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From CFT to CFT?

Matilde Marcolli

Banff, April 2012

Matilde Marcolli From CFT to CFT?

Disclaimer: this is a largely speculative talk, meant for an informal discussion session at the workshop “Novel approaches to the finite simple groups” in Banff

Matilde Marcolli From CFT to CFT?

Quantum Statistical Mechanics and Class Field Theory

QSM approach to Class Field Theory: number field K

• QSM system (A, σt) • partition function Z (β) is Dedekind zeta function ζK(β). • phase transition spontaneous symmetry breaking at pole β = 1 • unique equilibrium state above critical temperature • quotient CK/DK (idèle class group by connected component) acts by symmetries • subalgebra A0 of A: values of extremal ground states on A0 are algebraic numbers and generate K ab

• Galois action by CK/DK via CFT isom θ : CK/DK → Gal(K ab/K )

Matilde Marcolli From CFT to CFT?

Quantum Statistical Mechanics A = algebra of observables (C∗-algebra) State: ϕ : A → C linear

ϕ(1) = 1, ϕ(a∗a) ≥ 0

Time evolution σt ∈ Aut(A) rep π : A → B(H) on Hilbert space H

Hamiltonian H = d

dt σt |t=0

π(σt(a)) = e itHπ(a)e−itH

Symmetries • Automorphisms: G ⊂ Aut(A), gσt = σtg ; inner: a 7→ uau∗ with u = unitary, σt(u) = u, • Endomorphisms: ρσt = σtρ e = ρ(1) (need ϕ(e) 6= 0)

ρ∗(ϕ) = 1

ϕ(e) ϕ ◦ ρ

inner: u = isometry with σt(u) = λ itu

Matilde Marcolli From CFT to CFT?

Equilibrium states (inverse temperature β = 1/kT )

1

Z (β) Tr ( a e−βH

) Z (β) = Tr

( e−βH

) More general: KMS states ϕ ∈ KMSβ (0 < β 0 simplex KMSβ ; extremal Eβ (points of NC space) At T = 0: KMS∞ = weak limits of KMSβ

ϕ∞(a) = lim β→∞

ϕβ(a)

Matilde Marcolli From CFT to CFT?

The Bost–Connes system Algebra AQ,BC = Q[Q/Z] oN generators and relations

µnµm = µnm µnµ

∗ m = µ

∗ mµn when (n,m) = 1

µ∗nµn = 1

e(r + s) = e(r)e(s), e(0) = 1

ρn(e(r)) = µne(r)µ ∗ n =

1

n

∑ ns=r

e(s)

C ∗-algebra C ∗(Q/Z) oN = C (Ẑ) oN with time evolution

σt(e(r)) = e(r), σt(µn) = n itµn

Rep on `2(N), partition function Tr(e−βH) = ζ(β)

• J.B. Bost, A. Connes, Hecke algebras, Type III factors and phase transitions with spontaneous symmetry breaking in number theory, Selecta Math. (1995)

Matilde Marcolli From CFT to CFT?

KMS states of the BC system • Representations πρ on `2(N):

µn�m = �nm, πρ(e(r))�m = ζ m r �m

ζr = ρ(e(r)) root of 1, for ρ ∈ Ẑ∗ • Low temperature extremal KMS (β > 1) Gibbs states

ϕβ,ρ(a) = Tr(πρ(a)e

−βH)

Tr(e−βH) , ρ ∈ Ẑ∗

• phase transition at β = 1; high temperature: unique KMS state • Zero temperature: evaluations ϕ∞,ρ(e(r)) = ζr

ϕ∞,ρ(a) = 〈�1, πρ(a)�1〉

Intertwining: a ∈ AQ,BC , γ ∈ Ẑ∗

ϕ∞,ρ(γa) = θγ(ϕ∞,ρ(a))

θ : Ẑ∗ '→ Gal(Qab/Q) Class field theory isomorphism

Matilde Marcolli From CFT to CFT?

Gibbs states near the phase transition • Gibbs states of BC system are polylogs at roots of unity

ϕβ,ρ(e(r)) = ζ(β) −1 ∑ n≥1

ζnr nβ

= ζ(β)−1Liβ(ζr )

Lis(z) = ∞∑ n=1

zn

ns

• The zeta function ζ(β) has a pole at β = 1 • Behavior of these KMS states as β → 1: near criticality behavior of expectation values

Matilde Marcolli From CFT to CFT?

Useful polylogs identities • Fourier sums: ζ(s, a) = Hurwitz zeta function

Lis(e 2πim/p) = p−s

p∑ k=1

e2πimk/pζ(s, k

p )

• multiplication formula: p−1∑ m=0

Lis(ze 2πim/p) = p1−s Lis(z

p)

• Fermi–Dirac distribution −Lis+1(e−µ) • − log(−µ) limit as s → 1:

Lis(e µ) = Γ(1− s)(−µ)s−1 +

∞∑ k=0

ζ(s − k) k!

µk

lim s→k+1

( ζ(s − k)

k! µk + Γ(1− s)(−µ)s−1

) = µk

k! (Hk − log(−µ))

Hn = ∑n

k=1 1/k harmonic numbers

Matilde Marcolli From CFT to CFT?

QSM systems and complex multiplication (Connes–Marcolli–Ramachandran)

K = Q( √ −D) imaginary quadratic field

• 1-dimensional K-lattice (Λ, φ): fin gen O-submod Λ ⊂ C with Λ⊗O K ∼= K and O-mod morphism φ : K/O → KΛ/Λ (invertible is φ isom)

• (Λ1, φ1) and (Λ2, φ2) are commensurable if KΛ1 = KΛ2 and φ1 = φ2 modulo Λ1 + Λ2

• space of 1-dim K-lattices Ô ×Ô∗ (A ∗ K/K∗): adelic description of

lattices (Λ, φ) as (ρ, s), ρ ∈ Ô and s ∈ A∗K/K∗, mod (ρ, s) 7→ (x−1ρ, xs), x ∈ Ô∗

• commensurability classes: A·K/K∗ with A·K = AK,f × C∗ (nontrivial archimedean component)

Matilde Marcolli From CFT to CFT?

• Groupoid algebra of commensurability of 1-dim K-lattices C0(A·K) oK∗, up to scaling: Ô ×Ô∗ (A

∗ K/K∗) mod C∗ is

Ô ×Ô∗ (A ∗ K,f /K∗) and commensurability by (ρ, s) 7→ (sJρ, s

−1 J s),

J ⊂ O ideal adelically J = sJÔ ∩K

C (Ô ×Ô∗ (A ∗ K,f /K∗)) o J

+ K

• Class number from A∗K,f /(K∗ × Ô∗) = Cl(O)

• Ô∗ acts by automorphisms, semigroup Ô ∩ A∗K,f by endomorphisms, O× by inner • time evolution

σt(f )((Λ, φ), (Λ ′, φ′)) =

∣∣∣∣Covol(Λ′)Covol(Λ) ∣∣∣∣it f ((Λ, φ), (Λ′, φ′))

by n(J)it on commens class of invertible (positive energy)

Matilde Marcolli From CFT to CFT?

• Arithmetic subalgebra from modular functions evaluated at CM points in H (using restriction from GL2-system of Connes–Marcolli) • Partition function ζK(β) Dedekind zeta function • Gibbs states low temperature; phase transition at β = 1; unique KMS above

• At zero temperature intertwining of symmetries and Galois action (Class Field Theory)

1 // K∗ // GL1(AK,f )

��

' // Gal(Kab/K) // 1

1 // Q∗ // GL2(Af ) ' // Aut(F ) // 1.

using Shimura reciprocity and GL2-system

Matilde Marcolli From CFT to CFT?

General systems for number fields (Ha–Paugam)

AK := C (XK) o J+K , with XK := G ab K ×Ô∗K ÔK,

ÔK = ring of finite integral adeles, J+K = is the semigroup of ideals, acting on XK by Artin reciprocity

• Crossed product algebra AK := C (XK) o J+K , generators and relations: f ∈ C (XK) and µn, n ∈ J+K

µnµ ∗ n = en; µ

∗ nµn = 1; ρn(f ) = µnf µ

∗ n;

σn(f )en = µ ∗ nf µn; σn(ρn(f )) = f ; ρn(σn(f )) = fen

Matilde Marcolli From CFT to CFT?

• Artin reciprocity map ϑK : A∗K → G abK , write ϑK(n) for ideal n seen as idele by non-canonical section s of

A∗K,f // // JK s

]] : (xp)p 7→

∏ p finite

pvp(xp)

• semigroup action: n ∈ J+K acting on f ∈ C (XK) as

ρn(f )(γ, ρ) = f (ϑK(n)γ, s(n) −1ρ)en,

en = µnµ ∗ n projector onto [(γ, ρ)] with s(n)

−1ρ ∈ ÔK • partial inverse of semigroup action:

σn(f )(x) = f (n ∗ x) with n ∗ [(γ, ρ)] = [(ϑK(n)−1γ, nρ)]

• time evolution on J+K as phase factor N(n)it

σK,t(f ) = f and σK,t(µn) = N(n) it µn

for f ∈ C (G abK ×Ô∗K ÔK) and for n ∈ J + K

Matilde Marcolli From CFT to CFT?

Properties of QSM systems for number fields

• Partition function Dedekind ζK(β); symmetry action of G abK • Complete classification of KMS states (Laca–Larsen–Neshveyev): low temperature Gibbs states; phase transition at β = 1; unique high temperature state

• Reconstrution (Cornelissen–Marcolli): isomorphism of QSM systems (AK, σK) ' (AK′ , σK′) preserving suitable algebraic subalgebra determines field isomorphism K ' K′

• Arithmetic subalgebra for Class Field Theory via endomotives (Yalkinoglu)

Matilde Marcolli From CFT to CFT?

Gibbs states near the phase transition: • Partition function = Dedekind zeta function has pole at β = 1

• Residue given by class number formula

lim s→1

(s − 1)ζK(s) = 2r1(2π)r2hKRegK

wK √ |DK|

with [K : Q] = n = r1 + 2r2 (r1 real and r2 pairs of complex embeddings), hK = #Cl(O) class number; RegK regulator; wK numer of roots of unity in K; DK discriminant

• For a quadratic field ζK(s) = ζ(s)L(χ, s) with L(χ, s) Dirichlet L-series with character χ(n) = (DKn ) Legendre symbol

( a

p ) =

1 a square mod p (a 6= 0 mod p) −1 a not square mod p

0 a = 0 mod p

Matilde Marcolli From CFT to CFT?

• KMS states at low temperature

ϕγ,β(f ) = 1

ζK(β)

∑ n∈J+K

f (n ∗ γ) NK(n)β

• Also know (Cornelissen–Marcolli) L-functions with Grossencharacters related to values of KMS states: character χ ∈ Ĝ abK gives function

fχ(γ, ρ) :=

{ χ−1(γϑK(ρ

′)) if ∀v | fχ, ρv ∈ Ô∗K,v 0 otherwise,

with ρ′ ∈ Ô∗K such that ρ′v = ρv for all v | fχ

ϕ