Coulomb branches and KLR algebras - uwaterloo.ca

Physics Coulomb branches Representation theory Duality

Coulomb branches and KLR algebras

Ben Webster

University of WaterlooPerimeter Institute for Mathematical Physics

May 31, 2018

Ben Webster UW/PI

Coulomb branches and KLR algebras


Philosophy

What is the correct context for geometric representation theory?

I think I may be contractually obligated to now say that it’smathematical physics.

That’s fine, because I mostly think that’s true.

Ben Webster UW/PI

Symplectic duality and KLR algebras


Philosophy

Powerful aspects of physics (from the perspective of amathematician):

Natural geometric input (and by that, I mean PDEs!) suggestssurprising algebraic structures, especially when dimensionallyreduced, or sent to a limit. When algebras and categories arisethis way, the underlying quantum field theory provides insight ontheir structure.In particular, the same quantum field theory might have manydifferent realizations, and a given question might be easy in oneand hard in another. A particularly common version of this isvarious dualities.

Mirror symmetry relating Fukaya categories and coherent sheaves isan especially famous example of this, but various versions ofgeometric Langlands can also be thought of this way.

Ben Webster UW/PI



Gauge theories

Lots of interesting representation theory comes out this way. Thereare different configurations of the physics you can put in, but for mypurposes, I’m interested in a very specific example:

Fix a connected reductive C-algebraic group G, and let V be arepresentation. Out of this data, we can create a gadget called an“N = 4 supersymmetric 3-d gauge theory.”

You should think somewhere in the background, there is an actualstate space whose objects are sections of certain bundles on a3-manifold, and one calculates “expectations” of “observables” byintegrating a “probability measure” on this space of section. Youshould then forget you thought that.

Ben Webster UW/PI



Gauge theories

TheoremLet Z be a d = 3 TQFT. The state space Z(S2) is the homology of anE3-algebra, and thus has a commutative multiplication and a Poissonbracket {�,�} of degree �2.

If we “turn on” rotation of S2 by S1, then we obtain a deformation ofthis semi-classical structure to a filtered algebra A = Zh(S2).

The theory that interests me has two topological twists (i.e. associatedTQFTs); the local operators in this twists give two conic symplecticvarieties:

The Higgs branch MH = SpecZH(S2) and its quantization AH .The Coulomb branch MC = SpecZC(S2) and its quantization AC.

Ben Webster UW/PI


OEE.ee


Gauge theories





Ben Webster UW/PI


⇒Otto


Gauge theories





Ben Webster UW/PI



Higgs and Coulomb

The Higgs branch is the categorical quotient by G of the zero level ofthe moment map µ : T⇤V ! g⇤.

Its quantization is the non-commutative Hamiltonian reduction of thedifferential operators on V .

Many interesting symplectic varieties appear this way:the nilcones N of slN (U(sln))Slodowy slices of slN (W-algebras)Symn(C2/�) (symplectic reflection algebras)Nakajima quiver varietieshypertoric varieties

Ben Webster UW/PI



Higgs and Coulomb

The Coulomb branch is trickier. I’m only going to describe it in thecase of a quiver gauge theory, that is, for some quiver with verticesI ⇠= [1, n]:

V =M

i2I

Hom(Cvi ,Cwi)�M

i!j

Hom(Cvi ,Cvj) G =Y

i

GL(Cvi)

for dimension vectors v,w 2 ZI�0.

The corresponding Higgs branch is a Nakajima quiver variety forweights � and µ satisfying ↵_

i (�) = wi and µ = ��Pi vi↵i.

Ben Webster UW/PI



The definition of Coulomb branches

DefinitionLet a Coulomb diagram be a diagram in R/Z⇥ [0, 1] consisting of:

v =P

vi curves that map diffeomorphically to [0, 1] under projection to thesecond factor and all intersect generically (no tangencies or triple points)and with x = 0,a labelling of each strand with some i 2 I = [1, n]; this strand must beginand end at x = i/(n + 1).decorations of these strands with an arbitrary number of dots that avoidcrossings and y 2 [0, 1].

3

33

3

2

2

2

2

1

1

You should glue this intoa cylinder by attaching thefringed edges.

Ben Webster UW/PI





v =P


3

33

3

2

2

2

2

1

1


Ben Webster UW/PI





v =P


3

33

3

2

2

2

2

1

1


Ben Webster UW/PI





v =P


3

33

3

2

2

2

2

1

1


Ben Webster UW/PI





v =P


3

33

3

2

2

2

2

1

1


Ben Webster UW/PI




DefinitionThe (quantum) Coulomb branch AC of v,w is the quotient of theformal span of Coulomb diagrams, modulo a long list of relationswith ~ = 0 (resp. ~ = 1).

Multiplication in this algebra is given by stacking diagrams.

One subtlety

You might object that the join points won’t be generic if for some i,there are � 3 strands with this label. You should resolve this using the“half-twist” where all pairs of strands still cross.

ii i

=

ii i

One of the relations will assurethat this is well-defined.

Ben Webster UW/PI




These relations come in three families: “codimension 0”

i j

=

i j

unless i = j

i i

=

i i

+

i i

i j

=

i j

unless i = j

i i

=

i i

+

i i

= �~ = +~

Ben Webster UW/PI




The “codimension 1” relations: these depend on polynomials

qij(u, v) =Y

e : i!j

(v�u�ze)Y

e0 : j!i

(u�v+ze) pi,±(u) =wiY

k=1

(u±~/2�ri,k)

i i

= 0 and

i j

=

ji

qij(y1, y2)

i

= pi,+

i

!

i

= pi,�

i

!

Ben Webster UW/PI




Finally, there are “codimension 2” relations, which are required forconsistency of the ones we’ve seen.

ki j

=

ki j

unless i = k 6= j

ii j

=

ii j

+

ii j

qij(y3, y2)� qij(y1, y2)

y3 � y1

ii

=

ii

+

ii

pi,+(yn)� pi,�(y1)

yn � y1 � h

ji

=

ji

if i 6= j.

Ben Webster UW/PI



Simple examples

Easy special case: if I = {i}, and vi = 1,wi = 0 then we have a singlestrand. We have generators

y =

i

i

r+ =

i

i

r� =

i

i

If ~ = 0, then this is just C[T⇤C⇥]. The basic commutation relationsare those of multiplication by y sending r± to translation by ±~, thatis:

r+r� = r�r+ = 1 (y ± ~)r± = r±y

Ben Webster UW/PI



Simple examples

Now, assume wi = ` > 0. In this case:

r+r� = pi,�(y) r�r+ = pi,+(y) (y ± ~)r± = r±y.

TheoremIn this case, MC = C2/Z` and AC is isomorphic to the sphericalrational Cherednik algebra of Z`, sending r± 7! x`± and y to theEuler operator (have fun figuring out how the parameters relate toroots of pi,±(y)).

Note that these are all isomorphic to the w = 0 case if we tensor withrational functions in y via the map sending r� 7! r� · (pi,�(y))�1.

Ben Webster UW/PI



Simple examples

Minor simplification/complication: we can “zip” together k strandswith the same label for a portion of the diagram. These can carrysymmetric functions in k variables.

I’ll leave as an exercise to find the relations involving these.

Most important is sliding symmetric functions over zips:

e1

3(2)

3 3

=

3(2)

3 3

+

3(2)

3 3

Ben Webster UW/PI



The integrable system

This makes it much easier to write the identity in our algebra: we justnever unzip the strands.

This also allows us to see that our algebra contains a copy ofsymmetric functions in alphabets of size vi for each i 2 I.

3(2)

e1e2

3(2)

2(2)

h4

2(2)

1

1

Since these commute, they define an integrable system.Ben Webster UW/PI




In the general case, we can prove a similar birationality result:

TheoremIf we adjoin arbitrary symmetric rational functions on thick strands,then obtain a birational map from MC to

T⇤Cv1/Sv1 ⇥ · · ·⇥ T⇤Cvn/Svn .

In particular, MC is irreducible and has dimension 2P

vi.

This also shows that the polynomials on thick strands define a map

MC ! Cv1/Sv1 ⇥ · · ·⇥ Cvn/Svn

which is an integrable system with generic fiber (C⇥)P

vi .

Ben Webster UW/PI




Theorem (BFKKNWW)

If I is type ADE and �, µ are dominant, then MC is a slice to

Grµ = G[[t]] · tµ · G[[t]]/G[[t]]

inside Gr�, thinking of �, µ as coweights of the Langlands dual to thegroup with Dynkin diagram I.

There is a natural symplectic structure on this variety correspondingto the Manin triple (g((t)), g[[t]], t�1g[t�1]), which matches the onewe’ve discussed.

Ben Webster UW/PI



Dynkin and cyclic quivers

The theory of quantum groups suggests a quantization, called atruncated shifted Yangian, defined byKamnitzer-W.-Weekes-Yacobi.

Theorem (BFKKNWW)

If I is type ADE and �, µ are dominant, then AC is the appropriatetruncated shifted Yangian Y�

µ .

This has a Drinfeld-style presentation with modes corresponding to:

Ben Webster UW/PI


⇐tf¥#÷H÷d#



Theorem (Nakajima-Takayama)

If I is a cycle and �, µ are dominant, then MC is an bA quiver variety(for rank-level dual data). In particular, Symn(C2/Z`) arises from:

G = GLn V = gln � (Cn)�`.

In this case, the quantization AC is the spherical rational Cherednikalgebra, with dots going to the symmetric Dunkl-Opdam operators.

Ben Webster UW/PI




Common special case: we can think of dominant regular weights forpgln as partitions (normalized so that �n = 0, so

� = (w1 + · · ·+ wn�1,w2 + · · ·+ wn�1, . . . , 0)µ = (�1 � v1,�2 + v1 � v2, . . . ,�n + vn�1)

Theorem (Maffei + previous slide)

If I is of finite type A, then MC is the intersection of the Slodowy slicefor a nilpotent of Jordan type µ with the orbit closure of a nilpotent oftype �.

Note that the quiver variety for this data is given by the sameintersection for types �t and µt.

Ben Webster UW/PI



The extended category

Our description of AC naturally embeds it as an endomorphism ring ina bigger category whose:

Objects are all ways of positioning points on R/Z with vi ofthem having label i, and points only sitting on top of each otherif they have the same label.Morphisms a! b are Coulomb diagrams with bottom a and topb.

A simple computation with the nilHecke algebra shows that:

TheoremThe object where the labels 1, . . . , 1, 2, . . . , 2, . . . , n, . . . , n are inorder but at distinct points is isomorphic to the direct sum of

Qvi!

copies of our special object XC whose endomorphisms are AC.

Thus, up to Karoubi envelope, we don’t even need thick strands in thecategory.

Ben Webster UW/PI




You can make this category even bigger by expanding the seam x = 0to be wi red strands with label i (numbered 1, . . . ,wi at top).

DefinitionWe let C be the category whose objects are positionings of points asbefore avoiding these red lines and with morphisms Coulombdiagrams joining them, modulo relations before except that the bigonrelation changes to:

i

k

j

=

k

ji

� ri,k

k

ji

k

j i

=

i

k

j

� ri,k

i

k

j

Ben Webster UW/PI




Let X be the sum of all objects in C (up to isotopy), and e be theprojection onto XC. Let AC = EndC(X).

TheoremAssume I is a Dynkin diagram, and wi 6= 0 only on minuscule nodes(automatic for type A). At ~ = 0, the algebra AC and bimodule ACedefine a non-commutative resolution of MC.

I know this, because they correspond to a tilting bundle on an honestresolution.

For a cyclic quiver, this can be fixed, but you have make the categoryeven bigger, using ghosts.

Ben Webster UW/PI



Weight spaces

Now specialize ~ = 1, ze = 0, ri,k 2 Z+ 1/2. Let S ⇢ AC be thecommutative subring generated by symmetric functions on theidentity thick diagram. We can analyze the representation theory ofAC using the action of this subalgebra.

DefinitionGiven a maximal ideal m ⇢ S, we let

Wm(M) = {m 2 M | mNm = 0 for N � 0}.

We let the subcategory of weight modules be the finitely generatedAC modules which satisfy M ⇠=LWm(M).

Ben Webster UW/PI



Weight spaces

Let SX be the algebra of dots acting on the object X 2 Ob C.

We can extend the notion of a weight module to a representation M ofthe category C/algebra AC by analyzing the action of SX on the vectorspace assigned to each object/the image of eX , the identity on X.

You can think of each weight as labeling the points on your diagramwith complex numbers ak (which give the eigenvalue of the dot). Weassume these are integers.

Ben Webster UW/PI


jis jfIk= - 342B. ilk

i ;3• •

= Iiz=Z

ay • •

as = - 2

.-

--

.

X - 0• j=2

rz,

1=7 Yz•

•

ii. 2•

a ,=6 j

=1iz=3

mi= 4h

92=9


Weight spaces

Consider the category CZ with objects given by (X,m ⇢ SX) for m anintegral maximal ideal with:

Hom�(X,m), (X0,m0)

�

= lim �Hom(X,X0)/�(m0)NHom(X,X0) + Hom(X,X0)mN�

You can think of this as adding an idempotent projecting to eachweight space, and power series in the completion by m acting on it (inparticular, the inverse of any element of SX \m).

Proposition

There is an equivalence of the category of weight modules over AC Wecan think of weight modules as representations of with therepresentation sending M to

WX,m(M) = {m 2 eXM | mNm = 0 for N � 0}.

Ben Webster UW/PI



Weight spaces

This looks like it has many more objects, but actually there onlyfinitely many up to isomorphism:

Reordering points will preserve the isomorphism type unlessak = ak+1.Moving the first strand left over the seam will give anisomorphism unless ri1,m = a1 � 1/2 for some m. Similarly, withmoving the last (vth) strand right unless riv,m = a1 + 1/2.

Thus, if we move the black points to the real line around x = ak, usingposition on the circle as a tiebreak, and add red points at x = ri,k, theisomorphism type of the object only depends on the relative order ofthese points.

Ben Webster UW/PI


IEiz=z iE3jz3j=l ii. 2 j=2 iz=3

Jk ?=-342 as- - Zay'

'

rail"zri,i=4kai6 rz,

,=7yz92=9


wKLR algebras

There are too many formulas to write this down, but it’s not hard tocanonically identify the weight spaces related by isotopies on the realline.

Even when they’re not isomorphisms, when we do switch the order ofweight spaces, we still get morphisms. We’ll denote these by drawingdiagrams in R⇥ [0, 1] showing the interaction, and dots representingthe nilpotent part of the dot on the corresponding strand in C.

i j i�1 �2

Ben Webster UW/PI



wKLR algebras

The relations between these are readily calculated. They are given by:

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 �

=

ij k

+

ij k

�i,j,k

=

=

i �

=

ki

�ik

� i

=

ik

�ik

Ben Webster UW/PI



wKLR algebras

This is a weighted KLR algebra T which appeared previously in mywork.

TheoremThe category of weight modules over C is equivalent to therepresentations of T on which dots are nilpotent.

The category of weight modules over AC is a quotient of this categoryby the simples killed by e.

These categories are equivalent if ri,k are generic, but if they are closetogether, there are idempotents we can’t write using the object XC

(due to the tiebreak).

Ben Webster UW/PI



wKLR algebras

As Kevin discussed, AC has a category O . This is the subcategory ofweight modules killed by weight spaces such that

Pai � 0.

TheoremThe category O over C is equivalent to the representations of T, thequotient of T by the 2-sided ideal generated by diagrams where therightmost strand is black (rather than red). This is a tensor productcategorification for the tensor product

Nni=1 V($i)

⌦wi (with orderinduced by ri,k).

The category O over AC is a quotient of this category by the simpleskilled by e.

Ben Webster UW/PI



wKLR algebras

With a bit more combinatorics, you can prove:

Theorem (Kamnitzer-Tingley-W.-Weekes-Yacobi)

The highest weights in category O for the truncated shifted Yangian oftype ADE are canonical bijection with the product monomial crystalof Nakajima corresponding to the parameters ri,k.

Ben Webster UW/PI



All of these objects have interpretations in terms of the Nakajimaquiver variety MH . The parameters ri,k induce an action on theNakajima quiver variety as the eigenvalues of a cocharacter intoQ

GL(Cwi).The monomial crystal indexes fixed point components of thisC⇤-action on MH .T is a Steinberg algebra (in the sense of Sauter) given by theExt-algebra of certain semi-simple D-modules on the modulistack V/G.T is the Ext-algebra of the semi-simple object obtained byHamiltonian reduction of these D-modules to MH , the smoothquiver variety.

Ben Webster UW/PI



Combining these, we see that:

Theorem (W.)

The geometric category O of MH for the C⇤ action corresponding to ris Koszul dual to the category O for AC with ~ = 0 and parameters r.

Parabolic-singular duality for type A is a special case of this, as is therank-level duality for cyclotomic rational Cherednik algebras.

Ben Webster UW/PI



DefinitionWe call the Higgs and Coulomb branch of a single theory symplecticdual varieties. It’s not obvious why this would be a symmetricproperty, but physics suggests it actually should be (though maybethat duals are not unique!).

The nilcones Ng and NLg are dual.Special Slodowy slices and special nilpotent orbitscorresponding under the Spaltenstein involution are dual.Hypertoric varieties come in dual pairs indexed by a bijection ofunderlying combinatorial data: Gale duality.Quiver varieties in finite and affine types are dual to slices in theaffine Grassmannian.

Ben Webster UW/PI



Arguments much like those we’ve given show that:

Theorem (W.)

We have an isomorphism of OHiggs with the Koszul dual O!Coulomb in

all the cases we’ve discussed, with quantization and gradingparameters chosen to match.

In the case of flag varieties and Slodowy slices, this isparabolic-singular duality for the original category O.In the case of hypertoric varieties, this was proven byBraden-Licata-Proudfoot-W. by a more hands-on argument.The most interesting case is that of quiver gauge theoriesdiscussed above.

Ben Webster UW/PI



Thanks.

Ben Webster UW/PI


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Coulomb branches and KLR algebras - uwaterloo.ca