 NS punctures. Now Σ0 could also have a node which we can futher unglue to get a super Riemann surface Σ1 of genus g − 2 with 4 NS punctures. And so on.. If Σ instead has a node of type (b), then Σ decomposes into two super Riemann surfaces, Σ1, Σ2, of genera g1 and g2 such that g1 + g2 = g, each with 1 NS puncture. Now Σ1 could have a node of type (c) and then further break into a super Riemann surface Σ3 of genus g1 − 1 with 3 NS punctures. And so on...

(b) Can you give an example of how we add a divisors at inﬁnity to Mg corresponding to these nodes?

Sure, for example, to compactify M2, the moduli space of ordinary genus 2 Riemann surfaces, our compactifying divisor would be

(M¯1,1 × M¯1,1)/Z2 ∪ M¯1,2/Z2. Moreover, we have that

∂M¯ 2 = M¯1,1 × M¯1,1)/Z2 ∪ M¯1,2/Z2. ¯ ◦ M2 = M2.

Click here for a nice description of the compactification of Mg,n. (c) Ok. So what is the actual defintion of a super Riemann surface with NS punctures? Good point. Now that we know how/why NS punctures appear. Let’s look at how we actually define them.

Deﬁnition 6.3. A family of super Riemann surface with nNS Neveu- Schwarz punctures is a family of super Riemann surfaces with a choice of n = nNS sections si : S → Σ. Type 2: Let Σ be a super Riemann surface. However, now suppose that there exists a divisor F of Σ along which D fails to be nonintegrable. In 2 other words, along F, D 6= ∂z. We say such a super Riemann surface has a Ramond degeneration.

17 (d) How does a super Riemann surface with Ramond punctures appear from such a degeneration? I am not sure. Click here for some possible answers. (e) Can you at least give a deﬁnition of a super Riemann surface with Ramond punctures? Yes.

Deﬁnition 6.4. A family of super Riemann surfaces with nR Ra- mond punctures is a smooth, proper morphism Σ → S of superschemes of relative dimension 1|1 with the additional structure of rank 0|1 locally free subsheaf D ⊆ TΣ/S, an irreducible relative eﬀec- tive Cartier divisor F of degree n = nR on Σ/S, and an isomorphism ⊗2 [ , ]: D → (TΣ/S/D)(−F) of locally free sheaves on Σ. Here, as before, [ , ] denotes the super Lie bracket. ¯ 3. So what exactly is the boundary of Mg,nNS ,nR ?

¯ ¯ ¯ ¯ ¯ ∂Mg,nNS ,nR = ∪g1+g2=gMg1,1NS ×Mg2,1NS /Z2∪Mg−1,2NS /Z2∪Mg−1,2R /Z2. ¯ 4. What is the dimension of Mg,nNS ,nR ?

3g − 3 + nNS + nR|2g − 2 + nNS + nR/2.

6.2 A look at genus zero 1|1 supercurves Basic Idea: Underlying every genus zero SRS is a genus zero 1|1 supercurve. We can distill a lot of information about genus zero 1|1 supercurves using the theory of ordinary curves of genus zero. In other words, stuff we already know about the projective line can be used to figure out related stuff about genus zero supercurves 9. However, unlike ordinary genus zero curves, it turns out that not all genus zero supercurves are isomorphic. In particular, we have a distinct class of genus zero supercurve for every integer. Moreover, each distinct isomorphism class of genus zero supercurve is represented by a weird space called weighted super projective space.

The reduction of a genus zero supercurve (over a point) is an ordinary Rie- mann surface (or algebraic curve) of genus zero. We know from ordinary geometry, that any genus zero curve is isomorphic to the projective line, P1. We also know that any 1|1 supercurve (over a point) is split 10, therefore if Σ is a genus zero supercurve we must have that there exists m ∈ Z such that Σ = (P1, ∧•O(m)∗).

Important Fact: In ordinary geometry, any genus zero curve is isomorphic to P1. In supergeometry, we just saw that this is not the case. Instead, given

9I am going to start dropping the 1|1 designation and just make it assumed 10For the exact same reasont that any 1|1 supermanifold is split

18 any m ∈ Z there exists a genus zero supercurve Σ such that Σ = (P1, ∧•O(m)∗). Now let M0 denote the supermoduli space of genus zero 1|1 supercurves. We know from the above discussion that topologically M0 is isomorphic to Z and that each closed point x ∈ M0 represents a distinct isomorphism class of genus zero 1|1 supercurves. 11.

We just saw that genus zero 1|1 supercurves are in bijection with super ringed space of the form (P1, ∧•O(m)∗. It makes sense at this point to give spaces of this kind a name. Definition 6.5. If Σ = (P1, ∧•O(m)∗), then we say that Σ is isomorphic to the super ringed space WP1|1(1, 1|m). I have never heard of WP1|1(1, 1|m). How is it defined? . Let V = k2|1 − {(0, 0|0)}/ ∼, where (u, v|θ) ∼ (cu, cv|cmθ) for all (u, v|θ) ∈ k2|1 − {(0, 0|0)} and c ∈ k∗. Then define

1|1 ∗ WPk (1, 1|m) := Proj(Sym(V )). 1. Where does the weight (1, 1|m) come from? We should have actually, to be rigorous, deﬁned WP1|1(1, 1|m) as follows: Let V = k2|1 −{(0, 0|0)}/ ∼, where (u, v|θ) ∼ (cu, cv|cmθ) for all (u, v|θ) ∈ k2|1 − {(0, 0|0)} and c ∈ k∗. Then deﬁne

1|1 ∗ WPk (1, 1|m) := Proj(Sym(V )). It should be obvious from the discription where the weight comes from. 2. The ordinary projective line can be constructed by gluing two copies of A1 together. Is there an analogous gluing construction for (1, 1|m) weighted super projective space? Yes. There is. Let (u, v|θm) be a choice of homogeneous coordinates on X = WP1|1(1, 1|m). The notation θ(m) is used to signify that |θ| = m. ∼ ∼ Consider U = Spec k[z, ζu] and V = Spec k[w, ζv]. Let U ∩ V be the open subset ) of U where z 6= 0 and, similarly, let V ∩ U be the open subset of V where w 6= 0. Give both U ∩ V and V ∩ U their respective induced scheme structures. Then let ϕuv : U ∩ V → V ∩ U be an isomorphism −m of schemes defined by sending z → 1/z and ζu → ζuz . One can now readily check that ϕuv satisfies the properties of gluing. We now say that X is glued from the affine schemes U and V by ϕuv. 3. How do you go from the homogeneous coordinates u, v, θ to the affine coordinates z, ζu? n/2−1 n/2−1 Here z = v/u and ζu = θu . Similarly, w = u/v and ζv = θv . n/2−1 This implies that we have the identifications w = 1/z and ζv = ζuz .

11I guess I will start using the 1|1 designation

19 4. Wait, does this imply that underlying an genus zero SRS is one of these weird weighted superprojective spaces? No, not at all. If our genus zero SRS has no Ramond punctures, then it turns out that its underlying supercurve must be isomorphic to P1|1. If it does have Ramond punctures, say n of them, then its underlying 1|1 supercurve must be isomorphic to WP1|1(1, 1|1 − n/2).