theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from

MotivationRepresentation theory and CFT

Quantum groups

AM Semikhatov LCFT and representation theory


Quantum groups

Logarithmic Conformal Field Theory

AM Semikhatov

Lebedev Physics Institute

Dubna Workshop on LCFTetc, June 2007



Quantum groups

Logarithmic Conformal Field Theory:How far can we go with representation theory?

AM Semikhatov

Lebedev Physics Institute

Dubna Workshop on LCFTetc, June 2007



Quantum groups

Plan of the Talk

1 Motivation

2 Representation theory and CFT

3 Quantum groups



Quantum groups

Plan of the Talk

1 Motivation


3 Quantum groups



Quantum groups

Plan of the Talk

1 Motivation


3 Quantum groups



Quantum groups



Quantum groups

1 Motivation


3 Quantum groups



Quantum groups

A 2002 quotation

Logarithmic CFTs may be interesting from several standpoints.A Nichols, 2002:

LCFTs have now been studied for over ten years . . . in. . . : WZNWmodels [19–32], gravitational dressing [33,34], polymers andpercolation [35–38], 2d turbulence [39–43], certain limits ofQCD [44–46], the Seiberg–Witten solution of N = 2 supersymmetricYang–Mills [47,48], and the Abelian sand-pile model [49,50].. . . applications has been to disordered systems and the quantum Halleffect [18,51–60]. . . . to string theory [61–70] and in theAdS/CFT correspondence [71–78]. The holographic relation betweenlogarithmic operators and vacuum instability was considered in [79,80].An approach to LCFT using nilpotent dimensions was given in [81,82].. . . the appearance of a logarithmic partner of the stress tensor in c = 0LCFTs [37,83,84], . . . [85,86]. . . . of particular interest has been theanalysis of LCFTs in the presence of a boundary [87–91].



Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

A 2002 quotation





Quantum groups

And more recently

PA Pearce, J Rasmussen, and JB Zuber (“Temperly–Lieb” approach)Pierce and Rasmussen (dense polymers)M Jeng, G Piroux, P Ruelle (sand-pile model)F Lesage, P Mathieu, J Rasmussen, H Saleur (WZW models)vertex-operator algebras with nonsemisimple representation categories:YZ Huang, J Lepowsky, and L Zhang;J Fuchs; M Miyamoto; A Milas;Flohr, N CarquevilleV Schomerus and H Saleur (supergeometry =⇒ logs)M Gaberdiel and I Runkel (boundary logarithmic theories)Flohr and Gaberdiel (torus amplitudes)H Eberle and Flohr (fusion)not to mention S Hwang, J Fuchs, Semikhatov, I Tipunin and B Feigin,A Gainutdinov, Semikhatov, Tipuninhep-th/0306274, hep-th/0504093, math.QA/0512621, hep-th/0606196,

math.QA/0606506 [this talk]



Quantum groups

And more recently





Quantum groups

And more recently





Quantum groups

And more recently





Quantum groups

And more recently





Quantum groups

And more recently





Quantum groups

And more recently





Quantum groups

And more recently





Quantum groups

And yet more recently

T Creutzig, T Quella, and V Schomerus (boundary)N Read and H Saleur ×2D Adamovic and A Milas (Logarithmic intertwiners and W -algebras)Semikhatov (s`(2) model). . .



Quantum groups





Quantum groups





Quantum groups





Quantum groups





Quantum groups

Logarithms:

Logarithmic Conformal Field Theory:

nondiagonalizable action of a number of operators of the type of aHamiltonian



Quantum groups

Logarithms:



“Nonunitary evolution” etH =⇒ applications to models with disorder,systems with transient and recurrent states (sand-pile model), percolation,. . .



Quantum groups

Logarithms:



“Nonunitary evolution” etH =⇒ applications to models with disorder,systems with transient and recurrent states (sand-pile model), percolation,. . .



Quantum groups

Logarithms:



log: whence comest thou?

Let L0 ∼ z∂

∂zact nondiagonally:

zg ′(z) = ∆g(z),

zh ′(z) = ∆h(z) + g(z).

Solution:g(x) = B x∆,



Quantum groups

Logarithms:




Let L0 ∼ z∂


zg ′(z) = ∆g(z),

zh ′(z) = ∆h(z) + g(z).




Quantum groups

Logarithms:




Let L0 ∼ z∂


zg ′(z) = ∆g(z),

zh ′(z) = ∆h(z) + g(z).


h(x) = A x∆ + B x∆ log(x).



Quantum groups

Logarithms:



Logarithmic/nonsemisimple theories:

Being logarithmic/nonsemisimple is a property of representations chosen(even though algebras often get extended)



Quantum groups

Logarithms:



Logarithmic/nonsemisimple theories:

Being logarithmic/nonsemisimple is a property of representations chosen(even though algebras often get extended)



Quantum groups

1 Motivation


3 Quantum groups



Quantum groups

Rational models: basic representation-theory unput

Virasoro algebra [Lm, Ln] = (m − n)Lm+n + c12(m3 − m)δm+n,0

Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducible

Rational (p, p ′)-models at c = 13 − pp ′ −

p ′

p :

Kac table of “good” modules:12 (p − 1) × (p ′ − 1) nonisomorphicVirasoro irreps

p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1

These irreps have no extensions among themselves =⇒ semisimple(diagonalizable)

=⇒ chiral space of states =⊕

(irreps)

=⇒ numerous deep properties of rcft. . .



Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups



Highest-weight modules Ln > 1|∆〉 = 0, L0|∆〉 = ∆|∆〉Vermairreducibleindecomposable


p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups





p ′

p :


p ′−1

. . .

. . . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1



(irreps)




Quantum groups

LCFT: “minimal” extension

adding 1 row and 1 column:

p ′−1

. . .

. . . . . . . . . . . . . .. . .. . .︸︷︷︸

p−1

Also, a new possibility: (p, 1) modelswith the extended Kac table

. . .︸︷︷︸p

Drastic consequences

Representations admit indecomposable extensions

0→ X→ A→ Y→ 0, orY• −→ X•


(projective modules)

The symmetry extends from Virasoro to a larger W -algebra



Quantum groups


adding 1 row and 1 column:“only” p + p ′ − 1 new boxes

p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p

Nonlogarithmic content: void



0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p

Drastic consequences:


0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•



Projective modules: “maximally indecomposable”




Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•



Projective modules: “maximally indecomposable”They are home for logarithmic partners




Quantum groups

LCFT: “minimal” extension: New Hope?


p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•






Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•


(W -algebra projective modules)




Quantum groups



p ′−1

. . .. . .

. . . . . . . . . . . . . .. . .. . .. . .︸︷︷︸

p−1


. . .︸︷︷︸p



0→ X→ A→ Y→ 0, orY• −→ X•


(W -algebra projective modules)

The symmetry extends from Virasoro to a larger W -algebra(triplet W -algebra [(p, 1): Kausch, Kausch and Gaberdiel])



Quantum groups

Basic problem:

Virtually nothing is known about projective modules of Virasoro and“larger” algebras.

So —

in contrast to the rational case, representation theory fails?!



Quantum groups

Basic problem:


For the (p, 1) triplet W -algebra, projectivemodules must have the structure

Xas•

zz $$X−a

p−s◦��

X−ap−s◦

��Xa

s•

but none of theconstruction detailsare worked out

So —




Quantum groups

Basic problem:


For the (p, 1) triplet W -algebra, projectivemodules must have the structure

Xas•

zz $$X−a

p−s◦��

X−ap−s◦

��Xa

s•

but none of theconstruction detailsare worked out

So —




Quantum groups

Basic problem:


For the (p, p ′) triplet W -algebra, even the more complicated structure♣

vv

bdegikm ~~ ((

\ Z Y W U S Q ♦

��

N >/

))

♥roljgqq ec

N >��/ ))

♥--

LO R T W Y [

��

p�

� ((

♦��

ss

`aabbccddeeffgg uu♠�� %%

[ Y W T R O L

♣�� ))

♠))vv

��p

♠uu ((

/>

N

♣((

/>

Nss

`aabbccddeeffgg

♠��yy

cegjlor♦

((

\ Z Y W U S Q

♥

♥~~

♦vv

bdegikm♣

although involved in the true projective module, is insufficient.

So —

in contrast to the rational case, representation theory fails?!AM Semikhatov LCFT and representation theory


Quantum groups

Basic problem:


So —




Quantum groups

The Indecomposables Strike Back

Basic problem:


So —




Quantum groups

Return of the FreeField approach

Basic problem:


So —


Resort to:

1 Free-field construction

2 Kazhdan–Lusztig correspondence



Quantum groups

Return of the FreeField approach

Basic problem:


So —


Resort to:

1 Free-field construction

2 Kazhdan–Lusztig correspondence



Quantum groups

LCFTs in terms of free fields

1 Take screenings in a free-field realization: e.g., S±=

∫eα±ϕ(z)dz

2 rational models are the cohomology of (the differential associated with)screenings

3 Take the kernel of the screenings

4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.



Quantum groups



∫eα±ϕ(z)dz






Quantum groups



∫eα±ϕ(z)dz






Quantum groups



∫eα±ϕ(z)dz






Quantum groups



∫eα±ϕ(z)dz


3 Take the kernel of the screenings (“kernel > cohomology”)




Quantum groups



∫eα±ϕ(z)dz






Quantum groups



∫eα±ϕ(z)dz






Quantum groups



∫eα±ϕ(z)dz



4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.×

��

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W−

��

W+

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[r,s;0]

�

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[p+−r,s;1]

•W−

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W+

[r,p−−s;1]

◦

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Quantum groups



∫eα±ϕ(z)dz



4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.W -algebra generators for (3, 2)

W + =(

3527

(∂

4ϕ)2

+ 5627∂

5ϕ∂

3ϕ+ 28

27∂

6ϕ∂

2ϕ+ 8

27∂

7ϕ∂ϕ− 280

9√

3

(∂

3ϕ)2∂

2ϕ

− 70

3√

3∂

4ϕ(∂

2ϕ)2

− 280

9√

3∂

4ϕ∂

3ϕ∂ϕ− 56

3√

3∂

5ϕ∂

2ϕ∂ϕ− 28

9√

3∂

6ϕ(∂ϕ)2

+ 353

(∂

2ϕ)4 + 280

3∂

3ϕ(∂

2ϕ)2∂ϕ+ 280

9

(∂

3ϕ)2(

∂ϕ)2

+ 1403∂

4ϕ∂

2ϕ(∂ϕ

)2+ 56

9∂

5ϕ(∂ϕ)3

− 140√3

(∂

2ϕ)3(∂ϕ)2

− 560

3√

3∂

3ϕ∂

2ϕ(∂ϕ)2

− 70

3√

3∂

4ϕ(∂ϕ)4

+ 70(∂

2ϕ)2(∂ϕ)4

+ 563∂

3ϕ(∂ϕ)5 − 28√

3∂

2ϕ(∂ϕ)6

+(∂ϕ)8 − 1

27√

3∂

8ϕ)e2√

3ϕ,



Quantum groups



∫eα±ϕ(z)dz



4 The kernel is a representation space of a W -algebra— the maximum local algebra acting in the kernel.W -algebra generators for (3, 2)

W −=

(217192

(∂

5ϕ

)2− 2653

3456∂

6ϕ∂

4ϕ− 23

384∂

7ϕ∂

3ϕ− 11

1152∂

8ϕ∂

2ϕ− 1

768∂

9ϕ∂ϕ− 1225

64√

3∂

4ϕ

(∂

3ϕ

)2

− 13475576√

3

(∂

4ϕ

)2∂

2ϕ+ 2695

64√

3∂

5ϕ∂

3ϕ∂

2ϕ+ 2555

192√

3∂

5ϕ∂

4ϕ∂ϕ− 2891

576√

3∂

6ϕ

(∂

2ϕ

)2− 1351

192√

3∂

6ϕ∂

3ϕ∂ϕ

− 103192√

3∂

7ϕ∂

2ϕ∂ϕ− 13

384√

3∂

8ϕ

(∂ϕ

)2+ 3535

32

(∂

3ϕ

)2(∂

2ϕ

)2− 735

16

(∂

3ϕ

)3∂ϕ− 3395

54∂

4ϕ

(∂

2ϕ

)3

+ 24524∂

4ϕ∂

3ϕ∂

2ϕ∂ϕ+ 12635

576

(∂

4ϕ

)2(∂ϕ

)2+ 245

12∂

5ϕ

(∂

2ϕ

)2∂ϕ+ 105

32∂

5ϕ∂

3ϕ

(∂ϕ

)2

− 2443288

∂6ϕ∂

2ϕ

(∂ϕ

)2− 19

96∂

7ϕ

(∂ϕ

)3− 13405

144√

3

(∂

2ϕ

)5+ 8225

24√

3∂

3ϕ

(∂

2ϕ)

3∂ϕ− 105

√3

4

(∂

3ϕ

)2∂

2ϕ

(∂ϕ

)2

+ 66524√

3∂

4ϕ

(∂

2ϕ

)2(∂ϕ

)2+ 245

2√

3∂

4ϕ∂

3ϕ

(∂ϕ

)3− 245

8√

3∂

5ϕ∂

2ϕ

(∂ϕ

)3− 91

24√

3∂

6ϕ

(∂ϕ

)4+ 16205

144

(∂

2ϕ

)4(∂ϕ

)2

+ 3854∂

3ϕ

(∂

2ϕ

)2(∂ϕ

)3+ 525

8

(∂

3ϕ

)2(∂ϕ

)4+ 35

3∂

4ϕ∂

2ϕ

(∂ϕ

)4− 7∂5

ϕ(∂ϕ

)5+ 665

3√

3

(∂

2ϕ

)3(∂ϕ

)4

+ 105√

32

∂3ϕ∂

2ϕ

(∂ϕ

)5− 35

3√

3∂

4ϕ

(∂ϕ

)6+ 455

6

(∂

2ϕ

)2(∂ϕ

)6+ 5∂3

ϕ(∂ϕ

)7+ 25√

3∂

2ϕ

(∂ϕ

)8

+(∂ϕ

)10− 1

13824√

3∂

10ϕ

)e−2√

3ϕ,



Quantum groups



∫eα±ϕ(z)dz




Key differences from the rational case

The LCFT model may be dependent on the free-field representation taken,on the screenings chosen, etc.

The symmetry algebra of a LCFT model is larger than the “naive” algebra(e.g., Virasoro).

The space of states in a LCFT is not the direct sum of irreduciblerepresentations but the sum of all (finitely many) projective modules

P =⊕ι

Pι.



Quantum groups



∫eα±ϕ(z)dz








P =⊕ι

Pι.



Quantum groups



∫eα±ϕ(z)dz








P =⊕ι

Pι.



Quantum groups



∫eα±ϕ(z)dz








P =⊕ι

Pι.



Quantum groups



∫eα±ϕ(z)dz








P =⊕ι

Pι.



Quantum groups

W-algebras and their representations

(p, 1) models: the triplet W -algebra Wp

has 2p irreps X±r , r = 1, . . . , p.

∆X+(r) =(p − r)2

4p+

c − 1

24, ∆X−(r) =

(2p − r)2

4p+

c − 1

24.

(p, p ′) models: 2pp ′ irreps of the corresponding Wp,p ′ :X±r ,r ′ , r = 1, . . . , p, r ′ = 1, . . . , p ′,∆X+

r,r ′= ∆r,p ′−r ′;1, ∆X−

r,r ′= ∆p−r,r ′;−2,

∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2

4pp ′.

PLUS the 12(p − 1)(p ′ − 1) representations from the Virasoro

minimal model.

kernel of the screenings =

N⊕A

XA

— finite sum of irreducible representatons



Quantum groups



has 2p irreps X±r , r = 1, . . . , p.

∆X+(r) =(p − r)2

4p+

c − 1

24, ∆X−(r) =

(2p − r)2

4p+

c − 1

24.


r,r ′= ∆r,p ′−r ′;1, ∆X−

r,r ′= ∆p−r,r ′;−2,

∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2

4pp ′.


minimal model.


N⊕A

XA




Quantum groups



has 2p irreps X±r , r = 1, . . . , p.

∆X+(r) =(p − r)2

4p+

c − 1

24, ∆X−(r) =

(2p − r)2

4p+

c − 1

24.


r,r ′= ∆r,p ′−r ′;1, ∆X−

r,r ′= ∆p−r,r ′;−2,

∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2

4pp ′.


minimal model.


N⊕A

XA




Quantum groups



has 2p irreps X±r , r = 1, . . . , p.

∆X+(r) =(p − r)2

4p+

c − 1

24, ∆X−(r) =

(2p − r)2

4p+

c − 1

24.


r,r ′= ∆r,p ′−r ′;1, ∆X−

r,r ′= ∆p−r,r ′;−2,

∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2

4pp ′.


minimal model.


N⊕A

XA




Quantum groups



has 2p irreps X±r , r = 1, . . . , p.

∆X+(r) =(p − r)2

4p+

c − 1

24, ∆X−(r) =

(2p − r)2

4p+

c − 1

24.


r,r ′= ∆r,p ′−r ′;1, ∆X−

r,r ′= ∆p−r,r ′;−2,

∆r,r ′;n =(pr ′−p ′r+pp ′n)2 − (p−p ′)2

4pp ′.


minimal model.


N⊕A

XA




Quantum groups

W-algebra characters

(p, 1): The irreducible W -representation characters are given by

χ+r (q) =

1

η(q)

(r

pθp−r ,p(q) +

2

pθ ′p−r ,p(q)

),

χ−r (q) =

1

η(q)

(r

pθr ,p(q) −

2

pθ ′r ,p(q)

),

1 6 r 6 p.



Quantum groups


(p, p ′): The irreducible W -representation characters are given by

χr ,r ′(q) =1

η(q)(θpr ′−p ′r ,pp ′(q) − θpr ′+p ′r ,pp ′(q)), (r , r ′) ∈ I1,

χ+r ,r ′ =

1

(pp ′)2η

(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r

− (pr ′+p ′r)θ ′pr ′+p ′r + (pr ′−p ′r)θ ′pr ′−p ′r

+(pr ′+p ′r)2

4θpr ′+p ′r −

(pr ′−p ′r)2

4θpr ′−p ′r

), 1 6 r 6 p, 1 6 r ′6 p ′,

χ−r ,r ′ =

1

(pp ′)2η

(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r

+ (pr ′+p ′r)θ ′pp ′−pr ′−p ′r + (pr ′−p ′r)θ ′pp ′+pr ′−p ′r

+(pr ′ + p ′r)2−(pp ′)2

4θpp ′−pr ′−p ′r

−(pr ′ − p ′r)2−(pp ′)2

4θpp ′+pr ′−p ′r

), 1 6 r 6 p, 1 6 r ′6 p ′.



Quantum groups



χr ,r ′(q) =1


χ+r ,r ′ =

1

(pp ′)2η

(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r


+(pr ′+p ′r)2

4θpr ′+p ′r −

(pr ′−p ′r)2

4θpr ′−p ′r

), 1 6 r 6 p, 1 6 r ′6 p ′,

χ−r ,r ′ =

1

(pp ′)2η

(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r


+(pr ′ + p ′r)2−(pp ′)2


−(pr ′ − p ′r)2−(pp ′)2


), 1 6 r 6 p, 1 6 r ′6 p ′.



Quantum groups



χr ,r ′(q) =1


χ+r ,r ′ =

1

(pp ′)2η

(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r


+(pr ′+p ′r)2

4θpr ′+p ′r −

(pr ′−p ′r)2

4θpr ′−p ′r

), 1 6 r 6 p, 1 6 r ′6 p ′,

χ−r ,r ′ =

1

(pp ′)2η

(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r


+(pr ′ + p ′r)2−(pp ′)2


−(pr ′ − p ′r)2−(pp ′)2


), 1 6 r 6 p, 1 6 r ′6 p ′.



Quantum groups



χr ,r ′(q) =1


χ+r ,r ′ =

1

(pp ′)2η

(θ ′′pr ′+p ′r − θ ′′pr ′−p ′r


+(pr ′+p ′r)2

4θpr ′+p ′r −

(pr ′−p ′r)2

4θpr ′−p ′r

), 1 6 r 6 p, 1 6 r ′6 p ′,

χ−r ,r ′ =

1

(pp ′)2η

(θ ′′pp ′−pr ′−p ′r − θ ′′pp ′+pr ′−p ′r


+(pr ′ + p ′r)2−(pp ′)2


−(pr ′ − p ′r)2−(pp ′)2


), 1 6 r 6 p, 1 6 r ′6 p ′.



Quantum groups

W-Characters =⇒ Modular Group Representation

The need for generalized characters:

In LCFT, characters alone are not closed under SL(2,Z) action

(p, 1) models: The 2p characters give rise to a (3p − 1)-dimensionalSL(2,Z)-representation.

(p, p ′) models: The 2pp ′ + 12(p − 1)(p ′ − 1) characters give rise to

a 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation.



Quantum groups







Quantum groups







Quantum groups





It is highly probable that these dimensions 3p − 1 and 12 (3p − 1)(3p ′ − 1) are the

dimensions of the spaces of torus amplitudes.



Quantum groups





Theorem

The (3p − 1)-dimension SL(2,Z)-representation Zcft has the structure

Zcft = Rp+1 ⊕ C2 ⊗Rp−1,

Rp−1 is the [“sinπrsp ”] SL(2,Z)-representation realized in the s`(2)p−2

minimal model, Rp+1 is a [“cosπrsp ”] SL(2,Z)-representations of dimension

p + 1, and C2 is the defining two-dimensional representation of SL(2,Z).



Quantum groups





Theorem


Zcft = Rp+1 ⊕ C2 ⊗Rp−1,






Quantum groups





Theorem


Zcft = Rp+1 ⊕ C2 ⊗Rp−1,






Quantum groups





Theorem


Zcft = Rp+1 ⊕ C2 ⊗Rp−1,






Quantum groups





Theorem

The 12(3p − 1)(3p ′ − 1)-dimensional SL(2,Z)-representation Zcft has the

structure

Zcft = Rmin ⊕ Rproj ⊕ C2 ⊗ (R� ⊕ R�)⊕ C3 ⊗ Rmin,

Rmin is the 12 (p − 1)(p ′− 1)-dimensional SL(2,Z)-representation on the

characters of the (p, p ′) Virasoro minimal model, C3 ∼= S2(C2), and Rproj, R�,and R� are SL(2,Z)-representations of the respective dimensions12 (p + 1)(p ′+ 1), 1

2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).



Quantum groups





Theorem


structure




2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).



Quantum groups





Theorem


structure




2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).



Quantum groups





Theorem


structure




2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).



Quantum groups





Theorem


structure




2 (p − 1)(p ′+ 1), and 12 (p + 1)(p ′− 1).



Quantum groups





It is highly probable that these dimensions 3p − 1 and 12 (3p − 1)(3p ′ − 1) are

the dimensions of the spaces of torus amplitudes.



Quantum groups

Some details for (p, p ′)Generalized characters:

subrep. dimension basis

Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1

Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0

C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�

C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�

C3 ⊗ Rmin 3 · 12(p − 1)(p ′− 1) ρr,r ′ , ψr,r ′ , ϕr,r ′ , (r , r ′) ∈ I1



Quantum groups



Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1

Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0

C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�

C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�


κr,r ′ = χr,r ′ + 2χ+r,r ′ + 2χ−

r,p ′−r ′ + 2χ−p−r,r ′ + 2χ+

p−r,p ′−r ′ , (r , r ′) ∈ I1,

κ0,r ′ = 2χ+p,p ′−r ′ + 2χ−

p,r ′ , 1 6 r ′6 p ′− 1,

κr,0 = 2χ+p−r,p ′ + 2χ−

r,p ′ , 1 6 r 6 p − 1,

κ0,0 = 2χ+p,p ′ ,

κp,0 = 2χ−p,p ′



Quantum groups



Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1

Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0

C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�

C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�


ρ�r,r ′(τ) = p ′r−pr ′

2 χr,r ′(τ) + p ′(r − p)(χ+r,r ′(τ) + χ−

r,p ′−r ′(τ))

+ p ′r(χ+p−r,p ′−r ′(τ) + χ−

p−r,r ′(τ)), (r , r ′) ∈ I1,

ρ�r,0(τ) = p ′(rχ+

p−r,p ′(τ) − (p−r)χ−r,p ′(τ)), 1 6 r 6 p−1,

ϕ�r,r ′(τ) = τρ�

r,r ′(τ), (r , r ′) ∈ I�,



Quantum groups



Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1

Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0

C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�

C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�


ρ�r,r ′(τ) = pr ′−p ′r

2 χr,r ′(τ) − p(p ′−r ′)(χ+r,r ′(τ) + χ−

p−r,r ′(τ))

+ pr ′(χ+p−r,p ′−r ′(τ) + χ−

r,p ′−r ′(τ)), (r , r ′) ∈ I1,

ρ�0,r ′(τ) = p(r ′χ+

p,p ′−r ′(τ) − (p ′−r ′)χ−p,r ′(τ)), 1 6 r ′6 p ′−1,

ϕ�r,r ′(τ) = τρ�

r,r ′(τ), (r , r ′) ∈ I�,



Quantum groups



Rmin12(p − 1)(p ′− 1) χr,r ′ , (r , r ′) ∈ I1

Rproj12(p + 1)(p ′+ 1) κr,r ′ , (r , r ′) ∈ I0

C2 ⊗ R� 2 · 12(p − 1)(p ′+ 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�

C2 ⊗ R� 2 · 12(p + 1)(p ′− 1) ρ�

r,r ′ , ϕ�r,r ′ , (r , r ′) ∈ I�


ρr,r ′(τ) = pp ′((p−r)(p ′−r ′)χ+

r,r ′(τ) + rr ′χ+p−r,p ′−r ′(τ) −

(pr ′−p ′r)2

4pp ′ χr,r ′(τ)

− (p−r)r ′χ−r,p ′−r ′(τ) − r(p ′−r ′)χ−

p−r,r ′(τ)), (r , r ′) ∈ I1

ψr,r ′(τ) = 2τρr,r ′(τ) + iπpp ′χr,r ′(τ), (r , r ′) ∈ I1,

ϕr,r ′(τ) = τ2ρr,r ′(τ) + iπpp ′τχr,r ′(τ), (r , r ′) ∈ I1.



Quantum groups

Corollary: Several modular invariants

involving τ explicitly:

ρ�(τ, τ) =

p−1∑r=1

im τ |ρ�r,0(τ)|

2 + 2∑

(r,r ′)∈I1

im τ |ρ�r,r ′(τ)|

2,

ρ(τ, τ) =∑

(r,r ′)∈I1

ρr,r ′(τ)(8(im τ)2ρr,r ′(τ) + 4pp ′πim τ χr,r ′(τ))

+ χr,r ′(τ)(4pp ′πim τ ρr,r ′(τ) + (πpp ′)2χr,r ′(τ)).

A-series:κ[A](τ, τ) =

= |κ0,0(τ)|2 + |κp,0(τ)|

2 + 2

p−1∑r=1

|κr,0(τ)|2 + 2

p ′−1∑r ′=1

|κ0,r ′(τ)|2 + 4

∑(r,r ′)∈I1

|κr,r ′(τ)|2

D-series (in the case p ′ ≡ 0 mod 4):

κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +

p−1∑r=1

|κr,0(τ) + κp−r,0(τ)|2

+∑

2 6 r ′ 6 p ′−1r ′ even

|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +

∑(r,r ′)∈I1

r ′ even

2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.

E6-type invariant for (p, p ′) = (5, 12):

κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|

2 + |κ0,5(τ) − κ0,11(τ)|2

+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|

2 + 2|κ2,5(τ) − κ3,1(τ)|2

+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|

2 + 2|κ1,2(τ) − κ4,2(τ)|2.



Quantum groups



= |κ0,0(τ)|2 + |κp,0(τ)|

2 + 2

p−1∑r=1

|κr,0(τ)|2 + 2

p ′−1∑r ′=1

|κ0,r ′(τ)|2 + 4

∑(r,r ′)∈I1

|κr,r ′(τ)|2


κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +

p−1∑r=1

|κr,0(τ) + κp−r,0(τ)|2

+∑

2 6 r ′ 6 p ′−1r ′ even

|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +

∑(r,r ′)∈I1

r ′ even

2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.


κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|

2 + |κ0,5(τ) − κ0,11(τ)|2

+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|

2 + 2|κ2,5(τ) − κ3,1(τ)|2

+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|

2 + 2|κ1,2(τ) − κ4,2(τ)|2.



Quantum groups



= |κ0,0(τ)|2 + |κp,0(τ)|

2 + 2

p−1∑r=1

|κr,0(τ)|2 + 2

p ′−1∑r ′=1

|κ0,r ′(τ)|2 + 4

∑(r,r ′)∈I1

|κr,r ′(τ)|2


κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +

p−1∑r=1

|κr,0(τ) + κp−r,0(τ)|2

+∑

2 6 r ′ 6 p ′−1r ′ even

|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +

∑(r,r ′)∈I1

r ′ even

2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.


κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|

2 + |κ0,5(τ) − κ0,11(τ)|2

+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|

2 + 2|κ2,5(τ) − κ3,1(τ)|2

+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|

2 + 2|κ1,2(τ) − κ4,2(τ)|2.



Quantum groups



= |κ0,0(τ)|2 + |κp,0(τ)|

2 + 2

p−1∑r=1

|κr,0(τ)|2 + 2

p ′−1∑r ′=1

|κ0,r ′(τ)|2 + 4

∑(r,r ′)∈I1

|κr,r ′(τ)|2


κ[D](τ, τ) = |κ0,0(τ) + κp,0(τ)|2 +

p−1∑r=1

|κr,0(τ) + κp−r,0(τ)|2

+∑

2 6 r ′ 6 p ′−1r ′ even

|κ0,r ′(τ) + κ0,p ′−r ′(τ)|2 +

∑(r,r ′)∈I1

r ′ even

2 |κr,r ′(τ) + κr,p ′−r ′(τ)|2.


κ[E6](τ, τ) = |κ0,1(τ) − κ0,7(τ)|2 + |κ0,2(τ) − κ0,10(τ)|

2 + |κ0,5(τ) − κ0,11(τ)|2

+ 2|κ1,1(τ) − κ1,7(τ)|2 + 2|κ2,1(τ) − κ2,7(τ)|

2 + 2|κ2,5(τ) − κ3,1(τ)|2

+ 2|κ2,2(τ) − κ3,2(τ)|2 + 2|κ1,5(τ) − κ4,1(τ)|

2 + 2|κ1,2(τ) − κ4,2(τ)|2.



Quantum groups

1 Motivation


3 Quantum groups



Quantum groups

Quantum groups: Kazhdan–Lusztig correspondence

THE SAME SL(2,Z) REPRESENTATIONS ARE REALIZED ONCENTERS OF THE CORRESPONDING QUANTUM GROUPS



Quantum groups



Screenings =⇒ quantum group g (“Kazhdan–Lusztig-dual”)

At a root of unity =⇒ finite-dimensional

Center Z

Quantum group g is ribbon and factorizable



Quantum groups




At a root of unity =⇒ finite-dimensional

Center Z




Quantum groups




At a root of unity =⇒ finite-dimensional (q2p = 1, dim g = 2p3 andq2pp ′ = 1, dim g = 2p3p ′3)

Center Z




Quantum groups





Center Z




Quantum groups





Center Z: dim Z = 3p − 1 and dim Z = 12(3p − 1)(3p ′ − 1)




Quantum groups






Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation



Quantum groups






Quantum group g is ribbon and factorizable =⇒ its center carries anSL(2,Z) representation [Lyubashenko, Turaev, Kerler]



Quantum groups







Theorem

This SL(2,Z)-representation on Z coincides with theSL(2,Z)-representation generated by the LCFT characters



Quantum groups







The quantum group knows surprisingly much about the LCFT



Quantum groups







The quantum group knows surprisingly much about the LCFT

Anything else?



Quantum groups

More on KL: Grothendieck rings/Fusion

The “(p, 1)” quantum group has 2p irreps X±r , 1 6 r 6 p. Grothendieck ring:

Xαr Xα′

s =

r+s−1∑t=|r−s|+1

step=2

Xαα′

t ,

Xαr =

{Xαr , 1 6 r 6 p,

Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.



Quantum groups



Xαr Xα′

s =

r+s−1∑t=|r−s|+1

step=2

Xαα′

t ,

Xαr =

{Xαr , 1 6 r 6 p,

Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.



Quantum groups



Xαr Xα′

s =

r+s−1∑t=|r−s|+1

step=2

Xαα′

t ,

Xαr =

{Xαr , 1 6 r 6 p,

Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.

This nonsemisimple algebra G2p contains the ideal Vp+1 of projective modules;the quotient G2p/Vp+1 is a semisimple fusion algebra — the fusion of the unitary

s`(2) representations of level k = p − 2:

Xr Xs =

p−1−|p−r−s|∑t=|r−s|+1

step=2

Xt , r , s = 1, . . . , p − 1.



Quantum groups



Xαr Xα′

s =

r+s−1∑t=|r−s|+1

step=2

Xαα′

t ,

Xαr =

{Xαr , 1 6 r 6 p,

Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.

The triplet (p, 1) W -algebra has just 2p irreps



Quantum groups



Xαr Xα′

s =

r+s−1∑t=|r−s|+1

step=2

Xαα′

t ,

Xαr =

{Xαr , 1 6 r 6 p,

Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.


This “(p, 1)”-quantum-group Grothendieck ring IS A CANDIDATE FORFUSION in the (p, 1) LCFT model



Quantum groups



Xαr Xα′

s =

r+s−1∑t=|r−s|+1

step=2

Xαα′

t , CORROBORATED BY a derivationfrom characters [FHST (2003)]

Xαr =

{Xαr , 1 6 r 6 p,

Xα2p−r + 2X−αr−p, p + 1 6 r 6 2p − 1.


This “(p, 1)”-quantum-group Grothendieck ring IS A CANDIDATE FORFUSION in the (p, 1) LCFT model



Quantum groups


The “(p, p ′)” quantum group has 2pp ′ irreps X±r,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′.Grothendieck ring:

Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,

Xα2p−r,r ′ + 2X−αr−p,r ′ , p+1 6 r 6 2p−1, 1 6 r ′6 p ′,

Xαr,2p ′−r ′ + 2X−αr,r ′−p ′ , 1 6 r 6 p, p ′+1 6 r ′6 2p ′−1,

Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.

1 This algebra is generated by two elements X+1,2 and X+

2,1;

2 its radical is generated by the algebra action on X+p,p ′ ; the quotient over the

radical coincides with the fusion of the (p, p ′) Virasoro minimal models;

3 X+1,1 is the identity;

4 X−1,1 acts as a simple current, X−

1,1Xαr,r ′ = X−α

r,r ′ .



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.


2,1;





1,1Xαr,r ′ = X−α

r,r ′ .



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.


2,1;





1,1Xαr,r ′ = X−α

r,r ′ .



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.


2,1;





1,1Xαr,r ′ = X−α

r,r ′ .



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ ,

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.

Related to fusion in (p, p ′) LCFT models?!



Quantum groups



Xαr,r ′X

βs,s ′ =

r+s−1∑u=|r−s|+1

step=2

r ′+s ′−1∑u ′=|r ′−s ′|+1

step=2

Xαβu,u ′ , CORROBORATED BY computer

calculations [EF (2006)]

Xαr,r ′ =

Xαr,r ′ , 1 6 r 6 p, 1 6 r ′6 p ′,



Xα2p−r,2p ′−r ′ + 2X

−α2p−r,r ′−p ′

+ 2X−αr−p,2p ′−r ′ + 4X

αr−p,r ′−p ′ , p+1 6 r 6 2p−1, p ′+1 6 r ′6 2p ′−1.

Related to fusion in (p, p ′) LCFT models?!



Quantum groups

Example: (2, 3) model

The 2pp ′ = 12 representations:

X+1,1 (2), X−

1,1 (7) X+2,1 (1), X−

2,1 (5) X+3,1 ( 1

3), X−

3,1 ( 103)

X+1,2 ( 5

8), X−

1,2 ( 338) X+

2,2 ( 18), X−

2,2 ( 218) X+

3,2 (− 124

), X−3,2 ( 35

24)

X+1,2X

+1,2 = 2X

−1,1 + 2X

+1,1, X

+1,2X

+2,1 = X

+2,2, X

+1,2X

+2,2 = 2X

−2,1 + 2X

+2,1,

X+1,2X

+3,1 = X

+3,2, X

+1,2X

+3,2 = 2X

−3,1 + 2X

+3,1,

X+2,1X

+2,1 = X

+1,1 + X

+3,1, X

+2,1X

+2,2 = X

+1,2 + X

+3,2, X

+2,1X

+3,1 = 2X

−1,1 + 2X

+2,1,

X+2,1X

+3,2 = 2X

−1,2 + 2X

+2,2,

X+2,2X

+2,2 = 2X

−1,1 + 2X

−3,1 + 2X

+1,1 + 2X

+3,1, X

+2,2X

+3,1 = 2X

−1,2 + 2X

+2,2,

X+2,2X

+3,2 = 4X

−1,1 + 4X

−2,1 + 4X

+1,1 + 4X

+2,1,

X+3,1X

+3,1 = 2X

−2,1 + 2X

+1,1 + X

+3,1, X

+3,1X

+3,2 = 2X

−2,2 + 2X

+1,2 + X

+3,2,

X+3,2X

+3,2 = 4X

−1,1 + 4X

−2,1 + 2X

−3,1 + 4X

+1,1 + 4X

+2,1 + 2X

+3,1.

X−1,1 acts as a simple current, X−

1,1X±

r,r ′ = X∓r,r ′ .



Quantum groups

Example: (2, 3) model

The 2pp ′ = 12 representations:

X+1,1 (2), X−

1,1 (7) X+2,1 (1), X−

2,1 (5) X+3,1 ( 1

3), X−

3,1 ( 103)

X+1,2 ( 5

8), X−

1,2 ( 338) X+

2,2 ( 18), X−

2,2 ( 218) X+

3,2 (− 124

), X−3,2 ( 35

24)

X+1,2X

+1,2 = 2X

−1,1 + 2X

+1,1, X

+1,2X

+2,1 = X

+2,2, X

+1,2X

+2,2 = 2X

−2,1 + 2X

+2,1,

X+1,2X

+3,1 = X

+3,2, X

+1,2X

+3,2 = 2X

−3,1 + 2X

+3,1,

X+2,1X

+2,1 = X

+1,1 + X

+3,1, X

+2,1X

+2,2 = X

+1,2 + X

+3,2, X

+2,1X

+3,1 = 2X

−1,1 + 2X

+2,1,

X+2,1X

+3,2 = 2X

−1,2 + 2X

+2,2,

X+2,2X

+2,2 = 2X

−1,1 + 2X

−3,1 + 2X

+1,1 + 2X

+3,1, X

+2,2X

+3,1 = 2X

−1,2 + 2X

+2,2,

X+2,2X

+3,2 = 4X

−1,1 + 4X

−2,1 + 4X

+1,1 + 4X

+2,1,

X+3,1X

+3,1 = 2X

−2,1 + 2X

+1,1 + X

+3,1, X

+3,1X

+3,2 = 2X

−2,2 + 2X

+1,2 + X

+3,2,

X+3,2X

+3,2 = 4X

−1,1 + 4X

−2,1 + 2X

−3,1 + 4X

+1,1 + 4X

+2,1 + 2X

+3,1.

X−1,1 acts as a simple current, X−

1,1X±

r,r ′ = X∓r,r ′ .



Quantum groups

Beyond the Grothendieck ringThe TRUE tensor algebra of Uqs`(2)-representations [K Erdmann et al]:r + s − p 6 1, then

Xαr ⊗ Xβs =

r+s−1⊕t=|r−s |+1

step=2

Xαβt (min(r , s) terms in the sum).

r + s − p > 2, even: r + s − p = 2n with n > 1, then

Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=1

Pαβp+1−2a.

r + s − p > 3, odd: r + s − p = 2n + 1 with n > 1, then

Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=0

Pαβp−2a.

The same structure of “indecomposable-aware” fusion in (p, 1) LCFT?



Quantum groups


Xαr ⊗ Xβs =

r+s−1⊕t=|r−s |+1

step=2



Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=1

Pαβp+1−2a.


Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=0

Pαβp−2a.




Quantum groups


Xαr ⊗ Xβs =

r+s−1⊕t=|r−s |+1

step=2



Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=1

Pαβp+1−2a.


Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=0

Pαβp−2a.




Quantum groups


Xαr ⊗ Xβs =

r+s−1⊕t=|r−s |+1

step=2



Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=1

Pαβp+1−2a.


Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=0

Pαβp−2a.




Quantum groups


Xαr ⊗ Xβs =

r+s−1⊕t=|r−s |+1

step=2



Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=1

Pαβp+1−2a.


Xαr ⊗ Xβs =

2p−r−s−1⊕t=|r−s |+1

step=2

Xαβt ⊕

n⊕a=0

Pαβp−2a.




Quantum groups

Indecomposable modules: Was(n) and Ma

s(n)

1 6 s 6 p−1, a =±, and n > 2, module Was (n):

Xas• xa

1

��

Xas•xa

2

��

xa1

��

. . .xa2

��

xa1

��

Xas•xa

2

��X−a

p−s• X−ap−s• . . . X−a

p−s•1 6 s 6 p−1, a =± , and n > 2, module Ma

s (n):

Xas•xa

2

��

xa1

��

. . .xa2

��

xa1

��

Xas•xa

2

��

xa1

��X−a

p−s• X−ap−s• . . . X−a

p−s• X−ap−s•



Quantum groups

Indecomposable modules: Was(n) and Ma

s(n)

1 6 s 6 p−1, a =±, and n > 2, module Was (n):

Xas• xa

1

��

Xas•xa

2

��

xa1

��

. . .xa2

��

xa1

��

Xas•xa

2

��X−a

p−s• X−ap−s• . . . X−a

p−s•1 6 s 6 p−1, a =± , and n > 2, module Ma

s (n):

Xas•xa

2

��

xa1

��

. . .xa2

��

xa1

��

Xas•xa

2

��

xa1

��X−a

p−s• X−ap−s• . . . X−a

p−s• X−ap−s•



Quantum groups

Indecomposable modules: Oas(n, z)

1 6 s 6 p − 1, a =±, n > 1, and z ∈ CP1, module Oas (n, z):

Xas•xa

2

��

xa1

��

Xas•xa

2

��

. . . Xas• xa

1

��X−a

p−s• X−ap−s• . . . X−a

p−s•

Xas•

z2xa2

>>

z1xa1

ee

CP1 3 z = (z1 : z2)



Quantum groups

Indecomposable modules: Oas(n, z)

1 6 s 6 p − 1, a =±, n > 1, and z ∈ CP1, module Oas (n, z):

Xas•xa

2

��

xa1

��

Xas•xa

2

��

. . . Xas• xa

1

��X−a

p−s• X−ap−s• . . . X−a

p−s•

Xas•

z2xa2

>>

z1xa1

ee

CP1 3 z = (z1 : z2)



Quantum groups

Summary

We know

extended symmetry of the model:W -algebra

W -algebra irreps

Quantum-group projective modules

Quantum-group Grothendieckring/“fusion”

ZCFT: Characters and generalizedcharacters, modulartransformations on ZCFT

Modular transformations onquantum-group center Z

(some) modular invariants

We don’t

structure of W -algebra singularvectors etc.

W -algebra projective modules

honest W -algebra fusion

honest W -algebra torus amplitudes

full partition function

correlation functions



Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps




Modular transformations onquantum-group center Z = ZCFT


We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t









Quantum groups

Summary

We know


W -algebra irreps






We don’t







More than fifty percent success:



Quantum groups

Summary

We know


W -algebra irreps






We don’t







More than fifty percent success: Passed!



Quantum groups

Summary

We know


W -algebra irreps






We don’t







EVEN THOUGH I WAS CHEATING !!



Quantum groups

Thank You :)


More

4 More


More

4 More


More

From Free Fields to the Quantum Group

Consider the (p, 1) case. Screening: E =

∮e

−√

2pϕ

.

The Wp algebra:

W −(z) = e−√

2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W

0(z)],

where S+ =

∮e√

2pϕ. The W±,0(z) are primary fields of dimension 2p − 1

with respect to the energy-momentum tensor

T (z) =1

2∂ϕ∂ϕ(z) +

(√2p −

√2

p

)∂2ϕ(z).

On a suitably defined free-field space F,

Ker E∣∣∣F=

p⊕r=1

X+r ⊕ X−

r ,

a sum of 2p Wp-representations.


More



∮e

−√

2pϕ

.

The Wp algebra:

W −(z) = e−√

2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W

0(z)],

where S+ =

∮e√



T (z) =1

2∂ϕ∂ϕ(z) +

(√2p −

√2

p

)∂2ϕ(z).


Ker E∣∣∣F=

p⊕r=1

X+r ⊕ X−

r ,



More



∮e

−√

2pϕ

.

The Wp algebra:

W −(z) = e−√

2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W

0(z)],

where S+ =

∮e√



T (z) =1

2∂ϕ∂ϕ(z) +

(√2p −

√2

p

)∂2ϕ(z).


Ker E∣∣∣F=

p⊕r=1

X+r ⊕ X−

r ,



More



∮e

−√

2pϕ

.

The Wp algebra:

W −(z) = e−√

2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W

0(z)],

where S+ =

∮e√



T (z) =1

2∂ϕ∂ϕ(z) +

(√2p −

√2

p

)∂2ϕ(z).


Ker E∣∣∣F=

p⊕r=1

X+r ⊕ X−

r ,



More



∮e

−√

2pϕ

.

The Wp algebra:

W −(z) = e−√

2pϕ(z), W 0(z) = [S+,W−(z)], W +(z) = [S+,W

0(z)],

where S+ =

∮e√



T (z) =1

2∂ϕ∂ϕ(z) +

(√2p −

√2

p

)∂2ϕ(z).


Ker E∣∣∣F=

p⊕r=1

X+r ⊕ X−

r ,



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theor.jinr.rutheor.jinr.ru/~lcft07/lcft07/lectures/Semikhatov.pdf · Motivation Representation theory and CFT Quantum groups A 2002 quotation Logarithmic CFTs may be interesting from