coherent sheaves on coulomb branches...theorem (beilinson) the sum oo ( 1) o ( n) is a tilting...

Symplectic singularities Hypertoric varieties Coulomb branches

Almost commutative rings

Coherent sheaves on Coulomb branches

Ben Webster

University of WaterlooPerimeter Institute for Mathematical Physics

October 26, 2017

Ben Webster UW/PI




For a lot of history, it seemed as though commutative rings weremaybe the most natural framework in which to view mathematics.

Obvious context for number theory, algebraic geometry.In physics, observable quantities form a commutative ring.

Then quantum mechanics came along, and the picture looked a bitdifferent. The algebra of observables becomes non-commutative, butwith a classical limit (it’s “almost commutative”).

On the classical side, physicists had already noticed a hint of thenon-commutativity of quantum mechanics: the Poisson bracket(which is often called “semi-classical”)

Hamilton’s equation (for an observable):∂f∂t

= {H, f}

Heisenberg’s equation (for an operator): i~∂ f∂t

= [H, f ]

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commutative

semi-classical almost commutative

non-commutative

X

where I’d like to be

representation theoryalgebraic geometry

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commutative semi-classical almost commutative non-commutative

X



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commutative semi-classical almost commutative non-commutativeX



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This structure is surprisingly interesting in algebraic geometry:

Definition

A resolution of singularities M→ Y of a normal Poisson affinevariety is symplectic if M has a symplectic structure compatible withthe Poison structure.

We call this resolution conical if C[Y] has a positive grading (i.e. Y isthe cone over a weighted projective variety) making the Poissonbracket homogeneous.

Everyone’s favorite conical symplectic resolution is the Springerresolution:

T∗G/B→ N ⊂ g∗

but there are many others.

Ben Webster UW/PI



Tilting generators

Conic symplectic resolutions have a lot of nice properties:

They are always canonical and rational.

They are always (relative) Mori dream spaces.

They always possess a tilting generator: a vector bundle T suchthat Ext>0(T,T) = 0 and every coherent sheaf has a finiteresolution by summands of T .

The last of these properties is most interesting for a representationtheorist:

Corollary

The derived category of coherent sheaves on M is equivalent to thederived category of A = End(T)-modules.

Ben Webster UW/PI



Tilting generators

Since the cotangent space at ` ∈ Cn+1 to Pn is given byHom(Cn+1/`, `), the cotangent bundle T∗Pn can be identified withpairs (` ∈ Pn,X : Cn → Cn) where im(X) ⊂ ` ⊂ ker X.

The map T∗Pn → N≤1 to nilpotent (n + 1)× (n + 1) matrices ofrank ≤ 1 is a conic symplectic resolution.

Theorem (Beilinson)

The sum O ⊕O(−1)⊕ · · · ⊕ O(−n) is a tilting generator on T∗Pn.

The algebra A can be described by the quiver with certain relations:

O O(−1) · · · O(−n)...

......

Ben Webster UW/PI



Tilting generators

For other resolutions, there is an abstract construction of A:

Construct a quantization of a good reduction of M tocharacteristic p for a large prime p.

Using a version of the Frobenius map, we can turn this into anAzumaya algebra on M. Paradigmatic example: differentialoperators Fp〈x, ∂x〉 thought of as a sheaf of algebras overSpecFp[xp, ∂p

x ].

This Azumaya algebra doesn’t split, but for silly characteristic preasons. Via some characteristic p magic, you can produce avector bundle that morally splits this algebra.

This results in the vector bundle T .

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Stadnik’s tilting generators

That’s not an especially helpful description. In some special cases(such as M = T∗G/B) there’s a reasonable perspective on things, butin general, this prescription isn’t so easy to follow.

One interesting exception: hypertoric varieties. These are defined byconsidering a torus T acting (diagonally) on An, and the moment map

µ : T∗An → t∗ (x, ξ) 7→ (∑

aijxiξi)

We define Y = Spec(k[T∗An]T/(µ(t)). This has a symplecticresolution M defined by taking a GIT quotient instead.

If T = C∗ acting diagonally, then M is T∗Pn−1 and Y = N≤1, then× n matrices which nilpotent and rank ≤ 1.

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For a hypertoric variety, we can exploit the fact that the constructionof the tilting bundle on T∗An can be done T-equivariantly.

Theorem (Stadnik, W.)

A hypertoric variety over a field of characteristic 0 or large p carriesa tilting generator which is a sum of line bundles. The number ofsummands in a minimal tilting generator is the number of subsets ofthe weights of An that form a basis of t∗.

So, for T∗Pn−1, each singleton is a basis, and as expected, the numberof summands is n.

In general, Pic(M) ∼= X∗(T), the character lattice of T via theassociated bundle construction.

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0→ T → (C∗)n → F → 0 0→ LF → (C∗)n → LT → 0

Let LFR be the unitary subgroup of LFC. Each weight Lεi of An

defines a map LFR → U(1).

Choose elements ai ∈ U(1) and consider TR cut into chambers byremoving Lε−1

i (ai). We associate a line bundle to each chamber:

Choose a fixed chamber (say, that containing 1) and fix a linebundle/character of T attached to it (say O).

Travel arround to other chambers, and when you pass εi(f ) = ai

in the positive direction, change the attached character by addingthe weight εi : T → C∗, i.e. tensor the line bundle with O(εi).

The fact that we are travelling around LFR and some basic linearalgebra shows that the line bundle attached to each chamber iswell-defined.

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This fixes a collection of line bundles up to tensor product with asingle line bundle.

Note, this only depends on a and Y , not on the choice of GITparameter.

Theorem

For generic ai, the sum of these line bundles is a minimal tiltinggenerator.

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Endomorphisms

Theorem

Consider the quiver ΓT with vertices given by chambers in LFR, andarrows joining every pair of chambers adjacent across a hyperplane.The algebra A is the quotient of the path algebra of this quiver by thequadratic relations:

The two paths around the intersection of two hyperplanes areequal.

The paths across a hyperplane and back to the same chambersatisfy the linear relations of the normal vectors to thosehyperplanes.

For T∗Pn−1, this gives the algebra described earlier.

Ben Webster UW/PI



Endomorphisms

Significance:

This algebra is independent of the choice of GIT parameter (i.e.the choice of crepant resolution). Strong form of D-equivalence⇔ K-equivalence.

This description makes it clear category is Koszul, and Koszuldual has similar nice description.

The Koszul dual matches Exts in Fukaya category ofmultiplicative hypertoric variety (McBreen-W.); this is a naturalmanifestation of homological mirror symmetry.

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Wall-crossing

In fact, the lack of uniqueness of tilting generators here is a feature,not a bug.

Definition

For two different tilting generators T and T ′, we have a wall-crossingbimodule BT,T′ ∼= Hom(T,T ′).

Theorem

Derived tensor product with BT,T′ is a derived equivalenceDb(A -mod)→ Db(A′ -mod). These functors compose to generate anaction on Db(A -mod) ∼= Db(Coh(M)) of π1(LTC \ E) where E is thelocus where ai define a non-generic subtorus arrangement in LFC.

It’s better to think of this as a representation of the fundamentalgroupoid where points in LTR are sent to Db(A -mod) for thecorresponding tilting generator, and generic points are sent toDb(Coh(M)) for the different resolutions.

Ben Webster UW/PI



Wall-crossing

Luckily, the descriptions we’ve given are well-adapted tounderstanding wall-crossing.

Theorem

There is larger quiver Γ with relations with path algebra AΓ, suchthat each tilting generator T induces an inclusion ΓT ↪→ Γcompatible with relations, that is, AT = eTAΓeT , where eT is the sumof vertices in the image.

The wall-crossing bimodule is just eT′AΓeT .

The underlying set of Γ is t∗Z = Zn/f∗Z. This has a quiver structurewith arrows connecting elements with distance 1 in taxicab metric;the relations we discussed before have a natural port:

two paths of taxicab distance 2 with the same endpoints agree.length two paths out and back satisfy the relations of the normalvectors.

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Wall-crossing

Theorem (Bezrukavnikov-Okounkov)

The action of π1(LTC \ E) categorifies the equivariant quantumconnection for M.

Presumably, we could just prove this by hand using the bimodulesintroduced above, but since this is a theorem now, I haven’t bothered.

Ben Webster UW/PI



A little physics

What makes hypertoric varieties so nice? Many years ago, I thoughtfor a bit about how to generalize this to other symplectic quotients,but could never get it to work.

The magic ingredient is a big commutative subalgebra, which at leastisn’t easy to find in GIT quotients by non-commutative algebras.

Many years later, I saw the error of my ways. The problem was that Ihad been thinking about hypertoric varieties on the wrong side of aduality in physics.

Ben Webster UW/PI



A little physics

“Coulomb branch” is a much broader term from physics, but for us,this means a particular algebra defined by Braverman, Finkelberg andNakajima based on the action of a reductive group G on a vectorspace V .

Hypertoric varieties are Coulomb branches for the action of LF on Cn.

For non-abelian connected groups, let Yab be the Coulomb branch forthe maximal torus H ⊂ G, that is, the hypertoric variety for whichH = LF.

Theorem

The Coulomb branch of G acting on V is birational to the quotient ofYab/W for the Weyl group W of G.

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Examples

Examples:If G = GLn and V = gln ⊕ (Cn)⊕` then the associated Coulombbranch is Symn(C2/(Z/`Z)).The slice to a type A nilpotent orbit of Jordan type µ inside oneof type λ is the Coulomb branch for G =

∏GLvi acting on

V = ⊕mi=1Hom(Cvi ,Cλi)

⊕⊕m−1

i=1 Hom(Cvi ,Cvi+1)

where vi =∑i

k=1 λk − µk.For an (affine) simply-laced Dynkin quiver I, the groupG =

∏i∈I GLvi and representation

V = ⊕i∈IHom(Cvi ,Cλi)⊕⊕i→jHom(Cvi ,Cvj)

have a Coulomb branch given by the slice in Grλ to Grµ in theaffine Grassmannian for GI .

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Examples

The Coulomb branch of G acting on V might not have a symplecticresolution, but there is still a generalization of the resolutions that givehypertoric varieties (they might never be smooth).

If such a resolution exists, we call it a BFN resolution.

The type A examples of the previous page have BFN resolutions(some of the D and E type ones don’t):

Among the BFN resolutions of Symn(C2/(Z/`Z)) are the

Hilbert scheme Hilbn( ˜C2/(Z/`Z)).

The resolutions of nilpotent orbit slices are induced by the mapsT∗G/P→ N which give symplectic resolutions of orbits.

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Tilting generators revisited

The fact that M is basically Mab with some fiddliness around theWeyl group action also manifests in its tilting generator.

Just as in the abelian case, we construct a chamber structure on HR bypulling out the subtori where weights agree with a fixed valueai ∈ U(1). In order to make things work, this choice of values shouldbe compatible with a decomposition into simple G-reps.

Theorem

If M is a BFN resolution, it carries a tilting generator given by a sumof rank #W vector bundles, one for each W-orbit of chambers in HR.Under the birational map to Mab/W, this agrees with the generatorchosen previously.

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Tilting generators revisited

We can give a presentation of the endomorphisms of this vectorbundle much like the one discussed above. In addition to theendomorphisms from the abelian case, we also have:

An action of the Weyl group, permuting chambers and stepsacross hyperplanes in the obvious way.

When reflection sα fixes a chamber, we formally adjoin theendomorphism 1

α(sα − 1).

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The quiver case

All the interesting non-abelian cases we know come from quivers. Inthis case, the algebra has a particularly nice description.

The scalars assigned to weights in Hom(Cvi ,Cλi) give us λi (distinct,WLOG) elements in ai,k ∈ U(1) for each i. There are also weights inHom(Cvi ,Cvj), but let’s assign them 1 for now.

Consider the cylinder U(1)× [0, 1]. We’ll draw red strands at x = ai,k

for each i, k, and label these with i.

Ben Webster UW/PI



The quiver case

Definition

Let A be the span of diagrams in U(1)× [0, 1] which consist of vi

black strands with label i for each i ∈ I, subject to rules:

Strands aren’t allowed to have minima or maxima (they projectdiffeomorphically to [0, 1]), and must intersect generically (notangencies or triple points with each other, with red strands, ory = 0, 1).

Black (but not red) strands can carry an arbitrary number ofdots, which avoid crossing points.

We only fix these diagrams up to isotopy fixing all these conditionsand the position of red strands.

We can multiply these diagrams via stacking, where the result is 0 ifwe cannot isotope the top and bottom to match.

Ben Webster UW/PI



The quiver case

We let A be the quotient by the local relations:

i j

=

i j

unless i = j

i j

=

i j

unless i = j

i i

=

i i

+

i i

i i

=

i i

+

i i

i i

= 0 and

i j

=

ji

Qij(y1, y2)

ki j

=

ki j

unless i = k 6= j

ii j

=

ii j

+

ii j

Qij(y3, y2)− Qij(y1, y2)

y3 − y1

ij k

=

ij k

+

ij k

δi,j,k

=

=

i k

=

ki

δik

k i

=

ik

δik

Ben Webster UW/PI



The quiver case

This result can be extended a few different ways:for Hilbert schemes, or generally type A, need to turn on weightsin Hom(Cvi ,Cvj). This means that we cross a hyperplane whenstrands with labels i and j are a fixed distance apart⇒ weightedKLR relations.wall crossing functors correspond to bimodules that arise inchanging the position of red strands. Similar picture withdiagrams in a cyclinder, but now red strands cross.in type A case, we can define a categorical action of sl` where` =

∑i∈I λi. This allows us to write the action of wall-crossing

functors as Rickard complexes, and thus compare their action inthe quantum Weyl group (and thus to the monodromy of thequantum connection).in two-row case, this also allows us to recover the work of Annoand Nandakumar on annular tangles and coherent sheaves.Correct generalization is annular foams.

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The quiver case

Thanks.

Ben Webster UW/PI


coherent sheaves on coulomb branches...theorem (beilinson) the sum oo ( 1) o ( n) is a tilting...

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