Birational Calabi-Yau 3-folds are known to have equivalent derived categories [Bri] . There also exist pairs of Calabi-Yau 3-folds which are not birational but may be proved to be derived equivalent. A much-studied class of examples comes from the 'Pfaffian-Grassmannian' correspondence, concerning pairs of 3-folds arising as linear sections of the Grassmannian G(2, 7) and its projective dual Pfaffian. Rødland conjectured that such pairs share a mirror [Rød] , leading to an expectation that they are derived equivalent: this was proved by Borisov and Cȃldȃraru [BC] , and Kuznetsov [K1] .
Meanwhile, Hori and Tong [HT] gave a physical explanation of how such pairs of 3-folds arise from the same gauged linear σ-model. In work of Addington, Segal, and the author [ADS] , a partial mathematical interpretation of this was given by constructing a particular derived equivalence for each pair using categories of matrix factorizations.
According to mirror symmetry, the derived symmetries of a variety may be determined by monodromy on a stringy Kähler moduli space (SKMS). Hori and Tong described this space for such 3-folds: implicit in this was a prediction of further equivalences, corresponding to 'grade restriction windows'. Later physics work made these windows explicit, and argued that differences between equivalences are given by spherical twists: see Eager, Hori, Knapp, and Romo [EHKR] , and Hori [H, end of Section 5] .
2010 Mathematics Subject Classification. Primary 14F08; Secondary 14J32, 14M15, 18G80, 81T30. I am supported by the Yau MSC, Tsinghua University, and the Thousand Talents Plan. I also acknowledge the support of WPI Initiative, MEXT, Japan, and of EPSRC Programme Grant EP/R034826/1.
In this paper, I interpret this physics work by constructing, for each pair of 3-folds coming from the Pfaffian-Grassmannian correspondence, a set of equivalences corresponding to the 'windows' above by extending the methods of [ADS] . I then show that these equivalences, along with appropriate spherical twists, may be organized into an action of the fundamental group of the relevant SKMS.
1.1. Calabi-Yau pairs. I recall the construction of the 3-folds Y G and Y P . Start with a 7-dimensional vector space V , and consider the following.
• The Grassmannian of 2-planes in its Plücker embedding G(2, V ) ⊂ P(∧ 2 V ).
• The Pfaffian of 2-forms on V of rank at most 4, denoted P(4, V ) ⊂ P(∧ 2 V ∨ ).
These varieties are projectively dual. The Pfaffian P(4, V ) is singular along the locus of forms of rank at most 2, but taking sufficiently generic hyperplane sections yields a smooth Calabi-Yau 3-fold Y P . Taking a dimension 7 subspace Π ⊂ ∧ 2 V ∨ and its annihilator Π • ⊂ ∧ 2 V , we then obtain smooth Calabi-Yau 3-folds as follows. 1 Y G = G(2, V ) ∩ PΠ • Y P = P(4, V ) ∩ PΠ 1.2. Equivalences. For each pair Y G and Y P , I construct a set of derived equivalences depending on a discrete parameter given as follows. m = (m 0 , m 1 , m 2 ) ∈ Z 3 such that m l ≤ m l+1 ≤ m l + 1 Each choice of m gives an exceptional collection on G(2, V ) by successive mutations of a collection due to Kuznetsov (Proposition 2.2) . This collection is determined by a Lefschetz block
where S is the rank 2 tautological subspace bundle on G(2, V ). Now, by extending the construction of [ADS] , I prove the following.
Theorem A (Theorem 5.7). For each m ∈ Z 3 as above, an equivalence
may be constructed using the exceptional collection on G(2, V ) given by m.
Section 3 outlines how Ψ m is constructed: it is a 'window equivalence', where generators are obtained from the collection given by m. The equivalence obtained in [ADS] corresponds to m = (6, 7, 8) (Remark 4.3).
1.3. Physics. The SKMS in our case is described in the physics literature as a sphere with five punctures: see [HT, Figure 1 ] and [EHKR, Section 4] . Our 3-folds correspond to 'large radius limits' near two of these punctures, shown below as the poles. In this picture, homotopy classes of paths between large radius limits are expected to correspond to derived equivalences, given by 'grade restriction windows'. Such equivalences are supplied by Theorem A. Monodromy around the other three punctures are then expected to correspond to spherical twists [H, end of Section 5] . Theorems B and C below confirm these expectations from the physics literature.
1.4. Groupoid action. I show that the fundamental groupoid of the SKMS acts on the derived categories D b (Y G ) and D b (Y P ).
Let M = S 2 − {5 points} where two of the punctures are the poles, and basepoints m G and m P are chosen near them. Following the physics analysis [EHKR, Section 4] , I choose a finite subset Ψ k , 0 ≤ k ≤ 3, of the equivalences Ψ m of Theorem A as follows.
The corresponding exceptional collections are illustrated in Section 2 as Collections 0-3. Combining the construction of Theorem A with standard techniques for manipulating window equivalences, I obtain the following.
Theorem B (Theorem 6.3). There is an action of the fundamental groupoid π 1 (M, {m G , m P }) on D b (Y G ) and D b (Y P ), given by the following diagram.
1.5. Mutations and group action. The exceptional collections for the Ψ k are related by mutations of exceptional objects as follows.
The restrictions of these three exceptional objects to Y G are the three spherical objects below (Proposition 7.6).
Remark. This phenomenon is analogous to exceptional objects restricting to spherical objects on an anticanonical divisor [ST2, Example 3.14(c) ], though here the codimension of Y G is 7.
I then show (Proposition 7.7) that the 'differences' between equivalences
are spherical twists around the objects ( * ). Combining this with Theorem B, I deduce the following.
Theorem C (Theorem 7.8). There is an action of the fundamental group
Here we let Ψ = Ψ 3 , and the loops around equatorial holes indicate spherical twists around the objects ( * ).
It would be interesting to recover M, and the above actions, from an analysis of Bridgeland stability conditions. 1.6. Contents. Section 2 explains the exceptional collections in Theorem A, Section 3 outlines the structure of the proof, and then Sections 4 and 5 give the details. Section 6 proves Theorem B, and Section 7 proves Theorem C.
Acknowledgements. In celebration of his 77th birthday, I am pleased to express my gratitude to Kyoji Saito for his kindness and interest over the years.
I also want to thank N. Addington and E. Segal for the great experience of working on our paper [ADS] , of which this work is a continuation. I am grateful for conversations with K. Hori, who made me aware of the windows used here, and for discussions with R. Eager, A. Kuznetsov, and M. Romo. In the last stage of this project, I was away from my home institution due to the coronavirus pandemic, so I am especially appreciative of hospitality and support from Kavli IPMU, in particular Y. Ito and M. Kapranov, and from B. Kim at KIAS, and M. Wemyss at Glasgow University.
In this section, I explain the exceptional collections in Theorem A, and convenient ways to visualize them.
The construction in [ADS] used the following full exceptional collection on the Grassmannian G(2, V ) for a 7-dimensional vector space V [K2, Theorem 4.1], where S denotes the rank 2 tautological subspace bundle.
(2.A) I depict the collection (2.A) as follows, where as usual we write O(1) for det S ∨ , and for convenience in this section put S 2 for Sym 2 S.
Remark 2.1. The diagonal display format matches with [H] and is used later to visualize the direction of Homs in the collection: see Remark 4.10.
It will be helpful instead to take a dual exceptional collection as follows. 2 Collection 0.
From this collection, we deduce others by mutation. Namely, fixing an order for the collection E 1 , . . . , E n , it is standard that mutating E n from right to left gives
where the R i denote right mutation functors, and S is the Serre functor
Mutating in this way at O, and noting that the property of being an exceptional collection is invariant under the shift [dim G], we obtain the following.
Mutating at S then gives:
We now have two choices for how to mutate, at O(−1) or S 2 .
Mutating at S 2 gives a twist by O(−1) of Collection 0:
Collection 3.
S(−7)
S 2 (−7)
· · · · · · O(−1)
Mutating at O(−1) gives:
Noting then that being a full exceptional collection is invariant under twisting by O(k), we obtain further collections summarized in the following. gives a full exceptional collection on G(2, V ), for dim V = 7.
3. Structure of equivalence proof I outline the proof of the derived equivalences Ψ m between Y G and Y P in Theorem A, before giving the proof in the following Sections 4 and 5.
These derived equivalences are obtained by showing that Y G and Y P are derived equivalent, in an appropriate sense, to Landau-Ginzburg models (X G , f ) and (X P , f ).
Here the space X G is a bundle over G(2, V ), the space X P is an Artin stack which is described in the next section, and f is a function defined on both these spaces: f arises by restriction of a function from an Artin stack X, of which X G and X P are open substacks. This follows a physics construction of Hori-Tong [HT] , where this function is a 'superpotential'.
Each equivalence Ψ m is a composition of three equivalences as below, where D b (X G , f ) is a category of matrix factorizations (see for instance [ADS, Section 2] ), and Br m (X P , f ) is a closely related 'B-brane category' (Definition 4.5).
The vertical arrows are constructed as in [ADS] , using that the spaces Y may be recovered from the spaces X. In particular, Y G is the critical locus of f in X G , giving the left-hand equivalence Ψ G via Knörrer periodicity [Shi] . The right-hand equivalence Ψ P is a generalized Knörrer periodicity, established in [ADS, Section 5] .
The horizontal equivalence Ψ m W is a 'window equivalence', which factors via a certain subcategory in D b (X, f ), determined by one of the exceptional collections from Proposition 2.2. This is explained and proved in the following Section 4. The rest of the proof is then given in Section 5.
Remark 3.1. There is a general theory of window equivalences [HL, BFK] , however it does not yet apply in this case. In particular, it gives exceptional collections related to that of Kapranov [Kap] rather than those in Proposition 2.2: this seems to make it unsuitable for giving an equivalence with the Calabi-Yau Y P . For more discussion, see [ADS, Remark 4.12] .
Remark 3.2. In the physics literature [EHKR, Section 7] a grade restriction window for m = (7, 7, 7) is associated with the derived equivalence of Borisov and Cȃldȃraru [BC] . It would be interesting to compare the latter with the Ψ m obtained in this paper.
In this section, I construct the window equivalences Ψ m W used to prove Theorem A, after reviewing the construction of X G and X P , the underlying spaces of the Landau-Ginzburg models discussed above.
Let S be a 2-dimensional vector space, and consider the quotient stack
The open substack of G of rank 2 homomorphisms is equivalent to the variety G(2, V ). Furthermore, the vector bundle on G induced by the representation S of GL(S) restricts to the tautological subspace bundle on G(2, V ). The representation det S ∨ similarly corresponds to the bundle O(1) on G(2, V ).
Take the quotient stack
Representations of GL(S) also give vector bundles on X, and its substacks.
In particular, the representation S gives a bundle on X, and the notation S will also be used to denote this bundle. For brevity, we write the line bundle corresponding to the representation (det S ∨ ) ⊗k as O(k), by analogy with the notation on G.
be open substacks of X. These have the structure of a vector bundle over G(2, 7) and an Artin stack P respectively, where
We let f denote a function on the stack X, and use the same notation for its restriction to substacks. As the particular form of f is not used in this section, we defer its definition to the following Section 5.
The following categories correspond to the collections of Proposition 2.2.
Definition 4.1 (window subcategories W m ). Let m = (m 0 , m 1 , m 2 ) be a non-decreasing sequence of integers such that m l+1 ≤ m l + 1. Then
gives a set of bundles on X, and we write W m for the full subcategories of D b (X) and D b (X, f ) generated by this set.
For brevity, we notate bundles as follows.
Notation 4.2. Let S l,m denote the bundle Sym l S(m) on X, or its restriction to a substack.
Remark 4.3. In [ADS] a set of generators T l,m = Sym l S ∨ (m) was used. Note that T l,m ∼ = S l,m+l , using that S ∨ ∼ = S(1) because S ∨ ∼ = S ⊗ det S ∨ . The window subcategory used in [ADS] , with generators given in (2.A) above, is therefore the window W m with m = (6, 7, 8) in our notation here.
We write embeddings of stacks as follows.
is an equivalence, and the derived functor
Proof. To first prove fully faithfulness, we generalize the argument given in [ADS, Lemma 4.3] . It suffices to check fully faithfulness on generators: take then two of the generators S l,m and S l ′ ,m ′ of W m . The Homs between these are equal to the Homs between their restrictions to X G and X P , respectively, because the complements of the latter have codimensions at least 2. Furthermore, there are no higher Exts between them because they are vector bundles on an affine stack, so it suffices to check that they do not acquire any higher Exts after applying the functors i * . Namely, we require the following.
These vanishings are proved in Subsections 4.1 and 4.2 respectively.
To complete the proof, we need that i * G is essentially surjective. For this, note that the generators S l,m when considered as sheaves on G(2, 7) give generators for the derived category by Proposition 2.2. Essential surjectivity then follows by a general argument, exactly as in [ADS, Lemma 4.6 ].
Using the above proposition we have the following.
Definition 4.5 (B-brane categories). We define a subcategory
Remark 4.6. In other words, Br m (X P , f ) is the full subcategory of D b (X P , f ) generated by the bundles given by (4.A).
Definition 4.7 (window equivalences). We write
4.1. Grassmannian side. I start with some vanishing results on G(2, V ), before proving the vanishing (4.B) on X G . I recall the following, again writing T l,m = Sym l S ∨ (m).
Lemma 4.8. [ADS, Lemma 4.5 
We quickly obtain the following, where again S l,m = Sym l S(m).
Corollary 4.9. In the setting of Lemma 4.8 above, so that in particular we have m ′ ≥ m,
Proof. For the first, we exchange l and l ′ , and note that the result is isomorphic to the Ext-group in Lemma 4.8 above, using local freeness of the Syms.
For the second, we use that T l,m ∼ = S l,m+l , and the local freeness of O(1).
Remark 4.10. In terms of the following picture, this corollary says that there are no higher Exts on G(2, V ) from an object on one the lines pictured, say L, to any object on a parallel line L ′ lying to the right of L.
· · · · · · I now show the vanishing (4.B). By a standard calculation using the structure of X G as a vector bundle over G, we have
By expanding the Sym n piece into irreducibles we see that this splits into summands RHom G S l,m , S l ′ ,m ′ +n where S l,m , S l ′ ,m ′ ∈ W k and n ≥ 0. It therefore suffices to know that there are no higher Exts from the generators S l,m of W m to other generators S l ′ ,m ′ of W m or to sheaves S l ′ ,m ′ +n 'to their right'. But this is clear from the form of the exceptional collections in Section 2, combined with Corollary 4.9 above.
4.2. Pfaffian side. We now show the vanishing (4.C). Similarly to the above, by a standard calculation we have
We check the following very straightforward bound. This will give the vanishing for the Sym 0 piece: the vanishing for the Sym n piece with n > 0 will then follow by induction.
Lemma 4.11. For two generators S l,m and S l ′ ,m ′ of W m we have
Notice then that
Recall that the m l form a non-decreasing sequence with m l+1 ≤ m l + 1. If l ′ ≥ l then m l ′ −m l ≤ l ′ −l, and otherwise m l ′ −m l ≤ 0, hence the claim.
Remark 4.12. The collections W (0,0,0) and W (−2,−1,0) (given as Collections 0 and 4 in Section 2) show that, amongst such bounds which are uniform for all the W m , the bound of Lemma 4.11 is the best possible.
For our induction, we now observe that
Now suppose we use this repeatedly to expand the Sym n piece, namely RHom P S l,m , S l ′ ,m ′ ⊗ Sym n S ⊕7 .
RHom P (S l,m , S l ′′ ,m ′′ ) (4.E)
for which the inequality m ′′ − m < max(l ′′ − l, 0) + 7.
(4.F) is satisfied. This follows by induction: the base case n = 0 is shown in Lemma 4.11, and the inequality still holds when we increment l ′′ or decrement m ′′ each time we apply (4.D).
Let us now calculate the cohomology of the summands (4.E). First note
Applying RΓ P to this, we will obtain RHom P ∼ = RΓ P •RHom P . Observe then that there is a morphism of stacks δ : P → P 6 induced by det : GL(S) → C * , and we have RΓ P ∼ = RΓ P 6 •δ * . The functor δ * takes ker(det)-invariants where ker(det) = SL(S). Expanding Sym l S ⊗ Sym l ′′ S into irreducibles, the only sheaf that survives this is (∧ 2 S) ⊗l ∼ = O(−l) in the case that l = l ′′ . We thence deduce that the only contribution to the cohomology of (4.E) comes from RΓ P O(−m+m ′′ ) in the case l = l ′′ and that, setting n = m ′′ −m, this contribution is
(4.G)
But (4.F) implies that if l = l ′′ then n < 7, so the required vanishing of higher Exts in (4.C) follows from standard cohomology vanishing on P 6 . Remark 4.14. The last part of the argument above, the calculation of the cohomology of the summands (4.E), parallels the proof of [ADS, Lemma 4.3] . We include it here to explain some details, and for convenience.
We now show that the categories D b (X G , f ) and Br m (X P , f ) are equivalent, respectively, to the derived categories of our Calabi-Yau 3-folds Y G and Y P , completing the proof Theorem A.
In this section we require the definition of the function f on X from [ADS] , as follows. Take a surjective map A : ∧ 2 V → V such that the dual subspaces Π and Π • used to constructed the 3-folds in Section 1.1 are given by Π = ker A ∨ and Π • = Im A.
Recalling that X = Hom(S, V ) ⊕ Hom(V, ∧ 2 S) GL(S) with x ∈ Hom(S, V ) and p ∈ Hom(V, ∧ 2 S) take a function on X defined by
via the canonical isomorphism Hom(∧ 2 S, ∧ 2 S) ∼ = C.
In addition to the data (X, f ), the Landau-Ginzburg model includes the data of a C * -action on X for which f has weight 2. For this, we let C * act with weight 0 on x, and weight 2 on p. Note that this action has been suppressed in our notation so far. Following physics terminology, it is sometimes know as the 'R-charge'. For more details, see [ADS, Section 2.1].
Recall that X G is by definition the Artin stack
which is isomorphic to a variety as follows
In other words, it is the total space of the vector bundle O(−1) ⊕7 over G(2, V ). Denote the projection morphism for this bundle by π. The Calabi-Yau 3-fold Y G defined in Section 1.1 is the zero locus of a transverse section of O(1) ⊕7 given by
Then we have f = sp, where p is the tautological section of π * O(−1) ⊕7 . The C * -action preserve each fibre of the bundle, and acts with weight 2 on those fibres. Given this geometric situation, an equivalence from D b (Y G ) to D b (X G , f ) follows from a version of Knörrer periodicity, as follows. Let X G | Y G denote the base change of the bundle X G over G to the base Y G . Then we have
where k is the inclusion.
Definition 5.1. We define an equivalence Ψ G by the composition
This is an equivalence by a result of Shipman [Shi, Theorem 3.4 ].
Calculations. Later, in Section 7, I study certain bundles on Y G to prove Theorem C: for use there, in this subsection I calculate the images of these objects under Ψ G . I first give the method in a simpler case, before applying it to Y G .
Example 5.2. Consider a line bundle L over a variety B, and a subvariety Y ⊂ B cut out by a transverse section s of L. Let X be the total space of L ∨ , with projection π, and p the tautological section of π * L ∨ . We make the setup C * -equivariant so that s and p have weights 0 and 2 respectively. Take notation as follows, where X| Y is the total space of the base change of the bundle L over B to Y , and j is the inclusion of the zero section B in X.
Then in D b (X, f ), where f = sp, we have an object K given by
where the shift [−1] denotes a change in C * -weight. Forgetting the rightand left-moving morphisms respectively in K gives Koszul resolutions for C * -equivariant sheaves
Note that these determine objects of D b (X, f ) because they are supported on the zero locus of f , namely X| Y ∪ B. Indeed, they may be written as pushforwards from this locus using isomorphisms
which follow by flat base change, and the projection formula for j, respectively. Now, by a standard argument with matrix factorizations, both the objects of (5.A) are isomorphic to K in D b (X, f ): see for instance [Shi, proof of Lemma 3.2] . Putting all this together, we obtain an isomorphism in D b (X, f ) as follows.
We generalize the method of this example to deduce the following.
Proposition 5.3. We have that
where j is the inclusion of the zero section of X G , and similarly with occurences of O replaced with S or Sym 2 S.
Proof. Take F = O(1) ⊕7 , a bundle on G(2, V ) of rank r = dim V = 7, and set notation for morphisms as follows.
Following the argument of the example above, we have an object K of D(X G , f ) written 
Recalling that j ! = j * (ω j [dim j] ⊗ −), and noting that ω j = det F ∨ and dim j = −r, the first claim follows. Twisting K by S or Sym 2 S then gives the further claim.
Recall that X P is the Artin stack
We then have that f (x, p) = ω p • ∧ 2 x by the definition of f .
In this subsection, we construct an equivalence
for each choice of m. In [ADS] , such an equivalence was constructed for m = (6, 7, 8) by considering isotropic subspaces for the ω p as p varies. An outline is given in [ADS, Section 5 .1] with details in the subsections following. The argument is lengthy and subtle, but to obtain an equivalence for general m I only need to modify the very last step of it: this subsection explains the modification.
The argument uses the following locus in X P . Note that X P is a bundle over P and also over P 6 by composition with the morphism δ : P → P 6 induced by det : GL(S) → C * .
Definition 5.4. [ADS, Definition 5 .5] Let Γ ⊂ X P be the closed substack with points (x, p) where p corresponds to a point of Y P ⊂ P 6 , and the composition of x with a quotient morphism as follows S x −→ V → V / ker ω p has rank at most 1.
Note that Γ is a flat family of stacks over Y P . The following proposition gives the construction of Ψ m P . Take notation as below, where j and k denote inclusions.
Proposition 5.5. There exists an equivalence
Proof. I explain how choices can be made so that the given composition has essential image in Br m (X P , f ). This refines the argument of [ADS, proof of Proposition 5.9 ]. I first explain why, for E a sheaf on Y P , we may make choices so that Ψ m P (E) ∈ Br m (X P , f ). The result then follows by generation. Using that πj is flat, Ψ m P (E) is a sheaf on X P . Furthermore, by the projection formula, Ψ m
where Q = V / ker ω p , by for instance [Wey, Section 6.1.6 ]. This may be made C * -equivariant by adding appropriate shifts of C * -weight, and it follows that Ψ m P (E) has a C * -equivariant resolution π * F 2 ⊗ Sym 2 S → π * F 1 ⊗ S → π * F 0 → π * F where F and F 0 , . . . , F 2 are sheaves in the image of j * : Coh(Y P ) → Coh(P 6 ). Using a Beilinson collection on P 6 , each of these F s can be replaced by a quasi-isomorphic complex whose terms are direct sums of line bundles O P 6 (−m) with m ∈ (n − 7, n], for any choice of integer n. Now π * O P 6 (−m) ∼ = O(m), with the sign appearing as in Remark 4.13. Therefore, we may obtain a resolution of Ψ m P (E) by vector bundles in the window subcategory W m , by letting n = m l for F l , and n = m 0 for F . Doing likewise for E ranging over a set of generators for D b (Y P ), it follows that Ψ m P may be constructed to have essential image in Br m (X P , f ). The proof of the equivalence property is exactly as in [ADS, Theorem 5.12 ].
Remark 5.6. A different construction of such a functor is given in [ST1, Section 4.2] , avoiding much of the difficult analysis in [ADS] . However, the discussion in the latter suffices for our purposes. 5.4. Equivalences. We now complete the proof of Theorem A.
As before, let m = (m 0 , m 1 , m 2 ) be a non-decreasing sequence of integers such that m l+1 ≤ m l + 1.
Theorem 5.7. For each m as above, there is an equivalence
which factors through the subcategory W m of D b (X, f ).
Proof. Recall that in the previous section we obtained (Definition 4.7) an equivalence as follows.
Composing this with the equivalences of this section, from Definition 5.1 and Proposition 5.5, we get
and the theorem is proved.
In this section I construct the groupoid action of Theorem B. Let us first recall the notation which was used to state this theorem in Section 1.4. Notation 6.1. For brevity, we write windows, and equivalences,
and similarly write Ψ l W for l = 0, . . . , 3.
Theorem B will follow from the fact that W 3 = W 0 ⊗ O(−1), and the following proposition.
Proposition 6.2. There are natural isomorphisms as follows
Proof. Recall from Definition 5.1 that Ψ G is the composition
where morphisms are as follows.
Now assume given sheaves L on X G and M on Y G such that k * L ∼ = π * M. Then, using the projection formula, we have isomorphisms as follows.
We immediately deduce the first statement, by taking L = O X G (1) and
From Proposition 5.5 we have that Ψ m P is isomorphic to the composition
where morphisms are as shown below. Here we write D b (Γ) rather than D b (Γ, f ) because f restricts to zero on Γ ⊂ X P by Definition 5.4.
Applying a similar argument to Ψ P using sheaves L = O X P (1) and M = O Y P (−1), we deduce the second statement. In this case it suffices that k * L ∼ = π * M, as this gives (kj) * L ∼ = (πj) * M after applying j * . The minus sign here arises as in Remark 4.13. Theorem 6.3. There is an action of the groupoid π 1 (M, {m G , m P }) for M = S 2 −{5 points} on D b (Y G ) and D b (Y P ), given by the following diagram.
Proof. Removing a point from the far side of the sphere, the claim is that the groupoid acts by the following diagram, subject to a relation: that monodromy starting at either D b (Y G ) or D b (Y P ) and then going around a large circle gives the identity up to isomorphism.
Now we have that W 3 = W 0 ⊗ O X (−1), and note that 
, noting the opposite signs in the line bundle twists. Combining with the above Proposition 6.2 then given an isomorphism
which yield the required relation, and completes the proof.
In this section I describe the differences between the equivalences Ψ l in Theorem B (Theorem 6.3 above) as spherical twists on the Calabi-Yau Y G , and thereby prove Theorem C.
To describe the difference Ψ ′−1 •Ψ between such equivalences Ψ and Ψ ′ , we will use a functor Tr that 'transfers' between the two corresponding windows W and W ′ . The difference is then described as a twist functor, denoted Tw. The following proposition gives the formal part of this procedure.
For convenience, take two functors from W = W m as follows
so that the factorization of Ψ = Ψ m via W takes the form
Proposition 7.1. [DS, Proposition 2.2] For windows W and W ′ , assume given a functor Tr : W → W ′ that intertwines with an autoequivalence Tw of D b (Y G ) and with the identity on D b (Y P ), namely
Then there is an isomorphism
where Ψ and Ψ ′ are the window equivalences associated to W and W ′ .
Proof. This is a diagram chase of functors. In the reference Tr was assumed to be a restriction of an endofunctor of the category containing W, in our case D b (X, f ), but the proof proceeds without this assumption.
To construct functors Tr we define endofunctors of D b (X, f ), and then show that they restrict to functors between windows. Recall that X is a vector bundle over a stack G = [ Hom(S, V ) / GL(S) ] and write j : G → X for the inclusion of the zero section. Recall also that
where we use that ω j is the determinant of the normal bundle of G.
Definition 7.2. Take endofunctors of D b (X, f )
Here, and elsewhere in this section, we again put S 2 for Sym 2 S.
Remark 7.3. The functorial cone above is shorthand for the usual Fourier-Mukai constructions.
I then claim the following.
Proposition 7.4. The Tr l restrict to functors Tr l : W l → W l+1 .
Proof. I first show this for Tr 0 , as the others are similar. I claim that the functor acts as the identity on all generators of the window W 0 except O. First observe that
This is zero except on O by inspection of Collection 0, noting the vanishing in Corollary 4.9. It also implies, using that objects in the collection are exceptional, that
A generator of this Hom may be seen explicitly by writing down the Koszul resolution of j ! O G , namely
We thus see that Tr
which, in particular, lies in the window W 1 given as Collection 1.
For the other two functors Tr l for l = 1, 2, we replace (7.B) with Hom X j ! S l , − ∼ = Hom G S l , j * − and repeat the same argument, where we tensor (7.C) by S l .
I now complete the proof of the assumptions of Proposition 7.1, in particular property (7.A), after constructing autoequivalences of D b (Y G ) as follows.
Definition 7.5. Take endofunctors of D b (Y G )
Proposition 7.6. Each of the Tw l is a twist by a spherical object, and thence an autoequivalence.
Proof. To see this for Tw 0 , write k : Y G ֒→ G and note that
. After taking a Koszul resolution of k * k * O G , this may be calculated by a spectral sequence from Ext
. By examination of the exceptional collectional given as Collection 0, the only non-trivial terms occuring are Ext 0 (O G , O G ) ∼ = C and 7)) ∼ = C[− dim G] by duality. Noting that Y G is Calabi-Yau, the result follows. A similar argument applies to the other Tw l .
Proof of intertwinement (7.A). The isomorphism Φ ′ P • Tr 0 ∼ = Φ P follows immediately from the cone construction of Tr 0 : recall that Φ P = (Ψ m P ) −1 • i * P and note that i * P j ! O G = 0 for support reasons. The same argument applies to the other Tr l .
The other intertwinements
are more involved, but essentially follow from the exceptional collection property. We give the argument for Tr 0 before explaining how it adapts to the other Tr l . As preparation, recall that Φ G = Ψ −1 G • i * G , and write i in place of i G for clarity. It will be more convenient to prove the following, which is equivalent to (7.D).
i
By standard facts about spherical twists Ψ G •Tw 0 •Ψ −1 G is given by the spherical twist around Ψ G (O Y G ). By Proposition 5.3 this is isomorphic to j ! O G , and therefore
and noting that the i are open immersions and therefore flat, we find by base change that
We thence have isomorphisms of functors
Comparing the two cones above suggests that the intertwinement may follow from an isomorphism Hom X (j ! O G , −) ∼ = Hom X G (i * j ! O G , i * −) on W 0 .
(7.F) This isomorphism does indeed hold: we follow an argument for this, which furthermore gives (7.E), in [DS, proof of Lemma 3.17 , end of Section 3.2.2].
(Note that the setting there is more general, involving a spherical functor, not just a spherical object.) According to the argument, it suffices if the natural morphism of functors from D b (G) τ : RΓ G → RΓ G • i * is an isomorphism on the subcategory j * W 0 of D b (G). To see why we take this subcategory, note that j * : D b (X, f ) → D b (G) because f | G = 0, and the left-hand side of (7.F) is given by
Now we have that j * − ∼ = Hom G (O, j * −). It therefore suffices to check if τ is an isomorphism on objects
where B is a generator of W 0 .
We determine τ A . Firstly, R >0 Γ G (A) = 0 because A is a vector bundle on an affine stack. Furthermore, R >0 Γ G (i * A) ∼ = Ext >0 G (O, i * j * B). But then this vanishes by the exceptional collection property, as O is on the right-hand side of the collection associated to W 0 , namely Collection 0. Finally, we have that H 0 τ A is an isomorphism by normality of G, as the codimension of G − G is greater than two. We deduce that τ A is an isomorphism, and thence τ is an isomorphism, thereby proving the intertwinement (7.D) for Tr 0 .
For the other transfer functors Tr 1 and Tr 2 , it now suffices that τ is an isomorphism on S ∨ ⊗ j * W 1 and S ∨2 ⊗ j * W 2 respectively. These categories appear because the role of (7.G) is replaced by Hom X (j ! S, −) ∼ = RΓ G (S ∨ ⊗ j * (−)), Hom X (j ! S 2 , −) ∼ = RΓ G (S ∨2 ⊗ j * (−)). But then the above argument suffices, using that S is on the right-hand side for Collection 1, and S 2 is on the right-hand side for Collection 2, and that Proposition 5.3 continues to hold with occurences of O replaced with S or S 2 .
Combining the above, and applying Proposition 7.1, we can describe differences between equivalences as follows.
Proposition 7.7. The twists Tw correspond to 'window shifts' as follows.
We may then complete the proof of Theorem C, as follows.
Theorem 7.8. There is an action of the fundamental group π 1 (M, m G ) on D b (Y G ) given by the following diagram, with Ψ = Ψ 3 .
Proof. We take the groupoid action of Theorem B (Theorem 6.3) and forget one of the basepoints m P , as follows. Restricting that action to a chart containing both basepoints, we obtain the left-hand picture below. Proposition 7.7 then says that monodromies at D b (Y G ) around equatorial holes are given by the right-hand picture.
The remaining two monodromies are immediate, giving the result.
Remark 7.9. I do not discuss whether the action of Theorem C is faithful. If it is, then the spherical twists by the objects ( * ) act as a free group: it could be interesting to try to prove this by some B-side analogue of results of Keating on free group actions by symplectic Dehn twists [Kea] .