Equivalences between GIT quotients of LandauGinzburg Bmodels
Abstract
We define the category of Bbranes in a (not necessarily affine) LandauGinzburg Bmodel, incorporating the notion of Rcharge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of LandauGinzburg Bmodels that arise as different GIT quotients of a vector space by a onedimensional torus, and show that for each such pair the two categories of Bbranes are quasiequivalent. In fact we produce a whole set of quasiequivalences indexed by the integers, and show that the resulting autoequivalences are all spherical twists.
Contents
1 Introduction
The starting point for this paper is the celebrated result of Orlov [13] that the derived category of a CalabiYau hypersurface in projective space is equivalent to the triangulated category of graded matrix factorizations for the homogeneous polynomial defining . In physicists’ language this is the statement that the categories of topological Bbranes are the same in the sigma model with target and in the LandauGinzburg model with superpotential . This is just part of a much deeper conjecture which goes back to Witten [17] (based on earlier observations by Vafa and others) and which states that the conformal field theories associated to these two models are different limit points in the same moduli space, the socalled Stringy Kähler Moduli Space (SKMS). The Btwist of a conformal field theory is expected to be independent of your position in the SKMS, so it follows that the Btwisted theories of each model should be equivalent and in particular that their Bbrane categories are the same.
The basics of Witten’s idea are easy to understand, even for a mathematician. A LandauGinzburg model is a Kähler manifold with a holomorphic function called the superpotential. From such a thing one can write down a standard supersymmetric Lagrangian, analogous to the Lagrangian in classical mechanics coming from a Riemannian manifold equipped with a realvalued potential function. We consider the LG model
where is a homogeneous degree polynomial in the ’s. Now we ‘gauge’ this theory, which means we try and divide by the symmetry under which each has weight 1 and has weight . The resulting theory should be the same, at least in some limit, as the theory coming from the quotient LG model. However we should take the quotient carefully, which means take a GIT quotient, and here we have two choices. One is the total space of the canonical bundle on
and the other is the affine orbifold
The function descends to either of these, so they are both LG models. Physically, there is a parameter (the complexified FayetIliopoulos parameter) in the Lagrangian of the gauged theory, and the theories on and are expected to appear at two different limits of this parameter.
The important thing about a LG model is the set of critical points for the function . On this will be the hypersurface . If we assume that this is smooth, then locally around the function is just quadratic in the normal directions, physically these directions are then ‘massive modes’ and can be ‘integrated out’ in the lowenergy theory. What this means is that we should expect the theory coming from to look essentially like a theory based just on , with no superpotential. We conclude that the theory on is connected, in the moduli space of theories, to the theory on the orbifold with superpotential . This is called the CalabiYau/LandauGinzburg correspondence.
On the level of conformal field theories, this story is beyond the reach of current mathematical technology. However, all the spaces here are CalabiYau, which means the theories admit ‘Btwists’ which are topological field theories (more precisely they are Topological Conformal Field Theories/Cohomological Field Theories) and these are much more tractable. And as mentioned above, the Btwisted theories should be independent of the FI parameter, and so the result to prove is that the Bmodel (the Btwisted TCFT) arising from is the same as the Bmodel arising from the LandauGinzburg model .
The open sector of a Bmodel is the category of Bbranes, these are a type of boundary condition for the CFT. This should be a CalabiYau dgcategory. For a LG model with , the category of Bbranes is (a dgenhancement of) the bounded derived category of coherent sheaves. When we need a generalization of this, the idea (due to Kontsevich) is to use objects that are like chaincomplexes of sheaves, except that we now have instead of . We define this category, denoted for a LG model , in Section 2. In particular, the homotopy category of is the category of matrix factorizations of on , and the homotopy category of is .
Orlov’s result is thus the statement that , i.e. the homotopy categories of the open sectors of the two Bmodels are the same. In fact this goes a long way to proving that the whole of the Bmodels are the same, to get the full statement one would have to show that the open sectors are equivalent as CalabiYau dgcategories, the closed sectors should then follow by Costello’s theorem [3]. However this is not the aim of the current paper. Rather, we wanted to try to reprove Orlov’s result following more closely the ideas in Witten’s construction, which does not appear in Orlov’s proof. In particular we wanted to see the equivalence as the composition of two equivalences:
This both clarifies the result and suggests how to generalise it. For the first equivalence, we assume that the fundamental relationship between and is that they are birational, being related by a change of GIT quotient is just a special case of this. Hence we conjecture (Conjecture 2.15) that the Bmodels associated to any two birational CalabiYau LG models are equivalent (to be more precise, we just conjecture that their open sectors are equivalent as dgcategories). This conjecture will be obvious to experts, it is a generalization of a theorem of Bridgeland [2] and lies in the same circle of ideas as Ruan’s Crepant Resolution conjecture [14].
Unfortunately we do not get as far as addressing the second equivalence in this paper. We’ll just remark that although it looks mysterious, it is just a global version of a fairly classical result by Knörrer [9], and a closely related result is proved by Orlov [12].
What we actually manage to prove in this paper is a slight generalisation of the first equivalence, and so a small step towards our general conjecture. We show (Theorem 3.3) that if and are two different GIT quotients of a vector space by , and is an invariant polynomial on , then
are equivalent dgcategories. Our proof borrows heavily from the work of Hori, Herbst and Page [5], in which they give a detailed physical argument for a generalisation of Orlov’s result. Their key idea is a grade restriction rule. Their reasoning involves Abranes and is mathematically rather mysterious, however the rule itself will be instantly familiar to anyone who knows Beilinson’s Theorem [1]. Our improvement on their result, other than making it mathematically rigourous, is that they work only with the objects of the Bbrane category whereas we include the morphisms as well (the massless open strings).
We want to explain one last aspect of the physics picture. The SKMS (the space in which the FI parameters live) can be explicitly described for our examples: it is a cylinder with one puncture, and the two GIT quotient LG models live at either end of the cylinder. The category of Bbranes is the same for all points in this space, however we cannot trivialise it globally, i.e. there is monodromy. Therefore to get an equivalence between the Bbrane categories of and we must pick a path between the two ends of the cylinder. Up to homotopy there are such paths, so we should find such equivalences. This is what we, and Orlov, find. By composing equivalences we get autoequivalences of the Bbrane category at either end, this is the monodromy around the puncture. The puncture is the limit where the mass of a particular brane goes to zero, and the monodromy should be a SeidelThomas spherical twist around this brane. There is also monodromy around each end of the cylinder, this should just be given by tensoring with the linebundle.
Now we can explain the layout of the paper.
In Section 1.1 we give a sketch of our method for the special case that . This means we don’t have to worry about LG models, we just deal with the more familiar case of derived categories of coherent sheaves. We show that we can construct many derived equivalences between and , and that the resulting autoequivalences are spherical twists.
In Section 2 we explain properly what a LG Bmodel is, and what the category of Bbranes is.
In Section 3 we describe the class of examples we will consider. These are pairs of LG Bmodels and that are different GIT quotients of a vector space by .
In Section 3.1 we prove our main result, that there are many quasiequivalences between the categories of Bbranes on and .
In Section 3.2 we describe the resulting autoquasiequivalences of the category of Bbranes on . We show (moreorless) that they are spherical twists.
The technology of LG Bmodels is in its infancy, so many of the arguments of the last two sections are rather messy and adhoc. In particular the ‘moreorless’ of the previous paragraph is because we do not have a proper theory of FourierMukai transforms. We apologise to the reader for this unsatisfactory stateofaffairs, and hope that later treatments will clean these results up a bit.
Acknowledgements. I’d like to thank Richard Thomas for helpful suggestions, Manfred Herbst for patiently explaining [5] to me, and the geometry department at Imperial College for sitting through some lectures on this material when it was in preliminary form.
Some results closely related to those of this paper (although using the ‘derived category of singularities’ description of the category of Bbranes) have been been found independently by [7] and [8, Section 7].
1.1 A Sketch Proof for
As we will see in Section 2, a special case of the category of Bbranes in a LandauGinzburg Bmodel is the category of perfect complexes on a smooth space , which is a dgmodel for the derived category . We thought it would be helpful to explain the proof of our results in this special case, as is probably more familiar than . Also the proof in this case is quite simple and still contains the important points for the more general case, the hard work in generalizing is mostly technicalities.
For this sketch, we’ll use the example of the standard threefold flop. This is of course well understood and we will say nothing particularly original, but we will indicate afterwards how to generalise.
Let with coordinates , and let act on with weight 1 on each and weight on each . There are two possible GIT quotients and , depending on whether we choose a positive or negative character of . Both are isomorphic to the total space of the bundle over .
Both are open substacks of the Artin quotient stack
given by the semistable locus for either character. Let
denote the inclusions. This stacky point of view makes it clear that there are (exact) restriction functors
By we mean the derived category of the category of equivariant sheaves on . This contains the obvious equivariant linebundles associated to the characters of .
The unstable locus for the negative character is the set . Consider the Koszul resolution of the associated skyscraper sheaf:
Then is exact, it is the pullup of the Euler sequence from . On the other hand is a resolution of the sky scraper sheaf along the zero section. Similar comments apply for the Koszul resolution of the set .
Let
be the triangulated subcategory generated by the line bundles and . This is the grade restriction rule of [5], we are restricting to characters lying in the ‘window’ .
Claim 1.1.
For any , both and restrict to give equivalences
To see that these functors are fullyfaithful it suffices to check what they do to the maps between the generating linebundles, so we just need to check that
for , i.e.
for , and this is easily verified. To see that they are essentially surjective we need to know that the the two given line bundles generate . This is essentially a corollary of Beilinson’s Theorem [1]. One way to see it is to first observe that the set generates because is quasiprojective, then use twists of the exact sequence repeatedly to resolve any by a complex involving only and .
So for any we have a derived equivalence
passing through . Composing these, we get autoequivalences
To see what these do, we only need to check them on the generating set of linebundles . Applying to this set is easy, it just sends them to the same linebundles on .^{1}^{1}1The easiest sign convention is to keep and as degree 1 on both sides, i.e. it’s the bundle on that has global sections. Otherwise sends to . To apply however, we first have to resolve in terms of and . We do this using the exact sequence . The result is that sends
Claim 1.2.
is an inverse spherical twist around .
A spherical twist is an autoequivalence discovered by [15] associated to any spherical object in the derived category, i.e. an object such that
for some (i.e. the homology of the sphere). It sends any object to the cone on the evaluation map
The inverse twist sends to the cone on the dual evaluation map
The object is spherical, and the inverse twist around it sends to itself and to the cone
which agrees with . To complete the proof of the claim we would just need to check that the two functors also agree on the Homsets between and .
Now instead let with coordinates . Let act linearly on with positive weights on each and negative weights on each . The two GIT quotients and are both the total spaces of orbivector bundles over weighted projective spaces.
We must assume the CalabiYau condition that acts through . Let be the sum of the positive weights, so the sum of the negative weights is . The above argument goes through wordforword, where now
2 LandauGinzburg Bmodels
A LandauGinzburg model is a Kähler manifold equipped with a holomorphic function . We are only interested in the Bmodel on , and this doesn’t need the metric, just the complex structure. Also we want to work in the algebraic world, so for us will be a smooth scheme (or stack) over .
When , it is a standard slogan that the category of Bbranes is the derived category of coherent sheaves on . However the category of Bbranes should really be a dgcategory, whose homotopy category is (for background on dgcategories, we recommend [16]). A good model is given by , the category of perfect complexes. The objects of are bounded complexes of finiterank vector bundles, and the morphisms are given by
(this is what we might call the ‘Dolbeaut’ version of , other versions are possible as we will discuss below). The differential here is a sum of the Dolbeaut differential and the differential on , which itself is the commutator with the differentials on and . The homology of this complex is
Futhermore since is smooth every object in is quasiisomorphic to a complex in , so as required.
We need to generalise this for . Kontsevich’s idea was to modify the definition of a chaincomplex, replacing with . This doesn’t make sense on a graded complex, so the usual procedure (at least in the mathematics literature) is to work instead with graded complexes. However there is another possibility, standard in the physics literature, which is to replace the ‘homological’ grading with the notion of Rcharge (strictly speaking, vector Rcharge). This is a geometric action of on , under which must have weight 2. Then we can define a Bbrane to be a equivariant vector bundle , with an endormorphism of Rcharge 1, and the condition makes sense. If the action is trivial then we are forced to take , and we recover the definition of a perfect complex. Also, the definition of the morphism chaincomplexes in adapts easily, as we shall see.
Definition 2.1.
A LandauGinzburg Bmodel is the following data:

A smooth dimensional scheme (or stack) over .

A choice of function (the ‘superpotential’).

An action of on (the ‘vector Rcharge’).
such that

has weight (‘Rcharge’) equal to 2.

acts trivially.
From now on we’ll call the acting in this definition to distinguish it from other actions that will appear later.
Remark 2.2.
In physics terms, Axiom 2 follows from the fact that the axial Rcharge symmetry is acting trivially. It implies that the sheaf of functions is supercommutative under the grading. We could relax it, but keep supercommutativity, by allowing to be a superspace.
Definition 2.3.
A Bbrane on a LandauGinzburg Bmodel is a finiterank vector bundle , equivariant with respect to , equipped with an endomorphism of Rcharge 1 such that .
If we wanted to be more pretentious we could say that is a space endowed with a sheaf of curved algebras ( is the curvature) and that a Bbrane is a locally free sheaf of curved dgmodules over .
We can shift the Rcharge on a Bbrane by tensoring with a line bundle associated to a character of . We denote these shifts by for . This agrees with the homological shift functor in the following special case:
Example 2.4.
Let and act trivally. Then a Bbrane is just a bounded complex of vector bundles.
Note that since acts trivially every Bbrane splits as a direct sum
of its eigenbundles, and exchanges these subbundles. There is a weaker definition of LandauGinzburg Bmodel where we keep only the trivial action of , thus only this grading remains. We shall make no use of this weaker definition, except for the following example.
Example 2.5.
Let and be any polynomial. This defines a LG Bmodel in the weak (graded) sense. Then for a Bbrane both and must be trivial bundles, so is given by a matrix
whose square is . This is a called a matrix factorization of .
We can’t in general add Rcharge to this example. But we can if we orbifold it, as follows.
Example 2.6.
Let , where (the gauge group) acts with weight 1 on each and weight on . This is equivalent as a stack to . Let act with weight 0 on each and weight 2 on . If we pick a superpotential where is a homogeneous degree polynomial in the ’s, then this defines a LG Bmodel (for it is the orbifold phase of the Witten construction described in the introduction). Every equivariant vector bundle on is the direct sum of equivariant linebundles, these are given by the lattice
This bijects with the subset . This means that we can consider a Bbrane on to be given by a pair of graded free modules over the ring where each has degree 1, and graded maps
with . This is called a graded matrix factorization.
Now we want to define the morphisms between two Bbranes. We will precisely mimic the construction of Perf, by first defining a homomorphism bundle and then taking derived global sections of it.
Recall that a Bbrane on the LG Bmodel is a equivariant bundle on equipped with an endomorphism of Rcharge 1 whose square is zero. Let be the category whose objects are Bbranes on and whose morphisms are all morphisms of vector bundles. This is a dgcategory, and when the action on is trivial it is just the category of complexes of vector bundles on . It is also a monoidal category, since we can tensor equivariant bundle and their endomorphisms in the usual way.
Now let be any LG Bmodel, and let , be two Bbranes on . We have a equivariant vector bundle
and this carries an endomorphism
of Rcharge 1. One can check that
(the two copies of that appear cancel each other). This means that the pair is an object of . Furthermore, given a third Bbrane , we have composition maps
and these are closed and of degree zero.
Definition 2.7.
Given an LGmodel we define a category enriched over the category . The objects of are the Bbranes on , and the morphisms between two branes and are given by
We need to fix a monoidal functor that sends a equivariant vector bundle to a bounded equivariant chaincomplex of vector spaces that computes its derived global sections. Since we are working with smooth spaces over we will use Dolbeaut resolutions, i.e. we define
but we could also use other models such as Čech resolutions with respect to some invariant open cover.
Now is an object in . This means that
is a bicomplex, graded by Rcharge and by Dolbeaut degree, with differential
As usual we may collapse this bicomplex to a complex. If we apply this to all pairs of branes simultaneously we get the following:
Definition 2.8.
Given an LGmodel we define the dgcategory of Bbranes to be
The monoidalness of ensures that this is indeed a category.
Example 2.9.
Let and act trivially on . Then , the category of perfect complexes. Since is smooth the homotopy category of this is
Example 2.10.
Remark 2.11.
should only depend on a (Zariski) neighbourhood of the critical locus of . This has been proved (without Rcharge and on the level of homotopy categories) by Orlov [11].
Remark 2.12.
As far as we are aware this definition is new in the mathematics literature, but it is almost classical in the physics literature, see e.g. [6].
Remark 2.13.
We could make the definition of a Bbrane more general by allowing the endomorphism to be derived, i.e.
with Rcharge plus Dolbeaut degree equal to 2. Similarly we could generalize the definition of LG Bmodel by allowing to be a closed element of . The advantage of this more general definition of Bbrane is that the resulting category contains mapping cones, i.e. it is pretriangulated. However notice that in Example 2.9 above is already pretriangulated, this leads us to suspect that at least when our more restricted category of Bbranes is in fact pretriangulated as well. When is affine this is obvious.
Remark 2.14.
Since the Hom sets are actually bicomplexes, and the Dolbeaut grading is bounded, we have a spectral sequence converging to the homology of whose first page is
A map of LG Bmodels is just a map from to commuting with the Rcharges and such that . Assuming that the derived global sections functors and are chosen compatibly we get a dgfunctor
Similarly a birational map between and is a birational map from to that commutes with Rcharge and sends to .
Conjecture 2.15.
Let and be birational LG Bmodels, and assume that and are CalabiYau. Then there is a quasiequivalence
In the next section we prove a special case of this conjecture.
As was explained in the introduction, this is a conservative version of the real conjecture, which is that the Bmodels associated to and are equivalent. We state this version since it is not yet proved that the Bmodel exists.
3 Quotients of a vector space by
Take a vector space , and equip it with a linear action of , which we’ll denote by (the ‘gauge group’). We require that acts through . We have a stack quotient
There are also two possible GIT quotients of by associated to the characters of . From the stacky point of view these are open substacks
consisting of the semistable loci given by either character. All of these spaces are CalabiYau.
Now choose an action of on that commutes with the gaugegroup action. Note that both GIT quotients are then preserved by . Let be a function on that is invariant with respect to and has Rcharge 2. Then we have three LandauGinzburg Bmodels
(3.1) 
From now on we’ll abuse notation and call both and just .
Both GIT quotients are the total space of orbivector bundles over weighted projective space. To see this, let
be the decomposition of into eigenspaces with positive, negative and zero weights. Then projects down to , and it is the total space of the vector bundle associated to the graded vector space . Similarly is the total space of over .
For our sign conventions, it is simplest if we agree that is Proj of a negatively graded ring, so that the line bundle on is the one that has global sections. If we don’t adopt this then whenever we restrict to we have to flip the signs of all linebundles.
Let be the sum of the positive eigenvalues of on , since acts through the sum of the negative eigenvalues is .
We’ll make repeated use of the following fairly classical fact:
Lemma 3.1.
Corollary 3.2.
for all , and
for and .
Proof.
By adjunction and affineness of the projection , we have
∎
3.1 Quasiequivalences
In this section we will prove
Theorem 3.3.
There is a natural set of quasiequivalences
parametrised by .
The key idea of the proof of this Theorem comes from [5]. Using restriction functors shown in 3.1, we will identify both and with one of a set of full subcategories parameterized by .
Note that every vector bundle on is a direct sum of the obvious line bundles . Let
be the full subcategory consisting of Bbranes where all the summands of come from the set
We will show that the functors
become quasiequivalences when restricted to any of the subcategories , thus proving Theorem 3.3.
Recall that a dgfunctor between dgcategories is a quasiequivalence if the induced map on homotopy categories is an equivalence. This means that it must be a quasiisomorphism on Hom sets (quasifullyfaithful) and surjective on homotopyequivalence classes of objects (quasiessentiallysurjective).
Lemma 3.4.
For any , both functors
are quasifullyfaithful.
Proof.
Obviously we need only show the proof for . Let and be any two Bbranes in . We get corresponding Bbranes and on . Then
and
We wish to show that the map is a quasiisomorphism between these two complexes. Recall (Remark 2.14) that the homology of both complexes can be computed by spectral sequences whose first pages are
On , taking global sections just means taking invariants, which is exact, so for any linebundle ,
and by Corollary 3.2 this is also true on when . Since is a direct sum of linebundles from the set
the induced map
is an isomorphism between the first pages of the two spectral sequences. Hence is a quasiisomorphism. ∎
We will deduce quasiessentialsurjectivity from the following lemma, which is essentially Beilinson’s Theorem [1].
Lemma 3.5.
For any , any equivariant vector bundle on has a finite equivariant resolution by direct sums of shifts of linebundles from the set
Proof.
Recall that all vector bundles on are direct sums of the character line bundles. Since is quasiprojective, is a quotient of for some vector bundle on , and we can choose this quotient to be equivariant. Then we have a map which is surjective on . Since is smooth, the kernel of this map has a finite resolution by vector bundles, which we again may choose to be equivariant. The restriction of this resolution to , together with , give a finite equivariant resolution of by direct sums of character linebundles. Thus it is sufficient to prove the lemma for the linebundles .
On we have the Euler exact sequence
which resolves in terms of , and the action on means that it is equivariant. Pull this up to . By repeatedly using twists of this exact sequence we see that any linebundle has a equivariant resolution by shifts of line bundles from the set .
∎
Lemma 3.6.
For any , both functors
are quasiessentiallysurjective.
Proof.
Again we only show the proof for . Let be a Bbrane on . By Lemma 3.5 we can equivariantly resolve by a complex
where every term is a direct sum of shifts of line bundles with . If we let
then is an endomorphism of with Rcharge 1. We’re going to show that we can perturb to an endomorphism whose square is , and that the resulting Bbrane is homotopic to . To see that this proves the lemma, let be the vector bundle on given by the same direct sum of linebundles as . Then
(see Corollary 3.2), so is the restriction of an endomorphism of , so we have a Bbrane that restricts to give . So every Bbrane is homotopic to a Bbrane lying in , which is the statement of the lemma.
As well as the Rcharge, we will need to keep track of the grading on that comes from it being a complex, let’s call this the homological grading. Of course also has homological grade 1.
Now consider the complex and the bundle as objects in the usual derived category of sheaves on , which are quasiisomorphic under the map . The line bundles making up have no higher Ext groups between them (Cor. 3.2 again), so we have quasiisomorphisms
(3.2) 
Here we are using the homological grading on the LHS and the Dolbeaut grading on the RHS, but the quasiisomorphims are also equivariant with respect to Rcharge. This means we can find an element which is closed with respect to , has Rcharge 1, and maps to the endomorphism of , i.e.
We can use to perturb the endomorphism of . Unfortunately this does not yet make it a Bbrane for , rather we have
for some element which has homological grade 1 and Rcharge 1. Here we write to denote the supercommutator with respect to the Rcharge grading, strictly speaking this is the differential on that comes from considering as a Bbrane on rather than as a complex of sheaves in , but the difference is irrelevant and the signs are more convenient this way.
If we perturb further by we get
and notice that now all the unwanted terms have homological degree at most 1. We claim we can iterate this process, and since the homological degree is bounded it will terminate. Indeed, we wish to solve
where
is a series of terms of decreasing homological grade and Rcharge 1. The piece of this equation in homological grade is
Assume that we have found such that this equation holds in homological grades . By (3.2), has no homology in negative degrees, so we can find if is closed. But
so inducting on our solution exists. We let
so is a Bbrane on . It remains to show that it is homotopic to the brane . To see this we consider the dga
This carries its usual grading (the sum of Rcharge and Dolbeaut grade) and also the homological grading from . Its differential is a sum of terms induced from and the , these have homological grading and respectively. Thus we can filter this dga by defining
to be the sum of the bigraded pieces that have
then this filtration is compatible with the differential and the algbra structure. Also the filtration is bounded, in the sense that the induced filtration on any (usual) graded subspace is bounded. This is a sufficient condition for the associated spectral sequence of dgas to converge [10]. To get page 1 of this spectral sequence we take the homology of the term of the differential which has bidegree , this is the term induced from . The diffential on page 1 is induced from and , and was chosen so that it induced on homology. So page 1 is
This is concentrated in homological grade zero, so the spectral sequence collapses at page 2. We deduce that in the homotopy category the objects and are isomorphic. ∎
3.2 Spherical Bbranes
We use the same setup as in the previous subsection, but from now on we assume that has no zero eigenvalues in , so
are the positive and negative eigenspaces.
The zero section gives an inclusion
and there is an associated skyscraper sheaf . This is a spherical object in the derived category . We are going to modify it so as to produce a spherical object in the category of Bbranes .
Under our definition a Bbrane is a vector bundle, so it is supported over the whole of (it is ‘spacefilling’). However a better definition should allow arbitrary coherent sheaves, which in particular can be supported just on subschemes. Then no modification of would be necessary, we could just equip it with the zero endomorphism, which does indeed square to because along the zero section.
We have not attempted to develop such a definition because the presence of local Ext groups makes defining the morphisms between such objects significantly more difficult. Instead we shall resolve by vector bundles, and deform the resolution. Nevertheless the resulting object does behave as if it was supported just on the zero section (Prop. 3.8).
Let be dual bases of and , and the corresponding coordinates. Consider the Koszul resolution of :
We will deform the differential to make it a Bbrane on , and show that it is still spherical.
Write as
(3.3) 
This is possible since is gauge invariant, and has Rcharge 2 so has no constant term. We define a Bbrane on by the equivariant vector bundle
and the endomorphism
It is easy to check that .
Proposition 3.7.
The Bbrane is independent, up to isomorphism, of the choice of splitting (3.3)
Proof.
Let be another choice of splitting, and the corresponding endmorphism of . It is sufficient to prove the lemma in the case that for . In that case we have