Quiver gauge theories and symplectic singularities Alex Weekes (UBC) June 6, 2020 Workshop on Lie Theory and Integrable Systems in Symplectic and Poisson Geometry (Fields Institute) 1
Introduction quiver gauge theories • Use mathematical construction of Braverman, Finkelberg and Nakajima • Plan: 1. Background 2. Coulomb branches, properties and examples 3. Discuss proof that they have symplectic singularities 2 • Investigate properties of Coulomb branches of 3 d N = 4 • Viewpoint is algebraic geometry over C
Background
Symplectic singularities • Very interesting algebraic varieties with algebraic Poisson structures, generically (holomorphic) symplectic • Movating examples: • Normalizations of nilpotent orbit closures • Interesting “representation theory” and enumerative geometry 3 • Nilpotent cone of a simple Lie algebra g over C , e.g. N sl n = { A ∈ M n × n ( C ) : det( t − A ) = t n } • Kleinian singularities, e.g. C 2 / / ( Z / n Z )
Symplectic singularities • Frequently arise in pairs, as Coulomb and Higgs branches Braden-Licata-Proudfoot-Webster algebras 4 of 3 d N = 4 gauge theories • Subject ∗ of symplectic duality program proposed by • N g and N g ∨ are dual, where g , g ∨ are Langlands dual Lie (Here the “representation theory” is of g and g ∨ , and more precisely of categories O )
Symplectic singularities extends to a regular 2–form on Y • Implies X has fjnitely many holomorphic symplectic leaves (Kaledin), and rational Gorenstein singularities (Beauville, Namikawa) 5 • A normal affjne variety X / C has symplectic singularities if: 1. Have a given symplectic form ω on smooth locus X reg 2. For some (any) resolution π : Y − → X of singularities, π ∗ ω • Coordinate ring C [ X ] gets Poisson bracket {· , ·} � N sl n = O λ , O λ = nilp. orbit of type λ λ ⊢ n
Coulomb branches
Quiver gauge theories • Associated to a quiver Q plus dimension vectors 4 1 2 3 1 2 9 6 v , w ∈ Z I ≥ 0 , where I is the set of vertices • For example: the A 4 quiver 1 → 2 ← 3 → 4, with v = ( 3 , 1 , 2 , 9 ) and w = ( 4 , 0 , 1 , 2 ) • To this data, physicists associate a 3 d N = 4 gauge theory. Its Higgs branch is the Nakajima quiver variety M H ( v , w )
The Coulomb branch • Braverman-Finkelberg-Nakajima have given a construction 7 of the Coulomb branch M C ( v , w ) • Let D = Spec C [[ t ]] . Defjne moduli space R , of data 1. Vector bundle E i over D of rank v i , for all i ∈ I 2. Trivialization ϕ i of E i on D × , for all i ∈ I 3. For all i ∈ I and edges i → j , s i ∈ Hom( O D ⊗ C W i , E i ) , s i → j ∈ Hom( E i , E j ) which remain regular under ϕ i • Action of G [[ t ]] = � i GL( v i )[[ t ]] changing trivialization
The Coulomb branch Right side carries “convolution product”, making it a commutative algebra C 8 • BFN defjne Coulomb branch as affjne scheme / C M C ( v , w ) = Spec H G [[ t ]] ( R ) ∗ • BFN show M C ( v , w ) is irreducible normal affjne variety, actually defjned over Z • Also show M C has a Poisson structure, symplectic on M reg
Type A 1 m 9 n • Consider A 1 quiver datum • M C ( m , n ) has description due to Kamnitzer: ( i ) A monic, degree m , � � A ( z ) B ( z ) ∈ M 2 ( C [ z ]) : ( ii ) degrees B , C < m , C ( z ) D ( z ) ( iii ) AD − BC = z n
Finite ADE type and affjne type A i variety. Theorem (Nakajima-Takayama) are cocharacters of G Q i i Theorem (Braverman-Finkelberg-Nakajima) 10 algebraic group (of adjoint type). Then Suppose Q is oriented fjnite ADE, and let G Q be the associated M C ( v , w ) ∼ λ = W µ is a generalized affjne Grassmannian slice for G Q , where � � w i ̟ ∨ v i α ∨ λ − µ = λ = i , If Q is oriented affjne type A, then M C ( v , w ) is a Cherkis bow
Finite ADE type 1 Nakajima-Takayama) and Kamnitzer-Webster-W.-Yacobi, affjne type A • In fjnite ADE and affjne A types, know decomposition of 2 11 n • For type A and µ dominant, then M C ( v , w ) ∼ = O λ ∩ S µ where O λ , S µ ⊂ gl N nilpotent orbit/Slodowy slice, and λ, µ ⊢ N partitions. · · · n − 1 gives M C ( v , w ) ∼ = N sl n M C ( v , w ) into symplectic leaves (fjnite ADE by Muthiah-W.
General quivers • Quivers without loops/multiple edges correspond to simply-laced Kac-Moody types Can defjne for Kac-Moody group G Q general 12 M C ( v , w ) =: (generalized) affjne Grassmannian slice • Upshot: affjne Grassmannian for G Q is not defjned in • BFN conjecture a version of the geometric Satake correspondence using M C ( v , w )
Symplectic singularities
Main result Theorem (W.) • This is conjectured by BFN for all Coulomb branches, not just quiver gauge theories • Known already for dominant fjnite ADE type by Kamnitzer-Webster-W.-Yacobi, and affjne type A by Nakajima-Takayama Corollary and rational Gorenstein singularities. 13 Let Q be a quiver without loops or multiple edges, and v , w be arbitrary. Then M C ( v , w ) has symplectic singularities. M C ( v , w ) has fjnitely many holomorphic symplectic leaves,
First ingredient: partial resolutions • There is a completely integrable system 2 • Coulomb branches admit partial resolutions 14 M κ C ( v , w ) − ։ M C ( v , w ) κ is cocharacter of certain “fmavour symmetry” group Special case: Springer resolution T ∗ Fl n → N sl n / W ∼ � ̟ : M C ( v , w ) − → t / = C i v i → H G [[ z ]] It is faithfully fmat, and comes from H ∗ G ( pt ) ֒ ( R ) . ∗ Special case: Gelfand-Tsetlin integrable system n − 1 n ( n − 1 ) � � � N sl n → C , A �− → coeffjcients of det( t − A i ) i = 1
Final ingredient: open subsets Using results of Beauville and Bellamy-Schedler, suffjcient to (iii) U is smooth and symplectic (i) diagram is Cartesian so that V U give open subsets 15 M κ C ( v , w ) M C ( v , w ) t / / W (ii) codim C V = 4, C ( v , w )) sing ≥ 4 Then codim C ( M κ
Second ingredient: integrable system Theorem (W.) product products are smooth and symplectic. Establishes diagram on previous page, so proves theorem. 16 1. Étale neighbourhood of any fjber of ̟ is isomorphic to a M κ C ( v ( 1 ) , w ( 1 ) ) × · · · × M κ C ( v ( ℓ ) , w ( ℓ ) ) 2. For generic κ , can choose V such that over V these
Questions • Enumerate symplectic leaves, and their transverse slices? When is it a resolution? • Quivers with loops and/or multiple edges? Symmetrizable types? • Other Coulomb branches? 17 • Is M κ C ։ M C a Q –factorial terminalization, for generic κ ?
Thank you for listening! I refuse to answer that question on the grounds that I don’t know the answer. - Douglas Adams 18
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