M 2018 Hecke Relations in Rational Conformal Field Theory (with - - PowerPoint PPT Presentation

Hecke Relations in Rational Conformal Field Theory

(with Y. Wu, arXiv:1804.06860)

Jeff Harvey University of Chicago

M ∩ Φ 2018

INTRODUCTION

This talk concerns a new relation between characters

• f two-dimensional rational conformal field theory

(RCFT) given by Hecke operators. The relation generalizes previously known Galois symmetry relations between the representations of the modular group provided by RCFT characters. I will discuss these new relations, explain how they explain a number of scattered results in the literature and present some possible applications.

Examples:

The Yang-Lee model is a non-unitary minimal model (RCFT) with two independent characters

M(5, 2)

χY L = q−1/60 X

n=0

cY L (n)qn χY L

1/5 = q11/60 X n=0

cY L

1/5(n)qn

Affine is a unitary rational CFT with two independent characters

G2

χG2 = q−7/60 X

n=0

cG2

0 (n)qn

χG2

17/60 = q17/60 X n=0

cG2

17/60(n)qn

Although apparently unrelated, there is a subtle relation between the character coefficients:

2 6 20 50 120 261 3858/7 1091 2108 3917 7118 12587 21854 260072/7 62202 102428 166450 266850 422966 662780 7198178/7 1579853 2406046 3633082 5443362 8094202 11952388 122716344/7

coefficients of q expansion of vacuum character of affine G2 at level one (c=14/5) divided by 7 coefficients of q expansion of h=1/5 character of Yang-Lee model c=-22/5

DATA

1/7 1/7 1/7 2/7 Discrepancy

Z(τ) = X

i∈I,¯ i∈¯ I

Ni,¯

iχi(τ)χ¯ i(τ)

RCFT

Hilbert space: Characters: Partition function: Examples: Ising model, Yang-Lee model, affine Lie algebras, Monster VOA, BM VOA

Representation of chiral algebra, e.g. Virasoro for minimal models

χi(τ) = TrViqL0−c/24, q = e2πiτ H = M

i,¯ i

Ni,¯

iVi ⊗ V¯ i

ρ : SL(2, Z) → GL(n, C)

C2 = 1

Γ(N) = ⇢✓ a b c d ◆ ∈ SL(2, Z)| a = d = 1 mod N, b = c = 0 mod N

• S =

✓0 −1 1 ◆

Modular Properties

Characters are weakly holomorphic weight zero vector- valued modular functions transforming according to T = ✓1 1 1 ◆ Γ(N) ⊂ ker(ρ) N = order(ρ(T)) (ρ(S))2 = (ρ(S)ρ(T))3 = C

(Bantay)

(τ → −1/τ) (τ → τ + 1)

χi(−1/τ) = X

j

ρ(S)ijχj(τ)

Example

Ising Model

χI

0 = 1

2 s θ3(τ) η(τ) + s θ4(τ) η(τ) ! = q−1/48(1 + q2 + q3 + 2q4 + 2q5 + · · · ), χI

1/2 = 1

2 s θ3(τ) η(τ) − s θ4(τ) η(τ) ! = q23/48(1 + q + q2 + q3 + 2q4 + 2q5 + · · · ), χI

1/16 =

1 √ 2 s θ2(τ) η(τ) = q1/24(1 + q + q2 + 2q3 + 2q4 + 3q5 + · · · ) .

N = 48

ρ(S) =   1/2 1/2 1/ √ 2 1/2 1/2 −1/ √ 2 1/ √ 2 −1/ √ 2  

ρ(T) = e−2πi/48diag(1, e2πi/2, e2πi/16)

χi(τ + 1) = X

j

ρ(T)ijχj(τ)

K ⊆ Q[ζN] ε = ±1

Galois Symmetry

K=Field extension of by matrix elements of ρ(γ) Q De Boer & Goeree plus Kronecker-Weber: N=conductor (also of RCFT). Gal(Q[ζN]) ∼ = (Z/NZ)×

Group of units in Ring Z/NZ

ρ(S) → G`ρ(S)

fN,` : ρ(S)i,j → ε`(i)ρ(S)⇡`(i),j = ε`(j)ρ(S)i,⇡`(j) fN,` : ρ(T) → ρ(T)` ` ∈ (Z/NZ)× Coste-Gannon: For each

permutation of indices Primitive Nth root of unity

1 12 2 3 4 5 6 11 10 9 8 7 Z/12Z (Z/12Z)× 5 7 11 f5 f7 f11

Examples of Galois RCFT Relations

f60,7 Similarly f60,13 Three character RCFT

ρ(S)Y L = −

1 2 sin(π/5) 1 2 sin(2π/5) 1 2 sin(2π/5) 1 2 sin(π/5)

!

ρ(T)Y L = diag(e2π11/60, e−2πi/60) ρ(S)G2 = ✓ −1 1 ◆ ρ(S)Y L

ρ(T)G2 = (ρ(T)Y L)7

Yang-Lee F4 Ising f48,47 Baby Monster

These are relations between modular representations. We will extend them to relations between characters.

ω0

1

T6 ω1

Modular forms can be thought of in two ways: Functions of τ ∈ H

f ✓aτ + b cτ + d ◆ = (cτ + d)kf(τ)

Functions of rank 2 lattices

F(λL) = λ−kF(L) F(Zω1 + Zω2) = ω−k

2 f(ω1/ω2)

Hecke operator:

(TnF)(L) = X

L0⇢L |L/L0|=n

F(L0)

Scalar Hecke

ω2

ω0

1 = aω1 + bω2

ω0

2 = cω1 + dω2

ω0

2

ad − bc = n

Hecke Operators

f = X

n

a(n)qn (Tpf)(τ) = X

n

a(p)(n)qn

a(p)(n) = ( pka(pn) if p - n, pk−1 p a(pn) + a(n/p)

• if p|n.

Action on Fourier coefficients:

When translated into an action on modular forms as a function of or this leads to τ q = e2πiτ

To generalize Hecke operators to RCFT characters: Use Hecke operators for (Rankin, Chap. 9) Γ(N) Use modular representation properties of RCFT characters. This leads to the following formula for Hecke images of RCFT characters for (p,N)=1: On Fourier coefficients:

χi(τ) = X

n

bi(n)qn/N

(Tpχ)i(τ) = X

n

b(p)

i

(n)qn/N

b(p)

i

(n) = ( pbi(np) p - n pbi(np) + P

b ρij(σp)bj(n/p)

p|n The are again modular forms for but transform under a different representation of (Tpχ)i Γ(N) SL(2, Z)

Example: N = 60, p = 7, ¯ p = 43 σ7 = ✓ 27343 −33780 480 −593 ◆ σp where is the pre-image of under the mod N map and ✓¯ p p ◆ SL(2, Z) → SL(2, Z/NZ) ¯ pp = 1 mod N Representation

• f Hecke image

ρ(p)(S) = ρ(σpS), ρ(p)(T) = ρ(T ¯

p)

The change of representation under Hecke is the same as that under Galois for .

The equivalence relies on the identities

fN,p(ρ(S)) = ρ(σ¯

pS)

fN,p(ρ(T)) = ρ(T p)

` = p, (p, N) = 1

ρY L(σ7) = ✓0 −1 1 ◆

Applications and Examples

The example from the first DATA slide:

χY L = q−1/60G(q) = q−1/60

∞

X

n=0

cY L (n)qn χY L

1/5 = q11/60H(q) = q11/60 ∞

X

n=0

cY L

1/5(n)qn

Then we have the Hecke relation χG2 = T7χY L

cG2

0 (n) =

( 7cY L

1/5(7n − 1)

if 7 - n, 7cY L

1/5(7n − 1) + cY L

( n

7 )

if 7|n; cG2

2/5(n) =

( 7cY L (7n + 2) if 7 - (n − 1), 7cY L (7n + 2) − cY L

1/5( n−1 7 )

if 7|(n − 1).

Mk(Γ) → Mk+2(Γ)

Modular Linear Differential Equation

Ramanujan-Serre: Dk = d/dτ − 1 6iπkE2 MLDEs are a useful tool for studying and classifying characters of RCFT (Anderson-Moore, Eguchi-Ooguri, Mathur, Mukhi, Sen; Kaneko-Zagier,…) Second order: D2f − E2 6 Df − µE4 4 f = 0 D = 1 2πi d dτ X = {Y L, A1, A2, G2, D4, F4, E6, E7, E7 1

2 }

Yang-Lee model

Affine level 1 characters Deligne exceptional series Characters of Intermediate Vertex Subalgebra (Kawasetsu)

quasi modular weight 2 Eisenstein series

:

Solutions ( ) completely classified by Mathur, Mukhi, Sen ` = 0

Three character theories: No classification I am aware

• f. Examples include

Two character theories:

More Examples

χF4 = T13χY L

χ

E7 1

2 = T19χY L

χE7 = T7χA1

Minimal models: M4,3 (Ising)

M⊗2

5,2 (YL⊗2)

M7,2

BabyMonster

, ,

Models w/o Kac-Moody symmetry (Hampapura-Mukhi): c = 236/7 = 59 × 4/7

c = 164/5 = 41 × 4/5

c = 47/2 = 47 × 1/2

“Duality” of H-M and relations implied by Hecke relations

χN,p = T60∗N+pχY L +

N−2

X

k=0

d(k)Tp+60kχY L One can also use Hecke images to construct families of possible RCFT characters. As an example, if p = 7, 13, 47, 53 mod 60 then obey TpχY L

• 1. Have non-negative integer coefficients in q expansion.
• 2. The vacuum appears with degeneracy one.
• 3. The fusion coefficients from Verlinde are non-negative.

Consistency with Virasoro requires

Many RCFT with the same n=dimension of modular rep, N=order of and with effective central charges have modular representations and characters related by Hecke operators This generalizes and extends Galois symmetry. ρ(T) c(1)

eff = pc(2) eff

Tp, (p, N) = 1

We have found and studied such relations for n=2,3 involving minimal models, affine Lie algebras, Gaussian models and their orbifolds and certain moonshine CFTs. These relations appear to be very common.

SUMMARY

Questions

Do these Hecke operators have a natural physical

• rigin?

Can one understand more systematically which RCFT with the same n,N have characters related by Hecke

• perators?

Does this indicate there are new symmetries acting

• n the space of RCFTs?

RCFTs provide building block for construction of SCFTs describing special points in CY moduli space. Are there new geometrical relations following from Hecke relations?

RCFTs and their fusion algebras, modular tensor categories, characters etc. appear in many places in condensed matter physics:

Boundary modes of QHE systems and topological insulators. Tool for computing and studying Entanglement Entropy. Quantum computation. It will be interesting to see if these new Hecke relations have implications in the real world.

THANK YOU

Extra Slides

ω

Hecke and MLDE

φk = (−1)kWk/W

Wk =

• f1

f2 · · · fn Df1 Df2 · · · Dfn . . . . . . . . . Dk−1f1 Dk−1f2 · · · Dk−1fn Dk+1f1 Dk+1f2 · · · Dk+1fn . . . . . . . . . Dnf1 Dnf2 · · · Dnfn

• ,

`(W) = 6 @1 2ordi(W) + 1 3ordω(W) + X

p2F

• rdp(W)

1 A

• rd∞(W) + `(W)

6 = k 12

Dnf +

n−1

X

k=0

φk(τ)Dkf = 0

W = Wn i

p 5 7 11 13 17 19 23 25 29 31 35 37 41 43 47 49 53 57 `I(p) 6 6 6 Table 1: Number of zeros in the modular Wronskian for Hecke images under Tp

• f Ising characters for small values of p.

Mathur, Mukhi & Sen classified n=2, solutions, but solutions exist for and are Hecke images

• f solutions since changes

`(W) = 0 `(W) > 0 `(W) = 0 Tp

• rd∞(W)

p 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 `Y L(p) 2 2 2 2 2 2 2 2 6 6 8 Table 1: Number of zeros in the modular Wronskian for Hecke images under Tp

• f Yang-Lee characters for small values of p.

These Hecke images have negative coefficients These Hecke images have positive coefficients and as RCFT characters appear in work of Naculich and Hampapurma & Mukhi

In math these are often applied to weight k>0 modular forms and used to construct cusp forms which are eigenfunctions of the Hecke operators. E.g. the unique weight 12 cusp form (i.e. vanishing as In RCFT we interested in the action on weight 0, weakly holomorphic modular functions which are never Hecke eigenfunctions.

∆ = η24 = q

∞

Y

n=1

(1 − qn)24 =

∞

X

n=1

c(n)qn τ → i∞ Tn∆ = c(n)∆

The modularity theorem used in the proof of Fermat’s last theorem associates a weight 2 Hecke eigenform to each elliptic curve over the rationals.