Algebraic Geometry 2 (3) (2015) 332–364
doi:10.14231/AG-2015-015

The Pfaffian-Grassmannian equivalence revisited
Nicolas Addington, Will Donovan and Ed Segal
Abstract
We give a new proof of the ‘Pfaffian-Grassmannian’ derived equivalence between certain
pairs of non-birational Calabi–Yau threefolds. Our proof follows the physical constructions of Hori and Tong, and we factor the equivalence into three steps by passing
through some intermediate categories of (global) matrix factorizations. The first step isglobal Knörrer periodicity, the second comes from a birational map between Landau–
Ginzburg B-models, and for the third we develop some new techniques.
1. Introduction
The ‘Pfaffian-Grassmannian equivalence’ refers to a relationship between two particular Calabi–
Yau threefolds: Y1 , which is a linear section of the Grassmannian Gr(2, 7), and Y2 , which isthe dual linear section of the Pfaffian locus in P(∧2 C7 ). The relationship was first conjectured
by Rødland [Rød00], who by studying their Picard–Fuchs equations observed that Y1 and Y2
appeared to have the same mirror. This means that the usual conformal field theories with thesetarget spaces should occur as different limit points in the Kähler moduli space of a single field
theory. By itself this is a fairly common phenomenon; the special feature of this example is thatY1 and Y2 are (provably) not birational to one another. This was the first example with thisproperty, and such examples remain extremely rare.
If we pass to the B-twist of this theory, this picture implies that the B-models defined on Y1
and Y2 are isomorphic, and in particular that their categories of B-branes are equivalent. The
category of B-branes on a variety is the derived category of coherent sheaves, so this suggeststhat we should have a derived equivalence
Db (Y1 ) ∼= Db (Y2 ) .

(1.1)

This is a precise mathematical prediction, and it was proven by Borisov and Căldăraru [BC09],
and independently by Kuznetsov [Kuz06] using his broader program of homological projective
duality.
Around the same time as these proofs of (1.1) appeared, Hori and Tong [HT07] wrote an
Received 17 April 2014, accepted in final form 22 November 2014.
2010 Mathematics Subject Classification 14F05, 14J32, 18E30, 81T30 (primary), 14M15 (secondary).
Keywords: Calabi–Yau threefolds, derived category, derived equivalence, matrix factorizations, Landau–Ginzburg
B-model, Pfaffian, Grassmannian, homological projective duality.
This journal is c Foundation Compositio Mathematica 2015. This article is distributed with Open Access under
the terms of the Creative Commons Attribution Non-Commercial License, which permits non-commercial reuse,
distribution, and reproduction in any medium, provided that the original work is properly cited. For commercial
re-use, please contact the Foundation Compositio Mathematica.
N.A. and W.D. are grateful for the hospitality of the Hausdorff Research Institute for Mathematics, Bonn.
W.D. is grateful for the support of Iain Gordon and EPSRC grant no. EP/G007632/1, and of the World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan. W.D. also thanks Paul Aspinwall for
supporting a visit to Duke University under NSF grant no. DMS-0905923. E.S. is supported by an Imperial College
Junior Research Fellowship.

The Pfaffian-Grassmannian equivalence revisitedimportant physics paper that gave an argument for Rødland’s full conjecture, by constructing thenecessary field theory containing Y1 and Y2 in its Kähler moduli space. The theory is a gauged
linear sigma model (GLSM), which is a standard idea, but the gauge group is non-abelian, andfurthermore the argument that Y2 occurs as a limit relies on some very original analysis of
non-perturbative effects.
In this paper we give a new mathematical proof of the derived equivalence (1.1), inspiredby the ideas of Hori and Tong. In particular we find that this derived equivalence is at heart abirational phenomenon, but the birationality is between two Landau–Ginzburg models(X1 , W )(X2 , W ) .

Here X1 and X2 are larger spaces containing Y1 and Y2 , and W is a holomorphic function definedon both of them. The space X1 is a variety and Y1 is the critical locus of W in X1 , so this we
can analyze by standard techniques. However, on the other side we encounter two rather novel
phenomena:– The space X2 is not a variety; it is an Artin stack. It seems that the category of B-branes
on an Artin stack is not the same as the derived category; indeed, the correct definition ofthis category is not known in general. (See Section 4.1 for more discussion of this point.)– The subspace Y2 ⊂ X2 is not the critical locus of W .
To handle these phenomena, we develop new mathematical ideas, which very roughly parallel
the new physics in [HT07].
The importance of abelian GLSMs is now fairly widely understood in the mathematics literature, since they are closely connected to toric varieties and complete intersections therein.
However, we are only just beginning to understand the world of non-abelian GLSMs. We hopethat the perspective and techniques of this paper will encourage others to explore it further.
For the remainder of this introduction we explain the constructions that we are going to use,and give an outline of the ideas involved in the proof.
1.1 Calabi–Yau threefolds
Let V be a seven-dimensional complex vector space, and fix a surjective linear map
A : ∧2 V → V .
From these data we will build two different Calabi–Yau threefolds:
Y1 : We consider the GrassmannianGr(2, V ) ⊂ P(∧2 V )in its Plücker embedding. Intersecting it with the seven hyperplanes given by the kernel ofA, we obtain the first Calabi–Yau threefold Y1 .
Y2 : We consider the projective space P(∧2 V ∨ ) of two-forms on V . Thinking of a two-form as
an antisymmetric matrix we see that its rank must always be even, so generically the rank
is six. The Pfaffian locus
Pf(V ) ⊂ P(∧2 V ∨ )is where the rank drops to four or less. Intersecting this with the linear P6 given by theimage of A∨ , we obtain the second Calabi–Yau threefold Y2 .

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Assumption 1.1. We choose A generically enough that the codimension seven space P(ker A) ⊂P(∧2 V ) is transverse to Gr(2, V ), so Y1 is smooth. The smoothness of Y2 is slightly more delicate, since Pf(V ) has singularities along the locus where the rank drops to two, that is, alongGr(2, V ∨ ) ⊂ Pf(V ). But if P(ker A) is transverse to Gr(2, V ) then P6 = P(Im A∨ ) avoids thissingular locus and is transverse to the smooth locus of Pf(V ), so Y2 is smooth. This follows fromthe fact that Pf(V ) is the (classical) projective dual of Gr(2, V ); details are given in [BC09, §§ 1–2
and especially Corollary 2.3].1
1.2 Relation with the Hori–Tong GLSM
Now we can explain our interpretation of Hori and Tong’s construction. Let S be a two-dimensional complex vector space, and consider the linear Artin stack
X = Hom(S, V ) ⊕ Hom(V, ∧2 S) GL(S) .
Notice that GL(S) acts trivially on the determinant of the vector space underlying X, so X is aCalabi–Yau stack.
For Hori and Tong, these data specify a GLSM, which is a certain kind of two-dimensional
supersymmetric gauge theory. It is conformal because of the Calabi–Yau condition. The Lagrangian for this field theory contains a certain parameter τ (the complexified Fayet–Iliopoulos(FI) parameter) which is essentially the Kähler modulus. The two limits |τ |  1 and |τ |  1
roughly correspond to the two possible geometric invariant theory (GIT) quotients of X.
(i) In the first limit |τ |  1, we choose a stability condition consisting of a positive characterof GL(S). The unstable locus is where x does not have full rank, and the GIT quotient isthe variety
X1 = Gr(2, V ) ×GL(S) Hom(V, ∧2 S) .
This is the total space of the vector bundle O(−1)⊕7 over Gr(2, V ), where we introduce theconvention that O(−1) := det(S) = ∧2 S. In this limit, the GLSM is expected to reduce toa sigma model with target X1 .
(ii) Now we look at the other stability condition |τ |  1, where we choose a negative characterof GL(S). At this point we have to be careful about our definition of the GIT quotient.
Conventionally, one deletes the unstable locus, then takes the scheme-theoretic quotientof the remaining semi-stable locus. For our purposes this is too destructive, and we will
instead take the stack-theoretic quotient of the semi-stable locus. (This is now quite astandard thing to do, particularly if the resulting quotient stack is only an orbifold.) For
this stability condition the unstable locus is p = 0, so we consider the complement
X2 := {p 6= 0} ⊂ X .
This space X2 is an Artin stack; we can think of it as a bundle overP Hom(V, ∧2 S) ∼= P6
whose fibres are the stacks


Hom(S, V )
SL(S) .

The classical GIT quotient is the scheme underlying X2 : this is singular, and we will makeno use of it. It appears that the Artin stack X2 is the correct space to consider in the1

We are referring to Borisov–Căldăraru only for this geometric fact, which does not depend on their proof that
Db (Y1 ) ∼= Db (Y2 ). Our proof of the latter is independent of theirs.

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The Pfaffian-Grassmannian equivalence revisited|τ |  1 limit. In physics terminology, the gauge group has been broken only to a continuous
subgroup. Notice that since the stack X is Calabi–Yau, so too are the open substacks X1
and X2 .
The GLSM has another ingredient, known as the ‘superpotential’. This is the (invariant)function W on X defined by
W (x, p) = p ◦ A ◦ ∧2 x .

(1.2)

Here x ∈ Hom(S, V ) and p ∈ Hom(V, ∧2 S), and A is our fixed linear map from above. We canrestrict W to either X1 or X2 : the three pairs (X, W ), (X1 , W ) and (X2 , W ) then all define
Landau–Ginzburg B-models (see Section 2).
The most important thing about a Landau–Ginzburg model is the critical locus of the superpotential W . We now indicate how an analysis of this locus for the Landau–Ginzburg B-models(X1 , W ) and (X2 , W ) will allow us to recover the Calabi–Yau threefolds Y1 and Y2 from theSection 1.1.
(i) In the case of the pair (X1 , W ), we claim that the critical locus of W is exactly our Grassmannian Calabi–Yau Y1 . To see this, pick a basis for V , so A defines seven sections a1 , . . . , a7of O(1) on Gr(2, V ), which we can pull up to X1 . On X1 we also have seven tautologicalsections p1 , . . . , p7 of the pullback of O(−1), and the superpotential is
W =

7
X

ai pi .

i=1

Because A is generic, the critical locus of this function is the set {ai = pi = 0, ∀i}, which
by definition is Y1 ⊂ Gr(2, V ).
(ii) Now consider the pair (X2 , W ). If we fix a point [p] ∈ P Hom(V, ∧2 S), then W restricts
to give a quadratic form Wp on the fibre Hom(S, V ). The rank of this quadratic form is
twice that of the (antisymmetric) form p ◦ A. So the Pfaffian Calabi–Yau Y2 is the locus ofpoints p where the quadratic superpotential Wp on the fibre drops in rank. As we shall see
in Section 5, this is contained in (but not equal to) the critical locus of W .
1.3 Outline of the proof
Associated to any Landau–Ginzburg B-model (Y, W ) there is a category, which we denote by
Db (Y, W ), whose objects are ‘twisted complexes’ or ‘global matrix factorizations’. In the special
case when W ≡ 0, this category is the usual derived category of coherent sheaves Db (Y ). We
will prove the derived equivalence (1.1) as a composition of three equivalences, as follows:
Db (X1 , W )BBr(X2 , W ) ⊂ Db (X2 , W )
∼

Ψ3

∼

Ψ1

Ψ2
∼

Db (Y1 )Db (Y2 )(1.3)

Let us say a few words about each step.
Ψ1 : This step is well known to experts; it is a generalization of Knörrer periodicity which has
been proved several times over in recent years. We explain this step in Section 3.

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N. Addington, W. Donovan and E. Segal
Ψ2 : Let us forget about W momentarily, and also forget that X2 is an Artin stack. Since
they are related by variation of GIT, the spaces X1 and X2 are birational Calabi–Yau’s.
Kawamata and Bondal–Orlov have conjectured that any two birational Calabi–Yau’s arederived equivalent, and this is known to be true in many cases. Putting W back in, a moregeneral conjecture is that birational Calabi–Yau Landau–Ginzburg models have equivalent
categories of global matrix factorizations. (This should follow fairly easily from the W = 0
case.)
However, our X2 is actually an Artin stack. This complicates things, and in fact Db (X2 )is much bigger than Db (X1 ). However, as we shall see, we can construct a fully faithfulembedding from Db (X1 ) into Db (X2 ). We denote its image by BBr(X2 ), and we postulatethat this is the correct category of B-branes for the stack X2 .
When we put W back in we have a corresponding equivalence from Db (X1 , W ) to a certain
subcategory BBr(X2 , W ) ⊂ Db (X2 , W ). We will explain this step in Section 4.
Ψ3 : For Hori and Tong, this is the stage that requires the most novel arguments, and thesame is true for us. We use a variation on the Knörrer periodicity argument (as in step 1)
to construct an embedding of Db (Y2 ) into Db (X2 , W ), and show that the image is thesubcategory BBr(X2 , W ). We explain this step in Section 5.
Remark 1.2. It would be nice to compare our derived equivalence to the ones found by Borisov–Căldăraru and Kuznetsov; unfortunately, we do not know how to do this.
Remark 1.3. It may be helpful to compare what we do here to the proof of the ‘Calabi–
Yau/Landau–Ginzburg correspondence’ for B-branes presented in [Seg11] and [Shi12]. The goal ofthat project was similarly to re-prove a known equivalence (due to Orlov [Orl09]) using methods
that were more faithful to the original physical arguments.
Orlov’s result is the equivalence
Db (Y ) ∼= Db [ Cn / Zn ] , f ,where f is a degree n polynomial in n variables, and Y ⊂ Pn−1 is the corresponding Calabi–
Yau hypersurface. In the new proof the equivalence is factored into two steps, by considering anabelian GLSM
 n+1  ∗ CCwith the superpotential W = f p, where C∗ acts with weights (1, 1, . . . , 1, −n) and p is the last
coordinate. There are two GIT quotients: the first one is the total space of the canonical bundle
KPn−1 , and the first step is to prove an equivalence
Db (Y ) ∼= Db KPn−1 , W .
This follows from a ‘global Knörrer periodicity’ theorem, and we will use exactly the same
theorem to deduce our equivalence Ψ1 .
The second GIT quotient is the orbifold [ Cn / Zn ], and the second step is to prove an equivalence
Db (KPn−1 , W ) ∼= Db [ Cn / Zn ] , f .
We will extend the methods of this proof to prove our equivalence Ψ2 .
Note that there is no analogue of our third step Ψ3 in this construction.
Remark 1.4. Another previous body of work which is relevant is the study of the derived categories of intersections of quadrics, particularly as retold in [ASS14]. There, one considers an

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abelian gauged linear sigma model 
C3n C∗ ,where the C∗ acts with weight 1 on the first 2n coordinates x1 , . . . , x2n , and with weight −2 onthe last n coordinates p1 , . . . , pn . We equip this with a superpotential
W =

n
X

fi pi ,

i=1

where each fi is quadratic in the x-variables. The first GIT quotient X1 is the total space ofO(−2)⊕n over P2n−1 , and global Knörrer periodicity gives an equivalence
Db (X1 , W ) ∼= Db (Y1 )where Y1 ⊂ P2n−1 is the Calabi–Yau formed by intersecting all the quadrics. The second GIT
quotient X2 is the total space of the (orbi-)vector bundle O(−1)⊕2n over the weighted projective
space Pn−1
2:2:...:2 , and one obtains an equivalence
Db (X1 , W ) ∼= Db (X2 , W ) ,by the same methods as before. So we have passed through two steps, which are essentially thesame as those in the previous remark.
For the third step, we view (X2 , W ) as a family of Landau–Ginzburg B-models over Pn−1 ,each of which is of the form [ C2n / Z2 ], Wp for some quadratic form Wp . (This point of view isan analogue of the physicists’ Born–Oppenheimer approximation.) Where Wp is non-degenerate,Knörrer periodicity tells us that the category of matrix factorizations on the fibre is equivalent to
the derived category of two points, so generically (X2 , W ) looks like a double cover of Pn−1 . Morecareful analysis at the degenerate points reveals that Db (X2 , W ) is actually a non-commutative
resolution of a ramified double cover of Pn−1 .
Our equivalence Ψ3 is partially based on the techniques of this third step.
Remark 1.5. It is reasonable to ask what happens if we vary the dimensions of S and V , saygiving them dimensions r and d, respectively, and correspondingly adapt the definitions of X,
X1 , X2 , and W . This affects the three steps as follows:Ψ1 : The definition of the first Calabi–Yau Y1 also adapts immediately, and the equivalence Ψ1continues to hold, as it is a consequence of a much more general theorem. Of course d must
be big enough compared to r for Y1 to be non-empty.
Ψ2 : If we keep r = 2 and d odd, then the correct definition of BBr(X2 , W ) is clear and theequivalence Ψ2 generalizes immediately. If we move beyond these cases, then there are
obvious guesses as to how to proceed mathematically (particularly when r = 2 and d iseven), but we encounter an apparent discrepancy with the physical results; see Remark 4.8.
Ψ3 : This step is the most delicate, and the only other case that we can handle completely is
r = 2, d = 5, which recovers the derived equivalence between an elliptic curve and its
dual. In the case r = 2, d = 6, we can recover most of Kuznetsov’s result on Pfaffian cubic
fourfolds [Kuz06], and for r = 2, d > 7 our construction suggests a possible homologicalprojective dual for Gr(2, d). See Remark 5.13 for more details.
Remark 1.6. More recently, Hori used a dual model to provide a second physical derivation of
the Pfaffian–Grassmannian equivalence [Hor13]; see also [HK13]. It would be very interesting tofind a mathematical interpretation of this duality.

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N. Addington, W. Donovan and E. Segal
2. Categories of matrix factorizations
In this section we recall some general background on ‘global’ matrix factorizations.
2.1 Landau–Ginzburg B-models and curved dg-sheaves
We make the following definition.
Definition 2.1. A Landau–Ginzburg (or LG) B-model consists of:– a smooth n-dimensional scheme (or stack) X over C;– a choice of function W ∈ ΓX (OX ) (the ‘superpotential’);– an action of C∗ on X (the ‘R-charge’).
We denote this copy of C∗ by C∗R . We require that(i) the function W have weight (‘R-charge’) equal to two;(ii) the element −1 ∈ C∗R act trivially.
We denote by (X, W ) a Landau–Ginzburg (LG) B-model, suppressing the R-charge data fromthe notation. In affine patches, OX is a graded ring (graded by R-charge, and concentrated ineven degree), and W is a degree two element. Such a thing is sometimes called a ‘curved algebra’;
it is a very special case of a curved A∞ -algebra.
Example 2.2. Any (smooth) scheme X defines an LG B-model, by setting W ≡ 0 and lettingC∗R act trivially. This is an important special case.
Example 2.3. Let X = C2x,p and W = xp. We let C∗R act with weight zero on x and weight two
on p. This is an LG B-model, and it is the basic example to which Knörrer periodicity applies(see Section 3.1).
Example 2.4. The example we care about in this paper is the linear Artin stack
X = Hom(S, V ) ⊕ Hom(V, ∧2 S) GL(S)
introduced in Section 1.2. We have already specified the superpotential W (1.2), but we need toalso specify the R-charge, which we do letting C∗R act on Hom(V, ∧2 S) with weight two, and on
Hom(S, V ) with weight zero. These data define an LG B-model.
We also care about the open substacks X1 , X2 ⊂ X. These have superpotentials given by the
restriction of W , and each one is C∗R -invariant, so they define LG B-models.
We now give the appropriate notion of a sheaf on an LG B-model.
Definition 2.5. A curved dg-sheaf on (X, W ) is a sheaf E of OX -modules, equivariant withrespect to C∗R , equipped with an endomorphism dE : E → E of R-charge 1 such that(dE )2 = W · idE .
Note that, in affine patches, (E, dE ) is simply a graded module equipped with a ‘curved
differential’.
Terminology 2.6. We will call (E, dE ) coherent (respectively, quasi-coherent) if the underlying
sheaf E is coherent (respectively, quasi-coherent). If E is actually a finite-rank vector bundle, wewill call (E, dE ) a matrix factorization.

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The Pfaffian-Grassmannian equivalence revisited
We are primarily interested in matrix factorizations and coherent curved dg-sheaves.
Remark 2.7. Notice that because −1 ∈ C∗R acts trivially on X, any curved dg-sheaf splits into‘even’ and ‘odd’ eigensheaves
E = Eeven ⊕ Eodd
and the differential dE exchanges the two. There is a weaker definition of LG B-model where
we neglect the R-charge and keep only this (trivial) Z/2-action; this results in a Z/2-gradedcategory, whereas with R-charge we can construct a Z-graded category.
There is a C∗R -equivariant line bundle on X associated to any character of C∗R . We denotethese line bundles by O[k]. For any curved dg-sheaf E, we can shift the equivariant structure bytensoring with O[k], and we denote the result by E[k].
Remark 2.8. Suppose W = 0 and that C∗R acts trivially, as in Example 2.2. Then a curved
dg-sheaf is precisely a complex of OX -modules, and a matrix factorization is a bounded complex
of vector bundles. In this case the shift functor [1] is the usual homological shift.
The following is a useful source of examples of curved dg-sheaves.
Example 2.9. Suppose that Z ⊂ X is a (C∗R -invariant) subvariety lying inside the zero locus
of W . Consider the skyscraper sheaf E = OZ , equipped with the zero endomorphism dE = 0.
This defines a curved dg-sheaf, concentrated in even degree.
2.2 Categories of curved dg-sheaves
Now we discuss the morphisms between curved dg-sheaves. Let (E, dE ) and (F, dF ) be curved
dg-sheaves, and let
HomX (E, F)
denote the usual sheaf of OX -module homomorphisms between the underlying sheaves E and F.
This sheaf is C∗R -equivariant, and carries a differential given by the commutator of dE and dF ,so it is a curved dg-sheaf on the LG B-model (X, 0). Its global sections
ΓX HomX (E, F)
form a complex of vector spaces, graded by R-charge. Consequently, we can try to build a dgcategory whose objects are matrix factorizations, or coherent curved dg-sheaves. Of course it
would be naive just to use the chain complexes above for morphisms; we have to do some morework to define the dg-category correctly. There are essentially two approaches:(i) Take as objects all matrix factorizations, and as morphisms the complexesRΓX HomX (E, F) ,where RΓX is a suitable monoidal functor that computes derived global sections. We mayfor example use Dolbeault resolutions, or C̆ech resolutions with respect to some fixed C∗R invariant affine cover of X, if one exists. We denote the resulting dg-category by Perf(X, W ).
This was the approach adopted in [Seg11]. It is fairly concrete, but it has the major disadvantage that we can only use matrix factorizations as objects; in the ordinary derived category
Db (X), it would be very frustrating if we could only use locally-free resolutions of coherent
sheaves and never the sheaves themselves. Consequently, it is helpful to have a second, more
technical approach. This was developed by Orlov [Orl12] and Positselski [Pos11].

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N. Addington, W. Donovan and E. Segal(ii) Let QCohnv
dg (X, W ) denote the dg-category of quasi-coherent curved dg-sheaves, with morphisms defined ‘naively’ as above. It is easy to check that this category contains mapping
cones, so if we have a bounded chain-complex of curved dg-sheaves
E• = . . . → E0 → E1 → E2 → . . . ,we can form the totalization Tot(E• ), and this is a curved dg-sheaf. We define a curved dgsheaf to be acyclic if it is (homotopy equivalent to) the totalization of an exact sequence.
Then we define QCohdg (X, W ) to be the quotient (as a dg-category) of QCohnv
dg (X, W ) by
the full subcategory of acyclic objects. Finally, we define Perf(X, W ) to be the full subcategory of QCohdg (X, W ) consisting of objects which are homotopy-equivalent to matrix
factorizations.
Fortunately, these two approaches define quasi-equivalent dg-categories; this was proven by
Shipman [Shi12, Proposition 2.9] for the case that X is a scheme, but the argument works for
quotient stacks without modification (see also [LP13, Proposition 2.11] for a similar statement
without R-charge). Equally, the choice of functor RΓX in the first construction is not important.
From the second construction, it is clear that Perf(X, W ) is pre-triangulated; that is, it contains mapping cones. The shift functor acts by shifting R-charge equivariance, that is, tensoringwith O[1].
We denote the homotopy category of Perf(X, W ) by Db (X, W ); this is a triangulated category.
We will adopt the convention that the set of morphisms between two objects in this category is
the graded vector space
Hom•Db (X,W ) (E, F)
given by all homology groups of the chain-complex Hom•Perf(X,W ) (E, F), not just the zeroth
homology. For the case W = 0, this means that we are using Hom•Db (X) (E, F) to denote thegraded vector space of all Ext groups between E and F.
Remark 2.10. Denoting the homotopy category of Perf(X, W ) by Db (X, W ) is only appropriate
when X is smooth; in the singular case the latter notation should mean something different. In
particular, in the special case that W ≡ 0 and the R-charge is trivial, Perf(X, W ) is precisely
the dg-category of perfect complexes on X, whose homotopy category coincides with Db (X) if
and only if X is smooth.
Remark 2.11. In the rest of the paper we will consider various functors between categories ofmatrix factorizations, and we will write everything at the level of the homotopy categories.
However, it will be clear from our constructions that everything is actually well-defined at thelevel of dg-categories.
2.3 Basic properties
We list some other basic properties of Db (X, W ) for later reference.
(i) If X is a scheme which admits a C∗R -equivariant ample line bundle, then every coherent
curved dg-sheaf is equivalent to a matrix factorization, and hence defines an object in
Db (X, W ) [Shi12, Lemma 2.12]. Presumably this fact is still true when X is one of thestacks considered in this paper, but we shall not attempt to prove it, since whenever weencounter a coherent curved dg-sheaf, we will be able to see explicitly that it is equivalent
to a matrix factorization.

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The Pfaffian-Grassmannian equivalence revisited(ii) Let E and F be two curved dg-sheaves in Perf(X, W ). We have discussed the ‘global derived
morphisms’
Hom•Perf(X,W ) (E, F)
which is a chain-complex of vector spaces, but we will also need the sheaf of ‘local derived
morphisms’. If U ⊂ X is a (C∗R -invariant) affine open set, then Hom•Perf(U,X) (E, F) is a
dg-module over the graded algebra OU , that is, a curved dg-sheaf on (U, 0). Gluing thesetogether over X gives us a curved dg-sheaf on (X, 0), which we denote by
RHomX (E, F) .
We have
Hom•Perf(X,W ) (E, F) = RΓX RHomX (E, F) .
In practice this sheaf is quite easy to compute: we do it by replacing E with an equivalentmatrix factorization E, and then we have
RHomX (E, F) = HomX (E, F) .
(iii) If E and F are matrix factorizations on an affine scheme, then it is a basic observation that
HomX (E, F ) is acyclic away from the critical locus of W , because multiplication by any
partial derivative ∂i W is zero up to homotopy. Consequently, for any two curved dg-sheaves
E and F, the derived morphism sheaf RHomX (E, F) is acyclic away from the critical locus,
so its homology sheaves are supported (set-theoretically) at the critical locus. So the wholecategory Db (X, W ) is in some sense supported on the critical locus of W ; cf. [Orl11].
(iv) Let Z be the zero locus of W and letζ : Z ,→ X
be the inclusion. We can extend Example 2.9 to conclude that any curved dg-sheaf on (Z, 0)
pushes forward to give a curved dg-sheaf on (X, W ), so we have a functor
ζ∗ : Db (Z, 0) → Db (X, W ) .
(Note that Z is typically singular so we must use a modified definition of Db (Z, 0) here.)If we neglect R-charge, it is well known (for example [Orl12]) that this functor is essentiallysurjective, and its kernel is the category of perfect complexes on Z. This gives an equivalentdefinition of Db (X, W ) as the ‘derived category of singularities’Dsg (W ) = Db (Z)/ Perf(Z) .
Presumably this is still true if we include R-charge, but we shall not bother to check the fullstatement here. We just note the easy fact that ζ∗ OZ is equivalent to the matrix factorization
O[1]

W
1

O,

and this is contractible. It follows quickly that if P • is any C∗R -equivariant perfect complex
on X, then ζ∗ ζ ∗ P • is contractible in Db (X, W ).
3. Knörrer periodicity and the Grassmannian side
One of the most important classical facts about matrix factorizations is Knörrer periodicity[Kno87]. We will briefly discuss this phenomenon, and various modern formulations of it that341

N. Addington, W. Donovan and E. Segal
have appeared in recent years [Orl06, Isi13, Shi12, Pre11], and conclude by showing our firstequivalence Ψ1 in Corollary 3.5.
3.1 Knörrer periodicity over a point
Consider an LG B-model X = C2 with the superpotential W = x1 x2 , and let Y be the subschemeof X consisting of just the origin (we neglect R-charge for the moment). In its simplest form,Knörrer periodicity states that we have an equivalence
Db (Y ) ∼= Db (X, W ) .
Remark 3.1. Since Y is the critical locus of W , this is a situation where we may take § 2.3(iii)very literally.
Finding such an equivalence is the same thing as finding a curved dg-sheaf E on (X, W ) which
generates the whole category and satisfies
Hom•Db (X,W ) (E, E) = C .
Recall that this space of morphisms is a graded vector space, so implicit here is the statementthat there are no morphisms in non-zero degree. Thus the object E behaves, homologically, like
an isolated point.
There are many possible choices for such an E; one is the skyscraper sheaf along the x2 -axis
E = O{x1 =0}
with dE = 0 (this is an instance of Example 2.9). Then we get an equivalence from Db (Y ) to
Db (X, W ) by mapping OY to E.
Remark 3.2. This choice of E breaks the symmetry between x1 and x2 . This is an importantfeature: there is a second choice where we let E be the skyscraper sheaf on the x1 -axis, which
produces a different equivalence, differing from the first one by a shift. A related fact is that if
we want to add R-charge to this construction, then we can do it by letting C∗R act with weight
two on x1 and weight zero on x2 , or vice versa, but this also breaks the symmetry.
This basic version of Knörrer periodicity can be generalized in various directions. Firstly, we
may replace X = C2 with X = C2n , and W with a non-degenerate quadratic function, so thecritical locus of W is still the origin. We replace the isotropic line {x1 = 0} ⊂ C2 with a choiceof maximally isotropic subspace M ⊂ C2n . Then one can check that E = OM is point-like, andgenerates Db (X, W ), so as above it gives us an equivalence between the derived category of apoint and Db (X, W ).
3.2 Knörrer perodicity in families: first version
Now we can try to formulate this construction in families. Most obviously we could choose X to
be the total space of an even-rank vector bundle
π: X → Y
and W to be a fibrewise non-degenerate quadratic form on X. Suppose that we can find a
subbundle M ⊂ X which gives a maximally isotropic subspace in each fibre. Then for each pointy ∈ Y we have a curved dg-sheaf Ey = OMy on the fibre over y, and these fit together into afamily E = OM on the whole space. We want to consider the functor whose Fourier–Mukai kernel

342

The Pfaffian-Grassmannian equivalence revisitedis E, that is, it sends each skyscraper sheaf Oy ∈ Db (Y ) to the corresponding Ey ∈ Db (X, W ),and sends the whole structure sheaf OY to E. In other words, we consider the diagramπιY ←− M −→ X
and the induced functorsπ∗ι

∗
Db (Y ) −→ Db (M ) −→Db (X, W ) .

It is proven in [Pre11, Theorem 9.1.7(ii)] that, given such an M , the composition π ∗ ι∗ gives us
an equivalence between Db (Y ) and Db (X, W ). Note that the existence of such an M is quite astrong condition; see [ASS14, § 4.3] for some discussion of this point.
Remark 3.3. In particular, π ∗ ι∗ is fully faithful. We pause to discuss this point in a little moredetail, since the reasoning used will be important in Section 5.
The functor π ∗ ι∗ is linear over the sheaf of functions on Y , so full faithfulness can be checked
locally on Y . Moreover, if we restrict to an affine neighbourhood in Y , then the derived category
is generated by the structure sheaf, so locally we only need to check full faithfulness on the
structure sheaf. Therefore it is enough to check that the endomorphisms ofπ ∗ ι∗ OY = E ∈ Db (X, W )agree with the endomorphisms of OY ∈ Db (Y ), as a sheaf over Y , that is, thatπ∗ RHomX (E, E) ∼= OY .
This statement is equivalent to the full faithfulness of π ∗ ι∗ ; in particular, it obviously implies
that
Hom• b
(Ey , Ey ) ∼= Hom• b (Oy , Oy )D (X,W )D (Y )

for all points y ∈ Y . Informally at least the converse implication also holds: if we have a familyof orthogonal objects Ey , and each one is ‘point-like’ in this sense, then the resulting kernel E
must give a fully faithful functor.
3.3 Knörrer periodicity in families: second version
There is a more general family version of Knörrer periodicity, based on the observation that we
do not actually need a projection π : X → Y , only a projection π : M → Y . Specifically, we
consider the total space of a vector bundle
π: X → B
over some base B, and let
Y ⊂B
be the zero locus of some transverse section f ∈ ΓB (X ∨ ). We can equip X with the superpotential
W = fp ,where p denotes the tautological section of π ∗ X. Since f is transverse, Y is smooth and isexactly the critical locus of W . The normal bundle NY /X to Y carries a non-degenerate quadratic
form given by the Hessian of W . Furthermore, this bundle has a canonical maximally isotropic
subbundle given by M = X|Y . So we should be able to get an equivalence between Db (Y ) andDb (X, W ) using the diagramπιY ←− X|Y −→ X .

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N. Addington, W. Donovan and E. Segal
Note that there is a more or less canonical way to add R-charge to this construction, by lettingC∗R act trivially on B and with weight two on the fibres of X.
Theorem 3.4 [Shi12, Theorem 3.4]. Consider an LG B-model (X, W ) of the form describedabove, with C∗R acting fibrewise with weight two. Assume that the base B is a smooth quasiprojective variety. Then the compositionπ∗ι∗
Db (Y ) −→ Db (X|Y ) −→Db (X, W )is an equivalence.
Similar theorems are proven in [Orl06] and [Isi13]. Note that C∗R is acting trivially on Y , so
Db (Y ) really does mean the usual derived category of Y .
3.4 Grassmannian example
Now consider the LG B-model (X1 , W ) discussed in Section 1.2, and described more precisely
in Example 2.4. This model is exactly of the form specified by Theorem 3.4: X1 is the total
space of the vector bundle π : O(−1)⊕7 → Gr(2, V ), and the R-charge is acting trivially on the
Grassmannian and with weight two on the fibres. Also, the superpotential is W = f p, wheref = A ◦ ∧2 x
is a transverse section of O(1)⊕7 on Gr(2, V ), and p is the tautological section of π ∗ O(−1)⊕7 .
The zero locus of f is the Calabi–Yau threefold Y1 ; hence Theorem 3.4 yields the following result.
Corollary 3.5. The derived category Db (Y1 ) is equivalent to Db (X1 , W ).
This concludes our discussion of the first equivalence Ψ1 .
4. Windows
In this section we will define the category BBr(X2 , W ) and the equivalence Ψ2 .
4.1 Without the superpotential
Let
ι1

ι2

X1 ,→ X ←- X2
be the three spaces considered in Section 1.2. For the purposes of this section we set the superpotential W to zero, and take the C∗R -action to be trivial, so Db (Xi ) and Db (X) are the usual
derived categories.
We are interested in the relationship between Db (X1 ) and Db (X2 ). If X1 and X2 were manifolds (or orbifolds), then we would expect them to be derived equivalent, since they are birationaland Calabi–Yau. What should we expect in this situation?
Physically, we can reason as follows. Using Hori and Tong’s construction, we know that thesigma models with targets X1 and X2 lie in the same Kähler moduli space of CFTs. (We gloss
over the fact that these targets are non-compact.) Consequently, the B-models associated to eachspace are the same. In particular, they have the same category of B-branes, and so we shouldhave two equivalent categories∼ BBr(X2 ) .
BBr(X1 ) =344

The Pfaffian-Grassmannian equivalence revisited
Since X1 is a manifold, we know that the category of B-branes BBr(X1 ) is Db (X1 ). However,X2 is an Artin stack. A “sigma-model” whose target is an Artin stack is really a gauge theory,and understanding the category of B-branes in a gauge theory is much more difficult. We will
not attempt to address this general question; instead, we will make an ad hoc definition of thecategory BBr(X2 ), constructing a fully faithful embedding
Db (X1 ) ,→ Db (X2 )and defining BBr(X2 ) as the image of this embedding. The main motivation for our definition isjust that it gives something equivalent to Db (X1 ), but we will give some a posteriori justification(see Remark 4.7).
To construct the embedding we will use the technique of ‘grade-restriction’, or ‘windows’,
introduced by the third-named author in [Seg11]. This was directly inspired by the physics paper[HHP08], but was also based on a long history of mathematical ideas by Beilinson, Kawamata,Van den Bergh, and others. What we do is to find a subcategoryG ⊂ Db (X)such that the restriction functor ι∗1 : G → Db (X1 ) is an equivalence and the other restriction
functor ι∗2 : G → Db (X2 ) is fully faithful. This technique has now been developed into an elegant
general theory [HL12, BFK12] which can be applied immediately in this example to show thatsuch a G exists. Unfortunately, the description that this theory gives of the image of G inside
Db (X2 ) is not explicit enough for our purposes, so we take a more hands-on approach.
Observe that any representation of GL(S) = GL(2) determines a vector bundle on each of
the spaces that we are considering. We will be interested in the ‘rectangle’ of representations
n
o
(4.1)
Syml S ∨ ⊗ (det S ∨ )m : l ∈ [0, 3) , m ∈ [0, 7) .
The associated vector bundles on Gr(2, V ) form a (Lefschetz) full strong exceptional collection
by [Kuz08, Theorem 4.1]. Let Tl,m denote the vector bundle Syml S ∨ (m) on X associated toSyml S ∨ ⊗ (det S ∨ )m , and let
DE
G = Tl,m : l ∈ [0, 3) , m ∈ [0, 7)⊂ Db (X)(4.2)
be the subcategory generated by this set of vector bundles. (Here, and throughout the paper, wemean ‘generated’ in the strong sense, by taking shifts and cones but not direct summands; that
is, G consists of those objects that have a finite resolution in terms of this set of bundles.)Proposition 4.1. The restriction functor
ι∗1 : G → Db (X1 )is an equivalence, and the restriction functor
ι∗2 : G → Db (X2 )is fully faithful.
Consequently, we obtain an embedding of Db (X1 ) into Db (X2 ), and its image is the subcategory
generated by the vector bundles associated to the representations (4.1). We define BBr(X2 ) tobe this subcategory.

345

N. Addington, W. Donovan and E. Segal
Remark 4.2. The reason we are not using the general theory of [HL12, BFK12] is that it fails to
identify this explicit set of generating bundles for the category G. This is because Kuznetsov’s
exceptional collection does not fit with the ‘grade-restriction rules’ for this GIT problem (forcomparison, Kapranov’s exceptional collection fits the Grassmannian side perfectly, but not thePfaffian side). It would be interesting to find a natural derivation of Kuznetsov’s exceptional
collection via GIT.
We split the proof of Proposition 4.1 into four lemmas.
Lemma 4.3. Both ι∗1 and ι∗2 are fully faithful.
Proof. It is enough to check this statement on the generators of G. On X, the higher Ext groups
between them vanish: since they are vector bundles, we have∨
ExtpX (Tl,m , Tl0 ,m0 ) ∼⊗ Tl0 ,m0 ) ,= Rp ΓX (Tl,mand the functor of taking GL(S)-invariants (that is, global sections) is exact. Also, the morphisms
between two generators will not change when we restrict to either X1 or X2 . To see this notethat the complements of both substacks have codimension at least two, so by Hartogs’ lemma∨ ⊗ T 0 0 does not change after restriction, and therefore
the space of all sections of the bundle Tl,ml ,mneither does the space of GL(S)-invariant sections.
So we need only check that the generators do not acquire any higher Ext groups after restriction, that is, that

∗
∗
Ext>0
Xi ιi Tl,m , ιi Tl0 ,m0 = 0
for all l, l0 ∈ [0, 3) and m, m0 ∈ [0, 7), for both i = 1 and i = 2.
For i = 1 we use the projection formula applied to the projection
q1 : X1 = Tot O(−1)⊕7 → Gr(2, V )to compute the cohomology of0
RHomX1 ι∗1 Tl,m , ι∗1 Tl0 ,m0 ∼= RHomX1 q1∗ Syml S ∨ (m), q1∗ Syml S ∨ (m0 )



0
∼
= RHomGr(2,7) Syml S ∨ (m), q1∗ q1∗ Syml S ∨ (m0 )0∼= RHomGr(2,7) Syml S ∨ (m), Syml S ∨ (m0 ) ⊗ Sym• O(1)⊕7 .
Our claim now follows from the vanishing result used in [Kuz08], which is stated below inLemma 4.4, and a minor extension of it, given in Lemma 4.5.
For i = 2 we work similarly, using the fact that X2 has a projectionq2 : X2 = Tot S ∨⊕7 → Pto an Artin stack P = ∧2 S ⊕7 − {0} GL(S) . There is a map δ : P → P6 induced bydet : GL(S) → C∗ and forgetting the isotropy groups. Now working as above for X1 , and usingthat the functor δ∗ is exact, we have



l ∨
l0 ∨
0
• ⊕7
RHomX2 ι∗2 Tl,m , ι∗2 Tl0 ,m0 ∼
RHom
Sym
S
(m),
Sym
S
(m
)
⊗
Sym
S
=
P


0
0
∼
= RΓP6 δ∗ Syml S ⊗ Syml S ∨ ⊗ (det S ∨ )m −m ⊗ Sym• S ⊕7 .
Now, using the Littlewood–Richardson rule [FH96, § A.1], we may decompose this last bundle
into direct summands corresponding to irreducible representations of GL(S). The summands we

346

The Pfaffian-Grassmannian equivalence revisited
obtain are Schur powers Sµ S ∨ with µ 6 (m0 , m0 + l0 ), with the maximal µ occurring being thehighest weight for the bundle Tl0 ,m0 . We evaluate δ∗ (Sµ S ∨ ). Every point of P has non-trivial
stabilizer SL(S) ⊂ GL(S), and Sµ S has non-trivial SL(S)-invariant vectors only if µ = (ν, ν). In
this case Sµ S ∨ ∼= (det S ∨ )ν ; hence δ∗ (Sµ S ∨ ) ∼= OP6 (−ν). This has no higher cohomology as longas ν 6 6, and so we are done because ν 6 m0 6 6 by construction.
The following two lemmas are calculations used in Lemma 4.3 above.
Lemma 4.4 [Kuz08, Lemma 3.5]. Let Gr = Gr(2, V ), with dim V = n odd. If 0 6 l, l0 6 21 n − 1
and 0 6 k 6 n − 1, then(0Syml −l S ∨ if l 6 l0 , k = 0 , p = 0 ,
0
p
l ∨
l ∨
∼
ExtGr Sym S , Sym S (−k) =0otherwise.
Proof. This is a specialisation of the result of [Kuz08, Lemma 3.5] to odd-dimensional V , asrequired in our case. The proof is combinatorial, using the Littlewood–Richardson rule to decompose a bundle on the Grassmannian into direct summands corresponding to irreducible representations of GL(S), and then the Borel–Bott–Weil theorem (as explained in [Kuz08, § 3]) to
calculate their cohomology.
Lemma 4.5. In the setting of Lemma 4.4 above, but with k < 0, we have

l ∨
l0 ∨
Ext>0
Gr Sym S , Sym S (−k) = 0 .
0

Proof. It suffices to check that Syml S ⊗Syml S ∨ (−k) on Gr has no higher cohomology. Following
the proof of [Kuz08, Lemma 3.5] we have000Syml S ⊗ Syml S ∨ (−k) ∼= Syml−1 S ⊗ Syml −1 S ∨ (−k) ⊕ Sl −k,−l−k S ∨ ,and so we may proceed inductively. We therefore need only check that the Schur power Sα S ∨has no higher cohomology on the Grassmannian Gr for
α = l0 − k, −l − k, 0, . . . , 0 .
The proof then follows by application of the Borel–Bott–Weil theorem, with the following twocases.
Case k 6 −l. In this case α is a dominant weight; hence there is no higher cohomology.
Case −l < k < 0. Using ρ to denote half of the sum of the positive roots of GL(n) as in[Kuz08], we have
α + ρ = n + l0 − k, n − l − k − 1, n − 2, n − 3, . . . , 1 .
Our assumptions give n − 1 > n − l − k − 1 > n/2 > 0; hence the second entry in this weight
coincides with one of the later ones. By the Borel–Bott–Weil prescription, it follows from thisthat no cohomology occurs in this case.
This completes the proof of the lemma.
The final stage in the proof of Proposition 4.1 is the following.
Lemma 4.6. The restriction functor ι∗1 : G → Db (X1 ) is essentially surjective.
Proof. This is the statement that the set of vector bundles on X1 corresponding to the set (4.1)
of GL(S)-representations generates the derived category Db (X1 ). This may be deduced from the347

N. Addington, W. Donovan and E. Segal
fact that the corresponding set of vector bundles on Gr(2, V ) generates the derived category
Db (Gr(2, V )) by Kuznetsov’s result [Kuz08, Theorem 4.1], as follows.
First note that any coherent sheaf E on X1 extends to a coherent sheaf E 0 on X, and since
X is smooth, this extension E 0 has a finite resolution by vector bundles. Furthermore, the only
vector bundles which occur are the Tl,m associated to GL(S)-representations, as X is a quotientof a vector space by GL(S). Restricting this resolution via the inclusion ι1 : X1 ,→ X, we obtaina finite resolution of E on X1 by the ι∗1 Tl,m . We haveι∗1 Tl,m = q1∗ Syml S ∨ (m) ,so that the ι∗1 Tl,m are pullbacks via the projection q1 : X1 → Gr(2, V ) of the bundles Syml S ∨ (m)on Gr(2, V ). These latter bundles are themselves resolved by Kuznetsov’s full exceptional collection corresponding to the set (4.1) of GL(S)-representations; hence we deduce the result.
This concludes the proof of Proposition 4.1.
Remark 4.7. Recall that we are making an ad hoc definition of the category of B-branes on X2
asDE
BBr(X2 ) := ι∗2 G = ι∗2 Tl,m : l ∈ [0, 3) , m ∈ [0, 7)⊂ Db (X2 ) .
Let us explain why this definition is not totally unreasonable. We have that X2 is a bundle overP6 , with fibres
F = Hom(S, V ) SL(S) ,and so we should expect BBr(X2 ) to be some kind of product of BBr(P6 ) = Db (P6 ) with somecategory BBr(F) of B-branes on the fibres. The derived category of P6 is generated by the
Beilinson exceptional collection ∗ι2 T0,m = O(m) : m ∈ [0, 7) ,so what we are implicitly doing is declaring that
BBr(F) = O, S ∨ , Sym2 S ∨⊂ Db (F) .

We do not have a justification for this definition either, but it does satisfy
rank K0 (BBr(F)) = 3
which matches Hori–Tong’s calculation of the Witten index for the gauge theory described by F;see [HT07, Table 1].
Remark 4.8. Let us briefly discuss how one might adapt this argument if we were to vary thedimensions of S and V , making them r and d, respectively. The general theory of [HL12, BFK12]
still gives us an embedding of Db (X1 ) into Db (X2 ), but as before it tells us very little about theimage. So we should ask to what extent our more explicit methods can be adapted.
If we keep r = 2 and d odd, then everything works essentially verbatim, using the rectangular
windowonSyml S ∨ ⊗ (det S ∨ )m : l ∈ 0, 21 (d − 1) , m ∈ [0, d) .
Now let us keep r = 2, but make d even. Something goes wrong even at the crude heuristic levelof Remark 4.7, because now d does not divide dr . Mathematically, it seems sensible to declarethat BBr(X2 ) is the subcategory generated by the rectanglenoSyml S ∨ ⊗ (det S ∨ )m : l ∈ 0, 12 d , m ∈ [0, d) .

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The Pfaffian-Grassmannian equivalence revisited
If we delete d/2 bundles from the corner of this rectangle then we get Kuznetsov’s (nonrectangular) Lefschetz exceptional collection on Gr(2, d), and we see that we obtain an embeddingof Db (X1 ) into BBr(X2 ), rather than an equivalence. This definition allows us to recover a resultof Kuznetsov in the case r = 2 and d = 6 (see Remark 5.13). Unfortunately, this definition does
not appear to be compatible with the results of [HT07]. It suggests that the category of B-braneson the fibre F should be generated bynoSyml S ∨ : l ∈ 0, 12 d,but Hori–Tong calculate the Witten index of the corresponding gauge theory to be (d/2) − 1, notd/2. It would be very interesting to understand why these two approaches seem to give different
answers.
If we make r > 2, then we can presumably make some mathematical progress using Fonarev’s
Lefschetz exceptional collections on Gr(r, d) [Fon13], but the discrepancy with Hori–Tong’s calculation becomes even worse.
4.2 With the superpotential
We will now explain how to modify the constructions of the previous section when we add in the
superpotential W and the non-trivial R-charge described in Example 2.4. Specifically, we willshow that we have an embedding
Db (X1 , W ) ,→ Db (X2 , W ) .
The construction of this embedding follows closely our construction of the embedding Db (X1 ) ,→Db (X2 ). Suppose that we have some matrix factorization E ∈ Db (X, W ) on the ambient Artinstack. The underlying vector bundle of E must be a direct sum of shifts of the bundles Tl,m ,since these are the only vector bundles on X. To define the analogue of the window G, we justrestrict which vector bundles Tl,m we are allowed to use. Namely, we define
GW ⊂ Db (X, W )to be the full subcategory whose objects are (homotopy equivalent to) matrix factorizations
whose underlying vector bundles are direct sums of shifts of the vector bundles Tl,m , wherel ∈ [0, 3) and m ∈ [0, 7), as in (4.2). We then have the following.
Proposition 4.9. The restriction functor
ι∗1 : GW → Db (X1 , W )is an equivalence, and the restriction functor
ι∗2 : GW → Db (X2 , W )is fully faithful.
Proof. This follows from Proposition 4.1, using the arguments from [Seg11, § 3.1]. Full faithfulness
is straightforward; we can use the proof of [ibid., Lemma 3.4] verbatim. The key point is thatmorphisms on any Landau–Ginzburg model (X, W ) can be computed, via a spectral sequence,from morphisms on the model (X, 0).
The essential surjectivity of ι∗1 follows from Lemma 4.10 below, since we proved in Lemma 4.6
that any sheaf on X1 can be resolved by vector bundles from the set (4.1), and this resolution
can evidently be chosen to be C∗R -equivariant.

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N. Addington, W. Donovan and E. Segal
Lemma 4.10. Let (X, W ) be a LG B-model, and let E0 , . . . , Ek be a collection of C∗R -equivariant
vector bundles on X such that
Ext>0
X (Ei , Ej ) = 0∀i, j

in the ordinary derived category of X (that is, ignoring the R-charge grading). Now let(E, dE ) ∈ Db (X, W )be an object such that the underlying sheaf E has a finite C∗R -equivariant resolution by copies ofshifts of the bundles Ei . Then (E, dE ) is equivalent to a matrix factorization whose underlying
vector bundle is a direct sum of copies of shifts of the Ei .
Proof. This is proved in [Seg11, proof of Lemma 3.6]. It is shown there that it is possible toperturb the differential in the resolution of E until it becomes a matrix factorization for W whichis equivalent to (E, dE ). (The proof in that paper is stated for the case that E is a vector bundle,
but it works for sheaves without modification. The argument is also independent of which dg
model we choose for Perf(X, W ).)We define the category
BBr(X2 , W ) ⊂ Db (X2 , W )to be the image of GW under ι∗2 , and we claim that this is the correct category of B-branes for
the LG model (X2 , W ).
This concludes our discussion of the second equivalence Ψ2 .
5. The Pfaffian side
In this final section we complete our proof that Db (Y1 ) ∼= Db (Y2 ) by establishing the equivalence Ψ3 . To do this we construct an embedding
Db (Y2 ) ,→ Db (X2 , W )whose image is the subcategory BBr(X2 , W ) defined in the previous section.
Recall that X2 is the Artin stack
h
.
i
X2 =
GL(S)
(x, p) ∈ Hom(S, V ) ⊕ Hom(V, ∧2 S) : p 6= 0
and that it is equipped with the superpotential
W (x, p) = p ◦ A ◦ ∧2 x ,where A : ∧2 V → V is a surjection satisfying Assumption 1.1 that we have fixed throughout the
paper. For this section, we will denote by π the projection
π : X2 → P Hom(V, ∧2 S) ∼= P6 .
(x, p) 7→ [p]This makes X2 into a Zariski locally-trivial bundle of stacks with fibreF := Hom(S, V ) SL(S) .
To see this, observe that the preimage of a standard affine chart A6 ⊂ P6 is the stack
X2 |A6 = Hom(S, V ) × C∗ × A6 GL(S)∼ Hom(S, V ) SL(S) × A6 .
=

350

(5.1)

The Pfaffian-Grassmannian equivalence revisited
For another point of view, we can consider X2 as a vector bundle
q2 : X2 → P
over the stackP=∧2 S ⊕7 − {0}



GL(S)



(this was mentioned briefly in the proof of Lemma 4.3). Then we can factor π as δ ◦ q2 , where δis the forgetful map
δ : P → P6
sending P to its underlying scheme. The map δ is a Zariski locally trivial bundle of stacks withfibre B SL2 .
Note that the C∗R -action on X2 preserves each fibre of π, and if we write a fibre using theatlas (5.1), then the action on F is just dilation (with weight one). This is because letting C∗R
act with weight two on p is equivalent to letting it act with weight one on x, up to the actionof GL(S).
5.1 Heuristics and strategy
Fix a point [p] ∈ P6 . On the fibre X2 |[p] ∼= F over this point, the superpotential is a quadraticform:
Wp (x) := p ◦ A ◦ ∧2 x .
Since the C∗R -action preserves the fibre, the pair (F, Wp ) is a LG B-model in its own right. If thequadratic form Wp were non-degenerate, then our discussion of Knörrer periodicity in Section 3.2
would lead us to study SL(S)-invariant, maximally isotropic subspaces
Mp ⊂ Hom(S, V )in order to understand Db (F, Wp ). In fact, Wp is degenerate, but previous experience [ASS14]
suggests that this is still a sensible thing to do.
To ensure SL(S)-invariance, we need to take Mp = Hom(S, Lp ), where Lp ⊂ V is maximallyisotropic for the two-form
ωp := p ◦ A
on V . The rank of this two-form is six for a generic [p], and it drops to four precisely when[p] ∈ Y2 . Since A is generic, it never drops to two. Thus if [p] ∈/ Y2 , then a maximal Lp has
dimension four and a maximal Mp dimension eight, but if [p] ∈ Y2 , then dim Lp jumps up to fiveand dim Mp to ten.
We will restrict attention to maximally isotropic subspaces Mp for [p] ∈ Y2 , for the reasons
we now explain. Our results from the previous section (see in particular Remark 4.7) suggestthat we should focus on the ‘window’ subcategory
BBr(F, Wp ) ⊂ Db (F, Wp )consisting of (objects isomorphic to) matrix factorizations built only out of the three vector
bundles O, S, and Sym2 S. This category is, in some sense, the fibre of the category BBr(X2 , W )at the point [p]. Consequently, we only care about those maximally isotropic subspaces Mp thatdefine objects in the subcategory BBr(F, Wp ).
The sheaf OMp has a Koszul resolution with underlying vector bundle∧• Hom(S, V /Lp )∨ .
(5.2)

351

N. Addington, W. Donovan and E. Segal
Perturbing the Koszul differential as in Lemma 4.10, we find that OMp ∈ Db (F, Wp ) is equivalent
to a matrix factorization with this same underlying vector bundle. Then we use the formula forthe exterior algebra of a tensor product [Wey03, Corollary 2.3.3] to find that the representationsof SL(S) occurring in (5.2) are Symt S, for
0 6 t 6 dim(V /Lp ) .
To get OMp ∈ BBr(F, Wp ) it appears that we need to have dim(V /Lp ) = 2, and hence [p] ∈ Y2 . So
if we believe this heuristic argument, the category BBr(X2 , W ) is concentrated over the Pfaffian
locus Y2 .
In the spirit of Section 3.2, a continuous choice of Lp for all [p] ∈ Y2 will give us a functorb
D (Y2 ) → BBr(X2 , W ) sending O[p] to OMp . We claim that this functor is fully faithful. This isessentially equivalent (see Remark 3.3) to the claim that each object OMp behaves like the pointsheaf O[p] , that is,
Hom• b
(OM , OM ) ∼(O[p] , O[p] ) ,= Hom• b
D (X2 ,W )pp

D (Y2 )or alternatively to the claim that the whole family OM behaves like the structure sheaf OY2 ,that is,π∗ RHomX (OM , OM ) ∼= OY .
2

2

A suitable version of this claim will be proved in Proposition 5.3, but let us briefly discuss why
it is true. If each quadratic form Wp were non-degenerate, then it would be standard Knörrer
periodicity, and each object OMp would be point-like in the fibrewise directions. However, since
Wp is degenerate, this is not true: viewed as an object on (F, Wp ), the curved dg-sheaf OMp is notpoint-like; it looks like the skyscraper sheaf along the kernel of Wp . Fortunately, this calculation
is misleading, because if we view OMp as an object on (X2 , W ), then we must also take accountof the derivatives of W in the directions transverse to the fibre. As we shall see, these transverse
directions exactly cancel the degenerate directions of Wp , leaving a suitably point-like object.
Next we face another issue, which is that the spaces Lp , and hence Mp , can be chosen locallyon Y2 but not globally. One approach to overcoming this would be to take local choices andglue them to give a global embedding. Instead we replace each OMp with an equivalent objectOΓp ∈ Db (F, Wp ) which involves no choices and thus is easy to globalize to a family Γ. We define
Γ in Definition 5.5 and show in Proposition 5.6 that OMp is equivalent to the new object OΓp .
In Section 5.5 we fill in the final details that Γ gives us an embedding Db (Y2 ) → Db (X2 , W )whose image is BBr(X2 , W ). We conclude with some remarks on varying the dimensions of S
and V , and on homological projective duality.
5.2 The critical locus
We start by analyzing the critical locus of W on X2 . This means that we take the critical locus
of W on the atlas

Hom(S, V ) × Hom(V, ∧2 S) − {0}and form the stack quotient of it by GL(S). (The result is independent of our choice of atlas.
If [Z/G] is a quotient stack with Z smooth, and W is a G-invariant function on Z, then at any
point z ∈ Z the derivative dW |z defines a closed element of the cotangent complex Lz = [T ∨ Z →g∨ ]. We are considering the substack where this element is zero, and this is invariant since thecotangent complex is an invariant of the atlas up to quasi-isomorphism.)
Proposition 5.1. Let x ∈ Hom(S, V ) and p ∈ Hom(V, ∧2 S) − {0}. Then (x, p) is a critical point

352

The Pfaffian-Grassmannian equivalence revisitedof W if and only if Im(x) ⊂ ker ωp and rank(x) 6 1.
Proof. In the x-directions W is a quadratic form, so its derivatives vanish exactly along its kernel,
which is Hom(S, ker ωp ). In the p-directions W is linear, so its derivatives vanish exactly whenW (x, q) = 0 for all q ∈ Hom(V, ∧2 S). Thus (x, p) is a critical point of W if and only if Im(x)is contained in ker ωp and is isotropic for all ωq as q varies over Hom(V, ∧2 S). Now we need
only argue that these imply rank(x) 6 1. If rank(ωp ) = 6 then dim(ker ωp ) = 1, so rank(x) 6 1
already, but if rank(ωp ) = 4 we need a further argument.
Consider the locus of ω ∈ Hom(∧2 V, ∧2 S) for which ω has rank four as a two-form on V . By
[Har95, Example 20.5], a line ω + tξ is tangent to this locus if and only if ker ω is isotropic for ξ;that is, the tangent space to this locus is the kernel of the natural map
Hom(∧2 V, ∧2 S) → Hom(∧2 ker ω, ∧2 S) .
ξ
7→
ξ|ker ωThus the normal space to this locus in Hom(∧2 V, ∧2 S) embeds into Hom(∧2 ker ω, ∧2 S), andsince both have dimension three they are isomorphic.
Now by Assumption 1.1, A gives an embedding Hom(V, ∧2 S) ,→ Hom(∧2 V, ∧2 S) which istransverse to the rank four locus, so the normal space to the cone on Y2 under this embedding is
identified with Hom(∧2 ker ωp , ∧2 S) in the same way. In particular, for every two-form η on ker ωpthere is a q ∈ Hom(V, ∧2 S) such that ωq |ker ωp = η. Now if Im(x) ⊂ ker ωp were two-dimensional
there would be an η for which it was not isotropic, hence a q such that Im(x) was not isotropicfor ωq , so (x, p) would not be a critical point of W . Thus if (x, p) is a critical point of W , thenrank(x) 6 1, as claimed.
We now focus on the part of the critical locus that lies over the Pfaffian Calabi–Yau Y2 . Let
K ⊂ OY2 ⊗ V
be the rank three bundle whose fibre over [p] ∈ Y2 is
Kp := ker ωp ⊂ V .
In the proof of the previous proposition we saw that dW induces an isomorphismdW : NY2 /P6 → Hom(∧2 K, ∧2 S)(5.3)

of vector bundles over Y2 . Notice that although S is not really a vector bundle on P6 (it is avector bundle on the stack P), its determinant ∧2 S really is a legitimate line bundle on P6 ; infact, it is OP6 (1).
We abuse notation slightly and let

Hom(S, K) ⊂ Hom(S, V ) × Hom(V, ∧2 S) − {0}denote the subvariety
Hom(S, K) = { (x, p) ∈ X2 : [p] ∈ Y2 , x ∈ Hom(S, Kp ) } .
This is a vector bundle over the punctured affine cone over Y2 , and [ Hom(S, K) / GL(S) ] is a
substack of X2 whose fibre over a point [p] ∈ Y2 is[ Hom(S, Kp ) / SL(S) ] ⊂ [ Hom(S, V ) / SL(S) ] = F .
Alternatively, we may view [ Hom(S, K) / GL(S) ] as a vector bundle over the stack P.

353

N. Addington, W. Donovan and E. Segal
Lemma 5.2. The underlying scheme of [ Hom(S, K) / GL(S) ] is the total space of the vector
bundle Hom(∧2 S, ∧2 K) over Y2 . The underlying scheme of Crit(W )|Y2 is Y2 .
The ‘underlying scheme’ of a stack is the universal scheme that receives a map from thatstack; for a quotient stack this is simply the scheme-theoretic quotient.
the fibre of [ Hom(S, K) / GL(S) ] over a point [p] ∈ Y2 is the stackProof. As we just mentioned,
Hom(S, Kp ) SL(S) . The scheme underlying this is the scheme-theoretic quotientSL(S)
Hom(S, Kp ) / SL(S) = Spec OHom(S,Kp ).
By [KP96, § 8.4] we have a closed embedding
Hom(S, Kp ) / SL(S) ,→ Hom(∧2 S, ∧2 Kp )which is an isomorphism since both spaces have dimension three. The first statement of thelemma follows immediately.
By Proposition 5.1, Crit(W )|Y2 is the substack of [ Hom(S, K) / GL(S) ] where x has rank
one. The image of this substack in the underlying scheme is the zero section.
One can argue similarly that the underlying scheme of the whole of Crit(W ) is P6 , but weshall not use this fact.
5.3 Point-like objects from maximally isotropic subspaces
We now show that maximally isotropic subspaces give point-like objects, as we outlined in § 5.1.
Recall that, roughly, we want to find a family Lp of maximally isotropic subspaces for the familyof (rank four) two-forms ωp over Y2 . Then we are going to look at the corresponding maximallyisotropic subspaces Mp = Hom(S, Lp ) for the associated family of quadratic forms. We do notknow whether we have such a family L globally on Y2 , but we can find one Zariski locally (seeRemark 5.4 below).
Let us begin by stating this local data precisely. Suppose we have an affine open set U ⊂ P6
such that over the corresponding open set Y 0 := Y2 ∩ U in Y2 we can find a bundle
L ⊂ OY 0 ⊗ V
of maximally isotropic subspaces for the family of two-forms ωp . As we did for Hom(S, K) above,
let us use the notation
M := Hom(S, L) ⊂ Hom(S, V ) × Hom(V, ∧2 S) − {0}to denote the subvariety
M := Hom(S, L) =

(x, p) : [p] ∈ Y 0 , x ∈ Hom(S, Lp ).

Notice that M is preserved by both GL(S) and C∗R .
The variety M is a vector bundle over the punctured affine cone on Y2 , and its fibres are
maximally isotropic subspaces Mp for the family of quadratic forms Wp . The stack [ M / GL(S) ]is a vector bundle over the stack P|Y2 , which is like Y2 but with SL2 isotropy groups at eachpoint.
Since M lies in the zero locus of W , the skyscraper sheaf OM is a curved dg-sheaf on the LG
B-model (X2 |U , W ). For each point [p] ∈ Y2 , it restricts to give a curved dg-sheaf OMp on thefibre (F, Wp ). As discussed in § 5.1, we claim that these objects are ‘point-like’ in the sense that354

The Pfaffian-Grassmannian equivalence revisitedthey behave like the skyscraper sheaves O[p] ∈ Db (Y2 ). This follows immediately from a slightly
stronger claim, which is that the skyscraper sheaf OM along the whole family behaves like thestructure sheaf on OY2 . This claim is our next proposition.
Recall that π : X2 → P6 is the projection sending (x, p) to [p] ∈ P6 . In our local situation, it
is a map π : X2 |U → U .
Proposition 5.3. Suppose that we have U , Y 0 , L and M as above. Then the natural map
OY 0 −→ π∗ RHomX2 |U (OM , OM )is a quasi-isomorphism.
Proof. The natural map in question is the composition of the natural map OY 0 → π∗ OM withthe pushdown of the identity for OM . This is necessarily non-zero everywhere, so it is enough toprove that π∗ RHomX2 |U (OM , OM ) is quasi-isomorphic to OY 0 .
The curved dg-sheaf OM is the skyscraper sheaf along a smooth subvariety lying in the zero
locus of W . For a curved dg-sheaf of this form, it is easy to show that
RHomX2 |U (OM , OM ) ∼= (∧• NM/X2 , dW ) ;(5.4)

see for example [ASS14, § A.4]. (We neglect some shifts in R-charge which will be irrelevant.) So
we take the sheaf of normal polyvector fields (which would be the correct answer if W were zero)and perturb it by contracting with the section∨dW : OM → NM/X,
2

which is well defined since W vanishes along M . This is not a transverse section, in the sensethat its intersection with the zero section is not transverse. However, we will split it into twopieces, one of which is transverse and the other of which we analyzed earlier.
Since M is a vector bundle over (the punctured affine cone over) Y 0 , we have a short exact
sequence∨
0 → π ∗ NY∨0 /U → NM/X
→ Hom(S, V /L)∨ → 0 .
2 |U

(5.5)

Since the open set U is affine, the total space of M is also affine, so the sequence (5.5) splits:∨∼
NM/X= π ∗ NY∨0 /U ⊕ Hom(S, V /L)∨ .
2

Write dW = (dW )1 ⊕ (dW )2 with respect to this splitting. Then the right-hand side of (5.4) isa tensor product of the Koszul complexes associated to (dW )1 and (dW )2 .
The section (dW )2 consists of the fibre-wise derivatives of the family of quadratic forms Wp , so
as in the proof of Proposition 5.1 it vanishes exactly along the kernel Hom(S, K) of the family of
quadratic forms. Since dim Hom(S, K) = dim Hom(S, L)−dim Hom(S, V /L)∨ , we see that (dW )2
is transverse, so the associated Koszul complex is exact, and we may replace it with OHom(S,K) .
Thus RHomX2 |U (OM , OM ) is quasi-isomorphic to the Koszul complex of the section(dW )1 : OHom(S,K) → π ∗ NY∨0 /U
on the total space of Hom(S, K) over Y 0 . This section is not transverse, but what we actually careabout is π∗ RHomX2 |U (OM , OM ), which we can compute by first pushing down to the underlying
scheme of [ Hom(S, K) / GL(S) ]. By Lemma 5.2 this is the total space of the vector bundle
Hom(∧2 S, ∧2 K) over Y 0 ⊂ Y2 , and now the section (dW )1 is essentially the transpose of (5.3).
By this we mean: the transpose of (5.3) is a map of vector bundles Hom(∧2 S, ∧2 K) → NY∨2 /P6 ,and the corresponding section of the pullback of NY∨2 /P6 to the total space of Hom(∧2 S, ∧2 K)|Y 0355

N. Addington, W. Donovan and E. Segalis (dW )1 . Since (5.3) is an isomorphism, this is a transverse section of the pullback of NY∨0 /P6 thatvanishes along the zero section of Hom(∧2 S, ∧2 K)|Y 0 , so its Koszul complex is quasi-isomorphic
to the structure sheaf of the zero section, and we conclude thatπ∗ RHomX | (OM , OM ) ∼= OY 0 .
2 U

Remark 5.4. A bundle L ⊂ OY2 ⊗ V of maximally isotropic subspaces for the two-forms ωp
may be constructed Zariski-locally on Y2 as follows. Fix a point [x] ∈ Y1 ; this determines a
two-dimensional subspace Im(x) ⊂ V which is isotropic for all ωp . Then over the Zariski open
set where Kp ∩ Im(x) = 0 we can take Lp = Kp + Im(x) ⊂ V . The complement of this open set,that is, the locus where Kp ∩ Im(x) 6= 0, is a curve in Y2 . We remark that this correspondence
between points in Y1 and curves in Y2 is the essential ingredient of [BC09].
We do not know how to find such a bundle L over the whole of Y2 , however, and indeed
we suspect that no such global bundle exists. Consequently, we cannot immediately use theconstruction of Proposition 5.3 to give a global generating object. Fortunately, we know anotherequivalent construction, one which does work globally, as we explain in the next section.
5.4 Another construction of point-like objects
Instead of using a maximally isotropic subbundle, we will use the following subspace.
Definition 5.5. Let Γ ⊂ X2 be the closed substack consisting of points (x, p) where [p] ∈ Y2
and the map
x̄ : S → V /Kp
has rank at most one.
The space Γ is a flat family of stacks over Y2 , as its fibres are all isomorphic. Observe that W
vanishes along Γ, and that Γ is a cone in each fibre of π, hence is C∗R -invariant; therefore OΓ isa curved dg-sheaf on X2 , restricting on each fibre to give a curved dg-sheaf OΓp on F.
As we shall show momentarily, the object OΓp is (approximately) equivalent to OMp , where
Mp is a maximally isotropic subspace of F as in the previous section. The proof is a little involved,
but let us first explain why the result is not so surprising.
The quadratic form Wp on Hom(S, V ) descends to a non-degenerate quadratic form Wp0 on
Hom(S, V /Kp ), so we have a pullback functor
Db (Hom(S, V /Kp ), Wp0 ) → Db (Hom(S, V ), Wp ) .
By definition, Γp is the preimage of the locus of rank one matrices in Hom(S, V /Kp ), and Mp isthe preimage of the maximally isotropic subspace
Hom(S, Lp /Kp ) ⊂ Hom(S, V /Kp ) ,where Lp /Kp is a Lagrangian in V /Kp . Consequently, both OΓp and OMp are pullbacks of objects in Db (Hom(S, V /Kp ), Wp0 ). (We assume for this rough argument that both of these curved
dg-sheaves are equivalent to matrix factorizations.) But Wp0 is non-degenerate, so by Knörrer
periodicity this category is equivalent to the derived category of a point. It is hardly surprising,then, that two natural objects in this category turn out to be isomorphic.
Proposition 5.6. Fix [p] ∈ Y2 . Let Lp ⊂ V be a maximally isotropic subspace for the two-form
ωp = p ◦ A. Then the curved dg-sheaf OΓp is homotopy-equivalent to the curved dg-sheaf
OMp ⊗ det S ⊗ det(Lp /Kp )−1 [−1]356The Pfaffian-Grassmannian equivalence revisited
in QCohdg (F, Wp ).
The term det(Lp /Kp )−1 is a trivial line bundle on F, but will be necessary later when welet p vary.
Recall from Section 2.2 that QCohdg (F, W ) is the dg-category of curved dg-sheaves localizedat ‘acyclic’ objects, and that Perf(X, W ) is the full subcategory of objects that are equivalent tomatrix factorizations. The curved dg-sheaf OMp certainly lies in Perf(F, Wp ), so this proposition
proves incidentally that OΓp also lies in Perf(F, Wp ).
Proof of Proposition 5.6. Consider the locus
Σp := { x ∈ Hom(S, V ) : Wp (x) = 0, dim(Lp + Im(x)) 6 6 } .
It contains both Mp and Γp . It is an intersection of two quadrics in Hom(S, V ): one cut out by
Wp and the other by the determinant of the 2 × 2 matrixxS−→ V → V /Lp .
In fact it is a complete intersection: Wp is a quadric of rank ten, hence is irreducible, and thesecond quadric has rank at most four, hence is different from Wp , so their intersection is complete.
Thus OΣp is the restriction to {Wp = 0} of a perfect complex on F; hence it is a contractible
curved dg-sheaf by § 2.3(iv). So to prove the lemma it is enough to show the equivalence of theideal sheavesIΓ /Σ ∼
(5.6)
= IM /Σ ⊗ det S ⊗ det(Lp /Kp )−1 [1]
p

p

p

p

as curved dg-sheaves on (F, Wp ).
We take the following (SL(S) × C∗R )-equivariant resolution of singularities of Σp :onΣ̃p := (x, l, H) ∈ Hom(S, V ) × PS × Gr(6, V ) : (Lp + Im x) ⊂ H, x(l) ⊂ H ⊥ ,with the evident projection map
φ1 : Σ̃p → Σp .
⊥Here the orthogonal H is taken with respect to the pairing ωp , and since Lp ⊂ H we haveH ⊥ ⊂ L⊥p = Lp ⊂ H. To see that Σ̃p is smooth, observe that the projection
φ23 : Σ̃p → PS × P(V /Lp ) ∼= P1 × P1
defined by φ23 (x, l, H) = (l, H/Lp ) is a vector bundle: the fibre is the ten-dimensional vector
space
n
o
x ∈ Hom(S, H) : x(l) ⊂ H ⊥ .
To see that φ1 is a resolution of singularities, observe that if x ∈ Σp is generic in the sense that
Im(x) 6⊂ Lp , then the fibre φ−1
1 (x) is a single point: clearly H is uniquely determined, but also
x−1 (H ⊥ ) must be one-dimensional, and this determines l.
Now we analyze the non-generic fibres of φ1 , over points x where Im(x) ⊂ Lp . There are threecases:– dim(Im(x) + Kp ) = 5: Then x has rank two. We can choose H freely, and must then set
l = x−1 (H ⊥ ). Thus the fibre is P(V /Lp ) ∼= P1 .
– dim(Im(x) + Kp ) = 4: The fibre has two irreducible components: either H = (Im(x) + Kp )⊥and we can choose l freely, or l = x−1 (Kp ) and we can choose H freely. Thus the fibre357

N. Addington, W. Donovan and E. Segal
is two copies of P1 meeting at a point, more precisely, PS meeting P(V /Lp ) at the point(x, x−1 (Kp ), (Im(x) + Kp )⊥ ).
– dim(Im(x) + Kp ) = 3: We can choose both l and H freely, so the fibre is PS × P(V /Lp ) ∼=P1 × P1 .
Consequently, Rφ1∗ OΣ̃p = OΣp ; that is, Σp has rational singularities.
Next, we consider the preimage of Mp in Σ̃p :noM̃p := (x, l, H) ∈ Σ̃p : Im(x) ⊂ Lp .
This is the union of all the non-generic fibres. The projection φ23 makes M̃p into a rank nine
vector bundle over P1 × P1 , so M̃p is smooth. From the analysis of the fibres given above we knowthat Rφ1∗ OM̃p = OMp . Also, M̃p ⊂ Σ̃p is a divisor, and it is the zero locus of the map
x

S/l = det S ⊗ l−1 −→ H/Lp
which is a section of the line bundle φ∗23 O(−1, −1) ⊗ det S −1 . So we have an exact sequence
0 → φ∗23 O(1, 1) ⊗ det S[−1] → OΣ̃p → OM̃p → 0 .
The R-charge shift occurs because the map x has R-charge 1. Applying Rφ1∗ to the exact sequence
above gives us
Rφ1∗ φ∗23 O(1, 1) ⊗ det S = IMp /Σp [1] .
The final variety we consider is the proper transform of Γp in Σ̃p :noΓ̃p := (x, l, H) ∈ Σ̃p : x(l) ⊂ Kp .
The projection φ23 makes Γ̃p into a rank nine vector bundle over P1 × P1 , so it too is smooth,and a similar inspection of the fibres of φ1 yields Rφ1∗ OΓ̃p = OΓp . The subvariety Γ̃p ⊂ Σ̃p is
also a divisor. It is the zero locus of the mapxl −→ (H ⊥ /Kp ) ∼= (H/Lp ) ⊗ det S −1 ⊗ det(Lp /Kp )(to see the equality here note that the two-form ωp takes values in det S, so H ⊥ is the kernel ofthe map V → H ∗ ⊗ det S induced by ωp , from which it follows that det(H ⊥ ) = det H ⊗ det S −1 ).
Thus Γ̃p is cut out by a section of the line bundle
φ∗23 O(1, −1) ⊗ det S −1 ⊗ det(Lp /Kp )having R-charge one. We take the exact sequence
0 → φ∗23 O(−1, 1) ⊗ det S ⊗ det(Lp /Kp )−1 [−1] −→ OΣ̃p −→ OΓ̃p → 0
and apply Rφ1∗ to get
Rφ1∗ φ∗23 O(−1, 1) = IΓp /Σp ⊗ det S −1 ⊗ det(Lp /Kp )[1] .
Next, take the exact sequence of bundles on PS × P(V /Lp )0 → O(−1, 1) → O(0, 1) ⊗ S → O(1, 1) ⊗ det S → 0
and apply Rφ1∗ φ∗23 to get an exact triangle on Hom(S, V ):IΓp /Σp ⊗ det S −1 ⊗ det(Lp /Kp )[1] −→ Rφ1∗ φ∗3 O(1) ⊗ S −→ IMp /Σp [−1] ,358

The Pfaffian-Grassmannian equivalence revisitedwhere φ3 : Σ̃p → P(V /Lp ) is projection onto the third component. Thus claim (5.6) reduces tothe claim that Rφ1∗ φ∗3 O(1) is a contractible curved dg-sheaf.
To prove this last claim, take the exact sequence of sheaves on P(V /Lp )0 → O → O(1) → OH0 /Lp → 0 ,where OH0 /Lp is the skyscraper sheaf at some point H0 /Lp ∈ P(V /Lp ), and apply Rφ1∗ φ∗3 .
Observe that φ3 is flat since φ23 is. We get an exact triangle on Hom(S, V ):OΣp → Rφ1∗ φ∗3 O(1) → Rφ1∗ φ∗3 OH0 /Lp .
We know that the first term is trivial in Db (F, Wp ), so to show that the second is trivial, it
is enough to show that the third is. Analyzing fibres again we find that Rφ1∗ φ∗3 OH0 /Lp is the
structure sheaf of the locus{ x ∈ Hom(S, V ) : Wp (x) = 0, Im(x) ⊂ H0 } .
This is the complete intersection of the quadric cut out by Wp with the two hyperplanes, so its
structure sheaf is indeed a contractible curved dg-sheaf (by § 2.3(iv) again).
Now suppose that we have a family of maximally isotropic subspaces, as in Proposition 5.3.
Corollary 5.7. Let U ⊂ P6 be a Zariski open set, let Y 0 = U ∩ Y2 , and let
j : X2 |0Y ,→ X2 |U
denote the inclusion. Assume that over Y 0 we have a bundle L ⊂ OY 0 ⊗ V of maximally isotropic
subspaces for ω, and let M = Hom(S, L) ⊂ X2 |Y 0 be the corresponding bundle of maximallyisotropic subspaces for W . Then (possibly after shrinking U ) we have a homotopy equivalenceof curved dg-sheaves
OΓ|Y 0 ' j∗ OM ⊗ det S ⊗ det(L/K)−1 [−1]in QCohdg (X2 |U , W ).
Proof. Assume that U is small enough that X2 |U ∼= F × U , so X2 |Y 0 = F × Y 0 . Then the proofof Proposition 5.6 works perfectly well over the base Y 0 . The only point to note is that we alsoneed to pick a bundle H0 of co-isotropics containing L: we can certainly do this if U is smallenough. Consequently these objects are equivalent as curved dg-sheaves on (X2 |Y 0 , W ), and we
deduce the statement of the corollary by applying the functor j∗ .
So OΓ gives us a global version of our generating object, but in local patches we can continue
to work with maximally isotropic subspaces.
Remark 5.8. A priori it might seem simpler to just use OΓ , and ignore the maximally isotropic
subspaces entirely. Unfortunately, since Γ is singular, it is prohibitively difficult to do the calculation of Proposition 5.3 directly for OΓ .
5.5 Completing the proof
For every [p] ∈ Y2 , we have a curved dg-sheaf OΓp supported on the fibre over p, and these fit
into a global family OΓ . We want to consider a functor
F : Db (Y2 ) → Db (X2 , W )which sends each O[p] to the corresponding OΓp , that is, the functor which has OΓ as its Fourier–
Mukai kernel. From the results in the previous two sections we know that each OΓp behaves359

N. Addington, W. Donovan and E. Segal
like the corresponding skyscraper sheaf O[p] ∈ Db (Y2 ), and this more or less guarantees that F
will be an embedding (cf. Remark 3.3). In this section we fill in the remaining details in thisargument and then show that the image of this embedding is exactly the category BBr(X2 , W )from Section 4.
First, we give the definition of F in full. We consider the diagram
Γ

i

π̂

X2 |Y2

j

π

π

Y2

X2

j

P6

and define
F := j∗ i∗ π̂ ∗ : Db (Y2 ) → [QCohdg (X2 , W )] ,where [QCohdg (X2 , W )] denotes the homotopy category of QCohdg (X2 , W ). Since Γ is flat overY2 , we see that F sends O[p] to OΓp , and it sends OY2 to OΓ .
We claim that F lands in the subcategory Db (X2 , W ) ⊂ [QCohdg (X2 , W )]; that is, everything
in the image is homotopy-equivalent to a matrix factorization. More importantly, F lands in the‘window’ subcategory
BBr(X2 , W ) ⊂ Db (X2 , W ) .
Recall from Section 4 that this is the subcategory where we only allow (objects homotopyequivalent to) matrix factorizations built out of a certain set of vector bundles, namely the ones
corresponding to the ‘rectangle’ (4.1) in the irreducible representations of GL(S).
Proposition 5.9. For all E ∈ Db (Y2 ), we have F E ∈ BBr(X2 , W ).
Proof. It is enough to prove the statement when E is a sheaf. In that case F E is a sheaf on X2 ,which we can write as
F E = j∗ i∗ π̂ ∗ E = j∗ (OΓ ⊗ π ∗ E) .
The sheaf OΓ on X2 |Y2 has an Eagon–Northcott resolution (see, for example, [Wey03, § 6.1.6])
0 → ∧4 (V /K)∨ ⊗ Sym2 S(1) → ∧3 (V /K)∨ ⊗ S(1) → ∧2 (V /K)∨ (1) → O → OΓ → 0 ,and we can make this C∗R -equivariant by inserting the necessary shifts. Consequently, the sheaf
F E has a C∗R -equivariant resolution on X2 of the form
0 → π ∗ F3 ⊗ Sym2 S → π ∗ F2 ⊗ S → π ∗ F1 → π ∗ F0 → F E → 0 ,where F0 , . . . , F3 are sheaves on P6 , supported on Y2 . Every sheaf on P6 can be resolved by theline bundles O, . . . , O(6), so F E has a resolution by vector bundles lying in our rectangle (4.1).
Now Lemma 4.10 implies that F E lies in the subcategory BBr(X2 , W ) ⊂ Db (X2 , W ).
Next, we need to establish that F has a right adjoint. The functor(ji)∗ : Db (Γ) → Db (X2 , W )has a right adjoint, namely(ji)! = RHomX2 (OΓ , −) : Db (X2 , W ) → Db (Γ) .
Note that this statement is local, and that locally OΓ generates Db (Γ), so to prove the adjunction
it is sufficient to observe that
RHomX2 ((ji)∗ OΓ , F) = RHomΓ (OΓ , RHomX2 (OΓ , F))

360

The Pfaffian-Grassmannian equivalence revisited
holds tautologically for any F ∈ Db (X2 , W ).
The right adjoint to the functorπ̂ ∗ : Db (Y2 ) → Db (Γ)
should be π̂∗ , but unfortunately Γ is not proper (not even equivariantly), so π̂∗ produces quasicoherent sheaves in general. Fortunately, we have the following result.
Lemma 5.10. For E ∈ Db (X2 , W ), the complex of sheaves π∗ RHomX2 (OΓ , E) has bounded andcoherent homology sheaves. Consequently,
F R := π∗ RHomX2 (OΓ , −) : Db (X2 , W ) → Db (Y2 )is right adjoint to F .
Proof. The homology of RHomX2 (OΓ , E) is a coherent sheaf whose support lies in the criticallocus Crit(W ) of W (see § 2.3(iii)), and also in π −1 (Y2 ). From Lemma 5.2, the map
π : Crit(W )|Y2 → Y2
is just passage to the underlying scheme, so if F is a sheaf supported in Crit(W ), then (locallyin Y2 ) we calculate π∗ F by just taking SL(S)-invariants. If Y 0 ⊂ Y2 is a sufficiently small open set,
then over π −1 (Y 0 ) we can find a finite presentation of the homology of RHomX2 (OΓ , E). Taking
SL(S)-invariants, which is an exact functor, gives us a finite presentation of the homology ofπ∗ RHomX2 (OΓ , E).
From their definitions, both F and F R are ‘local’ over P6 , that is, linear over the structure
sheaf of P6 . Consequently, if U ⊂ P6 is an open set, then we have functors
Db (Y2 ∩ U )F
FR

Db (X2 |U , W ) .

Proposition 5.11. The functor F is fully faithful.
Proof. We will show that for any E ∈ Db (Y2 ), the unit of the adjunction
E → F RF E
is an isomorphism. Then the composition F R F is naturally isomorphic to the identity functor,
and so F must be an embedding.
This statement is local in Y2 , so we can restrict to an affine open subset U ⊂ P6 and corresponding open set Y 0 = Y2 ∩ U . Then it is enough to check the statement on the structure sheaf
OY 0 , since this generates Db (Y 0 ). So the required statement is that
F : OY 0 → π∗ RHomX2 |U (OΓY 0 , OΓY 0 )is a quasi-isomorphism.
By making U smaller if necessary, we may assume that we have a bundle of maximallyisotropic subspaces L ⊂ VY 0 , and an associated bundle M = Hom(S, L) ⊂ X2 |Y 0 . Then by
Proposition 5.6 we may replace OΓY 0 with OM , up to a shift and twisting by a line bundle. Thenπ∗ RHomX2 |U (OΓY 0 , OΓY 0 ) ∼= OY 0by Proposition 5.3. Finally, F must be an isomorphism on homology because it must send theconstant section 1 to itself (it preserves identity arrows), and it is linear over sections of OP6 .

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N. Addington, W. Donovan and E. Segal
Theorem 5.12. The functor
F : Db (Y2 ) → BBr(X2 , W )is an equivalence.
Proof. By Proposition 5.11 and Proposition 5.9, the functor F is an embedding from Db (Y2 )into BBr(X2 , W ), and by Lemma 5.10 it has a right adjoint. We also know that BBr(X2 , W ) is
equivalent to Db (Y1 ), by composing our equivalences Ψ1 and Ψ2 . However, Y1 is Calabi–Yau andconnected, so BBr(X2 , W ) cannot have a non-trivial admissible subcategory, and we deduce theresult.
So the equivalence Ψ3 holds. This last step of the argument is rather unsatisfactory in that
we have to appeal to our other two equivalences, rather than giving a self-contained proof. But
presumably it is possible to prove directly that BBr(X2 , W ) is Calabi–Yau and connected; inparticular, the Calabi–Yau property should follow by an argument along the lines of [LP13, § 4].
Remark 5.13. We conclude with some remarks about how our results adapt when we change thedimensions of S and V to r and d, respectively.
– r = 2, d = 5. In this case Y1 is an elliptic curve and Y2 is the dual elliptic curve. We havea very similar definition of BBr(X2 , W ) (see Remark 4.8), we have equivalences Db (Y1 ) ∼=Db (X1 , W ) ∼= BBr(X2 , W ) as before, and the methods of this section can be used to show
Db (Y2 ) ∼= BBr(X2 , W ). In fact, this case is rather easier than the d = 7 case because it isvery easy to show that we have a global maximally isotropic subbundle L on Y2 , and so we
do not need any alternative construction as in Section 5.4.
– r = 2, d = 6. In this case Y1 is a K3 surface and Y2 is a Pfaffian cubic fourfold. We candefine BBr(X2 , W ) as in Remark 4.8, and the arguments of Sections 3 and 4 show that
Db (Y1 ) embeds into BBr(X2 , W ). The methods of this section apply essentially verbatim
to show that Db (Y2 ) also embeds into BBr(X2 , W ); the only change is that the bundle K
over Y2 now has rank two, and so
Hom(S, K)/ GL(S) ∼= Hom(∧2 S, ∧2 K)
is still a vector bundle, but of rank one. Fortunately Y2 now has codimension one in P5 , andso (5.3) is still an isomorphism. Unfortunately we do not have a proof that this second embedding is actually an equivalence; that is, we have no analogue of Theorem 5.12. If we did,we would recover Kuznetsov’s result that Db (Y1 ) embeds into Db (Y2 ) [Kuz06, Theorem 2].
– r = 2, d = 9. In this case both Y1 and Y2 are smooth Calabi-Yau fivefolds, and we have twothirds of a proof that they are derived equivalent. As before we have Db (Y1 ) ∼= Db (X1 , W ) ∼=BBr(X2 , W ), and Proposition 5.3 still holds, but we do not know how to construct a kernel
globally over Y2 (since OΓ only works for d = 6 or 7).
– r = 2, d = 8 or d > 9. We do have a category BBr(X2 , W ), but Y2 is necessarily singular,
so our calculations with maximally isotropic subspaces show that BBr(X2 , W ) is in somesense a non-commutative resolution of Db (Y2 ). Indeed, we speculate that the homologicalprojective dual to Gr(2, V ) is the non-commutative resolution of the Pfaffian locus Pf ⊂P Hom(∧2 V, ∧2 S) constructed as follows: take the stackh.
i(x, ω) ∈ Hom(S, V ) ⊕ Hom(∧2 V, ∧2 S) : ω 6= 0
GL(S)with the superpotential
W (x, ω) = ω ◦ ∧2 x ,362

The Pfaffian-Grassmannian equivalence revisitedand take the subcategory of matrix factorizations built from the vector bundles
h
n

 o

Syml S ∨ ⊗ (det S ∨ )m : l ∈ 0, 12 (d − 1) , m ∈ 0, d2
when d is odd, or
n

h
 o


Syml S ∨ ⊗ (det S ∨ )m : l ∈ 0, 12 d , m ∈ 0, d2

when d is even. Of course one would like to begin by checking that this is equivalent toKuznetsov’s non-commutative resolution of Pf when d = 6 and 7 [Kuz06].
– If r > 2, then it is not clear to us how to proceed.
Acknowledgements
We thank Matt Ballard for helpful discussions, Ronen Plesser for helpful comments, and theanonymous referee for a thorough reading and many useful suggestions.
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Nicolas Addington adding@math.duke.edu
Department of Mathematics, Box 90320, Duke University, Durham, NC, 27708-0320, USA
Will Donovan will.donovan@ipmu.jp
Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for
Advanced Study, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan
Ed Segal edward.segal04@imperial.ac.uk
Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom364