Knot invariants via quantizations of Hecke modifications

Transcription

1 Knot nvarants va quantzatons of Hecke modfcatons Unversty of Vrgna Unversty of Waterloo Permeter Insttute for Mathematcal Physcs Aprl 29, 2017

2 Two approaches to Jones polynomal (1984) Jones defnes the Jones polynomal, based on the acton of the Temperley-Leb algebra on subfactors ( ) Other combnatoral approaches to Jones polynomal and generalzatons developed (Kaufmann bracket, HOMFLY, Hecke algebras). All of these depend on a proecton of K (or at least a brad representatve). (1989) Wtten shows that there s an truly 3-dmensonal way of understandng the Jones polynomal: ts specalzatons at roots of unty are the expectaton value of the trace of the holonomy around the knot for an connecton on a trval bundle w/fber C 2. The (nfnte-dmensonal) space of connectons s gven a Chern-Smons weghtng whch depends on the root of unty. Ths ntegral makes sense for other 3-manfolds, but wll gve you functon that sn t the restrcton of a polynomal.

3 Two approaches to Jones polynomal So, we re left wth two approaches: one combnatoral and one geometrc. How do we reconcle them? In partcular, Wtten s approach nvolves an nfnte dmensonal ntegral whch doesn t obvously actually make sense. In fact, Wtten complans several tmes n hs paper about mathematcans dscomfort wth ths ntegral. The magc words are dmensonal reducton. In crude terms: we choose a dvdng surface nsde our 3-manfold, and break up the ntegral over connectons n terms of ther behavor near ths surface.

4 Dmensonal reducton For every 3-manfold wth boundary, and tangle n ths 3-manfold, we thus have the nformaton of the ntegral of the holonomy over the connectons wth fxed behavor on the boundary. Theorem Ths nformaton can be encoded n a vector n the fnte dmensonal Hlbert space H Σ of the marked surface Σ; from the physcs, we naturally dentfy H Σ wth the conformal blocks of Σ (for any fxed complex structure). A dagram of a knot bascally corresponds to a slcng of S 3 by S 2 s so that only one crossng, cup or cap occurs between each par. If we ust want to understand knots n S 3, then we can safely assume that Σ = S 2 wth some number of markngs.

5 Dmensonal reducton Now, quantum groups enter stage rght. They provde a more combnatoral way of understandng these conformal blocks. Theorem For k large, the Hlbert space H Σ wth Σ the sphere wth n marked ponts s the nvarants of (C(q) 2 ) n under the acton of U q (sl 2 ). For hgher rank groups and more nterestng marked ponts, the same result carres over: we always get the nvarants of a tensor product. One hnt that S 2 and thus S 3 are specal: for any other Σ, then the dmenson of the conformal blocks keeps growng as k grows, rather than convergng to a fxed space.

6 Dmensonal reducton The representaton (C(q) 2 ) n tself s attached to the dsk wth n marked ponts. The crcle boundary s corresponds to the U q (sl 2 )-acton. Glung dsks along the boundary corresponds to takng Hom of U q (sl 2 )-representatons. An unmarked dsk corresponds to the trval representaton, so closng up the dsk to a sphere corresponds to takng nvarants. Glung two dsks nto a bgger one corresponds to tensor product. The fact that these dsks can swtch places, but you have to choose whch drecton they rotate, corresponds to the fact that ths tensor product s braded.

7 Dmensonal reducton The full package we obtan s a TQFT: Attached to a 3-manfold wth knot, we have the expectaton of holonomy around the knot. For a 2-manfold wth markngs, we obtan the Hlbert space H Σ. attached to the crcle, we obtan the braded tensor category of (sem-smplfed) representatons of U q (sl 2 ) representatons.

8 Dmensonal reducton There s a slghtly dsatsfyng pont here: the constructon of the Jones polynomal n ths perspectve sn t obvously a polynomal wth nteger coeffcents. There should be a physcal explanaton of ths fact. Luckly, ths bothered Wtten too, and he found an explanaton. Theorem (Gaotto-Wtten) The coeffcents of the Jones polynomal (and more generally RT nvarants) are gven by sgned counts of g-valued connectons A and 1-forms φ solutons to the Kapustn-Wtten PDEs on S 3 R + F A φ φ = d A φ d A φ = 0 wth certan fxed sngulartes along the knot K embedded n S 3 {0}.

9 A heurstc for categorfcaton An dea at least gong back to Crane and Frenkel s that Chern-Smons theory s really a shadow of a 4-dmensonal theory. Ths would lead us to expect: Each dsk wth marked for the representatons V at varous ponts gves a categorfcaton of V 1 V n. Each tangle should act wth a functor on ths category. Most mportant peces are a brad group actons and obects correspondng to crossngless matchngs. We can buld a knot by startng wth a matchng A, applyng the brad σ to that, and then cappng off wth another matchng B. On the level of categores, we thnk of A and B as the correspondng obects, and consder Ext(B, σa).

10 A heurstc for categorfcaton Wtten s second descrpton gves a natural possblty for ths categorfcaton: Heurstc The count of solutons to an equaton can be categorfed to the vector space f they are the crtcal ponts of a Morse functon, or more generally, the ntersecton of two Lagrangans n a symplectc manfold. Luckly, the equatons dscussed above are the crtcal ponts of a collecton of equatons n 5 dmensons, called the Haydys-Wtten equatons. Conecture (Wtten) The Khovanov homology of a lnk K can be wrtten as Morse homology for these equatons.

11 A heurstc for categorfcaton How can we prove somethng lke ths, or at least use t for nspraton n mathematcal constructons? Agan, dmensonal reducton. Frst, let s thnk about countng solutons of the KW equatons usng dmensonal reducton: we can do ths by consderng the space of solutons on a tubular neghborhood of the surface C, count the ways of extendng each of these to two halves, and count when they match up. The solutons on the tubular neghborhood whch are constant n the normal drecton gve a symplectc manfold and those extendng over the two halves gve two Lagrangan subvaretes whch we are ntersectng.

12 A heurstc for categorfcaton On the tubular neghborhood, lookng at solutons of the KW equatons that are constant n the normal drecton, we obtan the Bogomolny equaton F = d A φ on C R, wth certan fxed sngulartes on the marked ponts. If we pck a complex structure on C, then d A = A + A nduces a holomorphc structure on the bundle on C {r} for r R, and the Bogomolny equaton says that ths structure s constant n r, except that the structure can change at the marked ponts (where φ has a pole). These changes are called Hecke modfcatons. Thus, the coeffcents of the Jones polynomal count the ntersectons of two Lagrangans n a space of Hecke modfcatons.

13 Symplectc Khovanov homology That s somethng we know how to upgrade! We can do Floer homology nstead! Ths was mplemented for Khovanov homology by Sedel-Smth. They never sad that they were lookng at Hecke modfcatons, but they study a varety of matrces Y m whch corresponds to 2m marked ponts.

14 Symplectc Khovanov homology Attached to a sphere wth 2m marked ponts, we have the Fukaya category of compact Lagrangans n Y m. Attached to a dsk wth 2m marked ponts, we have a larger Fukaya category where we allow non-compactness n one drecton. The acton of brads corresponds to a parallel transport functor. The acton of cups and caps s va push-pull on a Lagrangan correspondence I p Y m 1 Y m. Theorem (Sedel-Smth, Abouzad-Smth) Ths constructon gves us Khovanov homology.

15 Symplectc Khovanov homology We shouldn t gve Wtten too much credt for conecturng Sedel-Smth homology. Rather, knowng about t allowed Wtten to guess the formalsm above. However, n other types, especally outsde type A, we don t have a Sedel-Smth constructon, so ths can gve us nspraton about what to do. However, Fukaya categores are too hard for me; for types other than A, we don t even know how to get smooth spaces of Hecke modfcatons. However, we can replace them by lookng at deformaton quantzaton nstead.

16 Almost commutatve algebras commutatve non-commutatve

17 Almost commutatve algebras commutatve sem-classcal almost commutatve non-commutatve

18 Almost commutatve algebras algebrac geometry representaton theory commutatve sem-classcal almost commutatve non-commutatve

19 Almost commutatve algebras algebrac geometry representaton theory commutatve sem-classcal X almost commutatve non-commutatve where I d lke to be

20 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 A + [A, A ] A + n In partcular, the rng gr(a) = =0 A /A 1 s commutatve and Z 0 -graded.

21 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 + n /A + n 1.

22 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.

23 Hecke modfcatons Theorem (Kamntzer-W-Weekes-Yacob) The space of Hecke modfcatons for any choce of labels V λ on marked ponts has a quantzaton gven by a truncated shfted Yangan Y λ where λ = λ λ m. Ths quantzaton has a large polynomal subalgebra wth ρˇ(λ) generators; and we can study the representaton theory of Y λ, usng the theory of weghts for ths torus. We can wrte λ = v α as a sum of smple roots.

24 Hecke modfcatons A weght for ths torus s a choce of ntegers a,1,..., a,v for all. We can draw these on the real lne, but we have to remember whch node they come from, so we label the ponts. k There s a choce of parameters r wll also be relevant, so we should mark those n red.

25 Hecke modfcatons A weght for ths torus s a choce of ntegers a,1,..., a,v for all. We can draw these on the real lne, but we have to remember whch node they come from, so we label the ponts. k λ 1 λ 2 There s a choce of parameters r wll also be relevant, so we should mark those n red.

26 Hecke modfcatons Weght vectors for the adont acton of the polynomals defne maps between weght spaces: we stll have the acton of the z,m s. We ve fxed the sem-smple part of ther acton, but the nlpotent part s stll nterestng. we have monopole operators whch change the weght. when two ponts wth the same label get close (nothng n between), you can ht them wth a Demazure operator.

27 Dagrams After futzng wth nvertble factors, we can defne a set of (natural) maps between weght spaces whch we represent wth the dagrams proecton to weght space nlpotent part of z,m 1 2 n e demazure 1 n y monopole operator(sh) 1 n 1 λ λ 2 4 ψ

28 Dagrams The relatons are local, and gven by = unless = = + = Q (y 1, y 2 ) = unless = k = ± 1 k k = 0 = Q (y 3, y 2 ) Q (y 1, y 2 ) y 3 y 1

29 Dagrams The relatons are local, and gven by = unless = = + = Q (y 1, y 2 ) = unless = k = ± 1 k k = 0 = Q (y 3, y 2 ) Q (y 1, y 2 ) y 3 y 1

30 Dagrams = α (λ) λ = λ + a+b=λ 1 a λ b λ λ = = α (λ) λ λ λ λ = =

31 Dagrams Call the resultng algebra T r. Theorem (W.) If the choces r are generc ntegral then the category of generalzed weght modules of A r s equvalent to the category of fnte dmensonal T r -modules where the dots act nlpotently. Ths algebra has a quotent T r whch actually categorfes the tensor product V λ1 V λl ; ths quotent klls any dagram wth a black strand left of all red ones at any horzontal slce. Theorem Modules that factor through T r correspond under the equvalence to category O, the category of modules where the sums,m a,m for all non-zero weght spaces satsfy,m a,m > N for some N 0.

32 Brads and cups The general theory of modules over quantzatons provdes a natural brad group acton on the derved category of modules over A ξ, called twstng functors; these come from tensorng wth sectons correspondng to quantzed lne bundles on a resoluton. Put another way, these relate to changng the quantzaton parameters. Theorem (W.) Under the equvalence, the acton of twstng functors agrees wth derved tensor product wth bmodules B σ spanned by dagrams that look lke λ 1 λ 3 λ 2 λ 1 λ 2 λ 3

33 Brads and cups The modules correspondng to crossngless matchngs can be defned purely algebracally by a smlar ansatz: draw the tangle n red, and then let black strands do what they lke. For sl 2, the resultng module looks lke: Proposton Ths module L X categorfes the class n the tensor product of the correspondng terated coevaluaton correspondng to the crossngless matchng X. Ths corresponds to a fnte dmensonal A r -module. Flppng the dagram, I get a rght module over T r, denoted R X.

34 The defnton Theorem (W.) If I wrte a knot K by startng wth a matchng X, actng by a brad σ and then cappng wth a matchng Y, then the the homology of the complex of graded vector spaces K (K) = R Y L Bσ L LX gves a knot nvarant, categorfyng the Reshetkhn-Turaev nvarant, and agreeng wth Khovanov, Khovanov-Rozansky, Cauts-Kamntzer, etc. n approprate specal cases.

35 The defnton It s temptng to hope that f we ust lsten very carefully to Wtten, we wll fgure out the categores attached to other surfaces, and thus to arbtrary 3-manfolds. However, Wtten hems and haws a far amount f you ask hm about ths. There s always a very serous danger that you try to count an nfnte (or worse, non-compact) soluton set. Apparently the physcs suggests dsks really are specal. It s worth notng that ths physcal approach requres fxng a root of unty. Thus, we mght need to wat for some advances n root of unty categorfcaton, and to fnd out how these connect to physcs.