Representation theory of symplectic singularities

Transcription

1 Quantzatons Hggs and Coulomb branches Category O and the algebra T Representaton theory of symplectc sngulartes Unversty of Vrgna December 9, 2016 Representaton theory of symplectc sngulartes

2 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras For a lot of hstory, t seemed as though commutatve rngs were maybe the most natural framework n whch to vew mathematcs. Obvous context for number theory, algebrac geometry. In physcs, observable quanttes form a commutatve rng. Then quantum mechancs came along, and the pcture looked a bt dfferent. The algebra of observables becomes non-commutatve, but wth a classcal lmt (t s almost commutatve ). On the classcal sde, physcsts had already notced a hnt of the non-commutatvty of quantum mechancs: the Posson bracket (whch s often called sem-classcal) f Hamlton s equaton (for an observable): t = {H, f } Hesenberg s equaton (for an operator): Representaton theory of symplectc sngulartes ˆf t = [Ĥ, ˆf ]

3 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras commutatve non-commutatve Representaton theory of symplectc sngulartes

4 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras commutatve sem-classcal almost commutatve non-commutatve Representaton theory of symplectc sngulartes

5 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras algebrac geometry representaton theory commutatve sem-classcal almost commutatve non-commutatve Representaton theory of symplectc sngulartes

6 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras algebrac geometry representaton theory commutatve sem-classcal X almost commutatve non-commutatve where I d lke to be Representaton theory of symplectc sngulartes

7 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras Defnton An almost commutatve rng s a rng A wth a fltraton A 0 A 1 and an nteger n > 0 such that A A j A +j [A, A j ] A +j n In partcular, the rng gr(a) = =0 A /A 1 s commutatve and Z 0 -graded. Representaton theory of symplectc sngulartes

8 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras The rng gr(a) nherts a sem-classcal structure: Defnton A concal Posson rng s a Z 0 -graded commutatve rng R wth a second operaton {, }: R R R, homogeneous of degree n, that satsfes the relatons of a Le bracket (blnear, ant-symmetrc, Jacob) such that the Lebntz rule holds: {ab, c} = a{b, c} + b{a, c}. There s a classcal lmt functor A (gr(a), {, }) from almost commutatve algebras to concal Posson algebras, wth the Posson bracket gven by {ā, b} = [a, b] A +j n /A +j n 1. Representaton theory of symplectc sngulartes

9 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras The most basc case s when A = C x, d dx s the algebra of polynomal dfferental operators. Ths s fltered wth ( A 1 = span 1, x, d ) dx A n = A n 1 Ths s almost commutatve (n = 2) wth gr(a) = C[x, p]. {f, g} = f g p x g p f x {p, x} = 1 Smlarly, U(g) for any Le algebra g s almost commutatve, wth classcal lmt C[g ] wth the KKS Posson structure. Representaton theory of symplectc sngulartes

10 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras Defnton If R s an concal Posson algebra, then a quantzaton of R s an almost commutatve algebra A whose classcal lmt s R. You can easly check that A s the unque quantzaton of C[x, p]. (Hnt: A 1 = C {x, p, 1} as 3-dmensonal Le algebras). What happens when we consder other Posson varetes? In general, fndng all quantzatons s not easy; Kontsevch got a Felds Medal n large part for dong so for a real Posson structure on R n. Representaton theory of symplectc sngulartes

11 Quantzatons Hggs and Coulomb branches Category O and the algebra T Almost commutatve algebras Defnton If R s an concal Posson algebra, then a quantzaton of R s an almost commutatve algebra A whose classcal lmt s R. You can easly check that A s the unque quantzaton of C[x, p]. (Hnt: A 1 = C {x, p, 1} as 3-dmensonal Le algebras). What happens when we consder other Posson varetes? In general, fndng all quantzatons s not easy; Kontsevch got a Felds Medal n large part for dong so for a real Posson structure on R n. Representaton theory of symplectc sngulartes

12 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes We call an affne varety Y concal Posson f ts coordnate rng has that structure. Defnton We call Y a concal symplectc varety (.e. concal varety w/ symplectc sngulartes) f the Posson bracket nduces a symplectc structure on the smooth locus (+slly techncal condtons). If C 2n has the usual symplectc structure, and Γ s a fnte group preservng ω, then Y = C 2n /Γ s an example. The varety of nlpotent matrces (and more generally, nlpotent cone of a sem-smple Le algebra g) has a natural symplectc structure. The correspondence between almost commutatve and sem-classcal s partcularly nce n ths case. Representaton theory of symplectc sngulartes

13 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes Theorem (Namkawa) Every conc symplectc varety Y has a unversal deformaton Y that smoothes t out as much as possble whle stayng symplectc (whch s thus rgd). The base of ths deformaton s a vector space H. For example, g comes from the nlcone N. The smplest example s: Y = C 2 /(Z/lZ) = {(u = x l, v = y l, w = xy) uv = w l }} The unversal deformaton s gven by addng formal coordnates a on H of degree 2. For a general C 2n /Γ, there s a smlar deformaton drecton attached to every conjugacy class of symplectc reflecton (an element that fxes a codmenson 2 subspace). Representaton theory of symplectc sngulartes

14 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes Theorem (Namkawa) Every conc symplectc varety Y has a unversal deformaton Y that smoothes t out as much as possble whle stayng symplectc (whch s thus rgd). The base of ths deformaton s a vector space H. For example, g comes from the nlcone N. The smplest example s: Y = C 2 /(Z/lZ) = {(u = x l, v = y l, w = xy) uv = w l }} The unversal deformaton s gven by addng formal coordnates a on H of degree 2. For a general C 2n /Γ, there s a smlar deformaton drecton attached to every conjugacy class of symplectc reflecton (an element that fxes a codmenson 2 subspace). Representaton theory of symplectc sngulartes

15 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes Theorem (Namkawa) Every conc symplectc varety Y has a unversal deformaton Y that smoothes t out as much as possble whle stayng symplectc (whch s thus rgd). The base of ths deformaton s a vector space H. For example, g comes from the nlcone N. The smplest example s: Y = {(u, v, w, a 1,... a l ) uv = w l + a 1 w l a l } The unversal deformaton s gven by addng formal coordnates a on H of degree 2. For a general C 2n /Γ, there s a smlar deformaton drecton attached to every conjugacy class of symplectc reflecton (an element that fxes a codmenson 2 subspace). Representaton theory of symplectc sngulartes

16 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes So, how do we get an analogue of the unversal envelopng algebra? Theorem (Bezrukavnkov-Kaledn, Braden-Proudfoot-W., Losev) Every concal symplectc varety Y has a unque quantzaton A of ts unversal Posson deformaton Y; ths s an almost commutatve algebra such that gr A = C[Y]. The center Z(A) s the polynomal rng C[H]; the quotent A λ for a maxmal deal λ H has an somorphsm gr A λ = C[Y]. Ths gves a complete rredundant lst of quantzatons of C[Y]. If Y s a nlcone, then A s the unversal envelopng algebra. If Y = C 2n /Γ, then A s a sphercal symplectc reflecton algebra. Representaton theory of symplectc sngulartes

17 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes Symplectc sngulartes are the Le algebras of the 21st century. - Okounkov There s a very nterestng nterplay between the geometry of Y and the representaton theory of A, modeled on that of g and U(g): Geometry orbt method geometrc quantzaton flag varety and Schubert varetes localzaton, D-modules, ntersecton cohomology support varetes Algebra prmtve deals Harsh-Chandra (b)modules category O character formulae translaton/projectve functors Representaton theory of symplectc sngulartes

18 Quantzatons Hggs and Coulomb branches Category O and the algebra T Symplectc sngulartes We should really have a thrd column here: Combnatorcs: Coxeter groups, tableaux, cells, KL polynomals Culmnaton s the Soergel calculus of Elas and Wllamson. f g To algebrasts, descrbes category O and HC bmodules. To geometers, descrbes the B B equvarant D-modules on G. The frst descrpton s of the endomorphsms of a projectve generator, the second s of Ext of a smple generator. Thus, these are Koszul dual to each other. Representaton theory of symplectc sngulartes

19 Quantzatons Hggs and Coulomb branches Category O and the algebra T Gauge theores How do we generalze ths pcture? Unfortunately, the case of a UEA has a lot of specal structure we can t expect n other cases. I don t have a good general answer; I m pretty skeptcal about one exstng at all. I do know a very nterestng set of examples, though: Begnnng wth a connected reductve complex group G, and a representaton V, there s a 3-d N = 4 supersymmetrc feld theory you can buld from ths. Ths feld theory has a modul space of vacua (lowest energy states) whch s a bg reducble algebrac varety wth two dstngushed components: the Hggs and Coulomb branches. Representaton theory of symplectc sngulartes

20 Quantzatons Hggs and Coulomb branches Category O and the algebra T Gauge theores If you d really lke to know what those words mean, you re n the wrong place, but f you hurry you can get there before they close. Representaton theory of symplectc sngulartes

21 Quantzatons Hggs and Coulomb branches Category O and the algebra T Hggs vs. Coulomb The Hggs branch s the hyperkähler quotent {(v, ξ) T V g v = T v G v ξ} /G Examples of such varetes are Nakajma quver varetes and hypertorc varetes (when G s abelan). The Coulomb branch s gven by startng wth the varety T Ť/W (for T G a maxmal torus), and applyng quantum correctons. By Braverman-Fnkelberg-Nakajma, I know what these correctons are, but there s no room n ths margn, etc. Examples nclude slces between Gr λ and Gr µ n the affne Grassmannan, Â quver varetes, hypertorc varetes, and (conjecturally) G-nstantons on C 2. Representaton theory of symplectc sngulartes

22 Quantzatons Hggs and Coulomb branches Category O and the algebra T Hggs vs. Coulomb For Hggs branches, a quantzaton can be constructed by replacng T V wth dfferental operators, and performng non-commutatve Hamltonan reducton. For Coulomb branches, C[Y] s tself the homology of a space wth a convoluton gvng multplcaton, and the quantzaton s C -equvarant homology. A whole zoo of algebras appear n both Hggs and Coulomb presentatons: hypertorc envelopng algebras, parabolc W-algebras, ratonal Cherednk algebras. I ll focus on the last of these. Representaton theory of symplectc sngulartes

23 Quantzatons Hggs and Coulomb branches Category O and the algebra T Hggs vs. Coulomb How are we supposed to study these algebras? Wth Le theory, we re startng from havng a century of experence, and a rather specal set up. On the Hggs sde, the geometrc column works pretty well. The replacement for D-modules s quantum coherent sheaves on a resoluton of the Hggs branch. These are hard to work wth (no sx functor formalsm) but close enough to G-equvarant D-modules on V to make thngs work. Wth Coulomb branches, the algebrac column s more successful. There s a natural torus n the quantzaton A and you can analyze ts weght spaces, wth the structure of the representaton V nfluencng how they are related. Representaton theory of symplectc sngulartes

24 Quantzatons Hggs and Coulomb branches Category O and the algebra T Category O Luckly, the combnatoral column can te them together. We call ξ A λ a gradng element f [ξ, ]: A λ A λ s sem-smple wth ntegral egenvalues. Defnton Category O ξ λ for ξ over A λ s the subcategory of modules where ξ acts wth fnte length Jordan blocks, and f.d. egenspaces, and egenvalues bounded above. Note ths category depends on ξ and λ; t s rchest f λ s ntegral n some approprate sense. These swtch roles between Hggs and Coulomb: a gradng element on one sde corresponds to an ntegral quantzaton parameter on the other. Representaton theory of symplectc sngulartes

25 Quantzatons Hggs and Coulomb branches Category O and the algebra T Category O We can defne a graded algebra T (purely based on the combnatorcs of G, V, ξ, λ) such that: Theorem (W.) For ξ ntegral: 1 T s somorphc to the endomorphsms of a projectve generator n O ξ λ for A Coulomb. 2 T s somorphc to the Ext-algebra of a sem-smple generator n O λ ξ for A Hggs (assumng certan hypotheses on V and G). 3 The category O λ ξ s Koszul dual to Oξ λ. You can thnk of the algebra T as a replacement of Soergel calculus n the Hggs/Coulomb context. Representaton theory of symplectc sngulartes

26 Quantzatons Hggs and Coulomb branches Category O and the algebra T Why? Ths should reflect some underlyng geometrc connecton between the varetes (there are many other concdences of underlyng geometrc nformaton). Conjecture (Braden-Lcata-Proudfoot-W.) There s a dualty on the set of symplectc sngulartes whch swtches Hggs and Coulomb branches. Representaton theory of symplectc sngulartes

27 Quantzatons Hggs and Coulomb branches Category O and the algebra T Why? The algebra T s graded, whereas the Coulomb category O sn t. Ths provdes a graded lft, whch has good postvty propertes. Theorem (W.) There are Verma modules n O ξ λ, and the multplctes of smples n them are gven by a verson of Kazhdan-Lusztg polynomals/canoncal bases. In partcular, we get a new proof of Rouquer s conjecture that decomposton numbers for ratonal Cherednk algebras are gven by affne parabolc KL polynomals. In general, lots of canoncal bases (for reps of Le algebras, for the Hecke algebra, etc.) show up ths way. Representaton theory of symplectc sngulartes

28 Quantzatons Hggs and Coulomb branches Category O and the algebra T Why? If we start wth a quver Γ, and dmenson vectors v, w, then quver varetes are the Hggs branches n the case V = j Hom(C v, C v j ) Hom(C v, C w ) G = GL(v, C). Theorem (W.) The category T -gmod categorfes a tensor product of smple representatons (dependng on w and ξ) for the quantum group attached to Γ (q =gradng shft). There are natural functors categorfyng all natural maps n quantum group theory. Ths allows us to construct a categorfed Reshetkhn-Turaev knot nvarant for any representaton. Even the Le algebras not attached to graphs (types BDFG) have an assocated algebra T, and ths constructon can be carred through purely combnatorally. Representaton theory of symplectc sngulartes

29 Quantzatons Hggs and Coulomb branches Category O and the algebra T KLR algebras The algebra T has a purely combnatoral defnton n terms of the hyperplane arrangement of t nduced by ϕ = n for the weghts ϕ of V and certan scalars n, (together wth the acton of W). In the case of a quver varety, the famly of algebras whch show up already has a rch theory: they are (weghted) Khovanov-Lauda-Rouquer algebras. Lke the Soergel calculus, we can defne ths algebra n terms of dagrams modulo local relatons. Let s consder the specal case of G = GL(n, C) V = gl(n) (C n ) l. The correspondng Coulomb quantzaton s the ratonal Cherednk algebra of S n wr Z/lZ. Representaton theory of symplectc sngulartes

30 Quantzatons Hggs and Coulomb branches Category O and the algebra T KLR algebras In ths case, the dagrams of T look lke: The x-values of the strands n a horzontal slce gve an element of t. The weghts of C n correspond to the red lnes: you cross a correspondng hyperplane when a red and black lne cross. the weghts of gl(n) are the dstance between ponts, so you cross one of ther hyperplanes when two strands cross, or reach a fxed dstance from each other. Representaton theory of symplectc sngulartes

31 Quantzatons Hggs and Coulomb branches Category O and the algebra T KLR algebras = for j j j = = + = + j m j m = 0 and = = j j + k + k + k + k + k + k = for + k j = +. j j + k + k + k = for + k j = zk = j j j j + k = + k + k + h + k j m = j m + δ,j,m j m = = = + h + k + k + k + k Representaton theory of symplectc sngulartes

32 Quantzatons Hggs and Coulomb branches Category O and the algebra T KLR algebras Thanks for lstenng. Representaton theory of symplectc sngulartes