Mednykh s Formula. Kevin Donoghue. January 29, 2016

Transcription

1 Mednykh s Formula Kevn Donoghue January 29, 206 These are notes rom a talk gven at the Temple Unversty graduate student semnar I learned ths materal rom [Bae], [DW90], and [Fre94] although the orgnal source (n Russan) s [Med78] An Analogy I start not wth Mednykh s ormula but somethng analogous: n n = π2 2 6 It s analogous because t s a surprsng statement whose proo nvolves: changng the bass o a space o unctons takng an nner product Explctly, consder (x) = x on [0, 2π] Wrte x = n Z a n e nx One can compute va the standard methods that x = π 2 + n 0 n enx Now take the L 2 norm o each sde The let hand s a calc ntegral The e nx terms on the rght hand sde are orthogonal, so one gets ( (2π) 3 = 2π π 2 + ) 3 n 2 n 0 rom whch the result easly ollows Note that the sum n n 2

2 had no busness admttng an explct expresson, let alone one so smple For example I beleve that or n 3 n t s unclear such an expresson exsts 2 Mednykh s Formula I wll compute a smlarly mraculous expresson, except ths tme rom the realm o topology Let G be a nte group and Σ a closed surace o genus g I propose to compute Hom(π Σ, G) The undamental group o Σ s so π Σ = x, y,, x g, y g [x, x ] [x g, y g ] = Hom(π Σ, G) = {(x, y,, x g, y g ) G 2g [x, y ] [x g, y g ] = } Snce G s nte ths s a nte number and you could have a computer calculate t But snce G could be some very werd nte group, an explct expresson seems hopeless Except t sn t: Theorem (Mednykh) Hom(π Σ, G) = ( dm V ) 2g 2 Ths s called Mednykh s ormula Here the V are the rreps o G Ths s a cool ormula smply because t exsts It s doubly cool because o the appearance o V I don t have the space to ully dscuss ths, but there are nvarants n topology (speccally, nvarants o knots) that nvolve the representatons o Le groups, but that don t have a clear topologcal nterpretaton It s lke the rght-hand sde exsts but the let-hand sde doesn t So although all the topology n ths talk s completely trval (suraces) there are analogs where the topology s hghly nontrval One can also, wth some more work, come up wth nte group analogs o those nvarants whch elucdate ther structure (a concrete example: to a nte group you can assgn a modular Hop algebra and plug that n the Reshetkhn-Turaev machne when you do so, the R-matrx s mmedately recognzable rom the Wrtnger presentaton) 2

3 3 Revew o Representaton Theory I rst revew representaton theory a lttle bt To complete the statement o the theorem, I should say what the V are A representaton o G s a homomorphsm ρ : G GL(V ) Everythng here s assumed to be over C, so t s a act that you can nd a bass or V such that ρ can be put n the orm (ρ ) (ρ k ) and moreoever, the blocks come rom a standard lst ρ : G GL(V ) ρ 2 : G GL(V 2 ) etc These are the V appearng n Mednykh s ormula and they re called rreps (rreducble representatons) For example, consder the representaton S 3 GL(R 3 ) gven by S 3 permutng the coordnate axes So s t represents a smple transposton and c a cycle, then t One can nd a bass n whch c t c ζ 3 ζ 2 3 3

4 Indeed there are three rreps or S 3 and they are: { t () ρ : c () { t ( ) ρ 2 : c () t ρ 3 : ζ 3 c ζ3 2 An mportant uncton n representaton theory s For example or S 3, χ (g) := tr(ρ (g)) e χ : t c e χ 2 : t c e 2 χ 3 : t 0 c Characters, beng traces, are nvarant under conjugaton So they orm unctons on conjugacy classes o G In act, t s a act that they orm a bass or unctons on conjugacy classes For example, or S 3 : I should x some notaton: δ e = χ + χ 2 + 2χ 3 6 C[G] s the group rng e, combnatons (g + h)(k + l) = gk + hk + gl + hl It s a noncommutatve algebra Greek letters α, β, γ, etc stand or conjugacy classes n G I wll use the notaton α = a α a C[G] 4

5 Dual to C[G] s F(G), unctons on G The center o the group algebra Z(C[G]) s spanned by the α and dual to α s the element δ α whch s on α and 0 on other conjugacy classes The dual to Z(C[G]) s thus naturally dented wth the class unctons F(Cl(G)) The centralzer C g o an element n G s the subgroup o elements whch conjugate g to tsel I g α, I wll wrte C α to denote the somorphsm class o ths group It ollows rom orbt-stablzer (have G act on tsel by conjugaton) that = C α α As mentoned earler there are two bases or F(Cl(G)): {δ α } and {χ } The characters wll be preerable because they satsy some serendptous denttes, the ollowng two o whch are needed here: 4 The Proo xyz= Some notaton to smply thngs: α C α χ (α)χ (α) = δ =j χ (x)χ j (y)χ k (z) = dm V δ =j=k := Hom(π Σ, G) := Hom(π Σ, G)/G thus the pcture represents the set o representatons modulo conjugacy there s no dot (The reason or the dot/dotless notaton s that whle the undamental group depends on the baseponts, ts conjugacy classes don t) The notaton denotes those homomorphsms n that map the ndcated crcle to α 5

6 For varous reasons, I wll not compute Hom(π Σ, G) but I wll rather compute the sze o Hom(π Σ, G)/G wth respect to a partcular measure I p : Hom(π Σ, G) Hom(π Σ, G)/G s the natural projecton, then dene ([φ]) := #p (φ) ths s the pushorward va p o the measure whch assgns / to every element Then the total measure o s Hom(π Σ, G) so Mednykh s ormula reduces to = n 2g 2 = dm V The approach to provng ths dentty wll be n two steps the same two steps n whch vrtually everythng about suraces s proved: rst understand how the nvarant changes when cuttng along a crcle:,? and then understand the nvarant or the par o pants: =? To start attackng the rst problem, start by understandng those homomorphsms wth speced boundary behavor: = # a α I a and a 2 are conjugate then any g conjugatng a to a 2 nduces a bjecton o sets = so ( = α # ) ( = # C α ), any a α 6

7 Lemma = α C α Proo The let hand sde s tautologcally equal to # g G whch, by Seert-van Kampen (unversalty o the ree product!) s equal to # # = # # g G = α a α C α 2 α a α rom whch the lemma easly ollows All ths works or nonseparatng suraces as well, except there one needs to the use groupod Seert-van Kampen It should be remarked that ths s usually presented n the language o TQFTs In that settng, a vector space (precsely F(Cl(G))) s assgned to a crcle and numbers are assgned to closed 3-manolds It can hard to understand the motvaton or such assgnments Ths lemma shows an nner product structure that comes rom Seert-van Kampen and thus oers a topologcal explanaton o the apperance o vector spaces n the TQFT setup Indeed the smlarty to an nner product necessarly provokes the ollowng denton := α δ α F(Cl(G)) Wth ths n place then the lemma can be re-expressed as =, where, s the nner product on F(Cl(G)) gven by { C α α = β δ α, δ β = 0 otherwse Note that that by denton = 7

8 Lemma = α,β,γ δ (αβγ)δ α δ β δ γ Proo The essental part o the proo s to note that only depends on the undamental group, so we can replace the par o pants by a trangle wth the three vertces all dented: = # a α b β c γ Now t should be clear how to compute Smply take the nner product o the par o pants wth tsel: = C 2 α C β C γ δ (αβγ) 2 Unortunately ths ormula s an absolute mess We re rescued by a cool act rom lnear algebra There s a canoncal somorphsm α,β,γ End(V ) = V V and End(V ) has a canoncal element d, whch corresponds to e e V V Here e s a bass o V and e s the dual bass n V : e j (e ) = δ j In partcular, or two bases e and, the ollowng are equal: e e = and so, n partcular, = δ (e e j e k )e e j e k or any bass {e } o F(Cl(G)) Usng the acts about characters mentoned beore, n partcular that α χ (α)χ j (α) C α 8 = δ =j,

9 one can see that e = α χ (α) C α δ α s orthonormal wth respect to,, and that e = α χ (α)α s dual But snce δ (e e j e k ) = xyz= χ (x)χ j (y)χ k (z) = dm V δ =j=k we have = dm V e e e and takng the norm squared o that gves Mednykh s ormula (or g = 2): = 2 dm V The ormula or general genus s analogous, just wth 2g 2 pars o pants nstead o 2 Remark Such a proo may seem hghly concdental but t s actually not The pont s that the assgnment X Hom(π X, G) s unctoral, so the process o pecng together pars o pants whch aesthetcally resembles the processes multplcaton n algebra nduces somethng lke an algebrac structure on the system o sets Hom(π X, G) When one orcbly starts workng over C, one gets an actual algebra structure Ths s the phlosophy behnd what s called topologcal quantum eld theory One can make somethng lke ths work or a general compact Le group G The set Hom(π Σ, G) s then possbly nnte, and possbly hghly sngular But t turns out the generc ponts orm a manold o dmenson (2g 2) dm(g) (at least or g 2) and there s a natural measure ( s essentally the Redemester torson o the twsted homology H (Σ; g Ad )) Up to a G-ndependent normalzaton, then = 2g 2 vol(g) dm V Ths s done n [Wt9] 9

10 Reerences [Bae] John Baez, Quantum Geometry Semnar, Fall baez/qg-all2004/ [DW90] Robbert Djkgraa and Edward Wtten, Topologcal gauge theores and group cohomology, Comm Math Phys 29 (990), no 2, [Fre94] Danel S Freed, Hgher algebrac structures and quantzaton, Comm Math Phys 59 (994), no 2, [Med78] A D Mednyh, Determnaton o the number o nonequvalent coverngs over a compact Remann surace, Dokl Akad Nauk SSSR 239 (978), no 2, [Wt9] Edward Wtten, On quantum gauge theores n two dmensons, Comm Math Phys 4 (99), no,