logarithmic conformal field theory, w-algebras and...

Why (L)-CFT? Why holography Results To do list

Logarithmic Conformal Field Theory,W-Algebras and Holography

Yannick Mvondo-She

Department of PhysicsUniversity of Pretoria

2nd Mandelstam Workshop Of Theoretical PhysicsJanuary 20, 2018

Durban, South Africa


Outline

Why (L)-CFT?

Why holography

Results

To do list


Why (L)-CFT?Because of appearance of Jordan cells...

RCFT

• Constructed by HighestWeight Representation ofthe Virasoro Algebra

• A highest weight state |h〉corresponding to a primaryfield of conformal dimensionh has the property:

Ln|h〉 = 0 (for n > 0),

L0|h〉 = h|h〉,

LCFT

• Primary states: L0|A〉 = h|A〉• Logarithmic partner:L0|B〉 = h|B〉+ |A〉

• Appearance of Jordan cells:

L0

(|A〉|B〉

)=

(h 01 h

)(|A〉|B〉

)


And?.... and logarithmic singularities in 2-point correlation functions

〈A(z1)A(z2)〉 = 0

〈A(z1)B(z2)〉 = 〈B(z1)A(z2)〉 =K

(z1 − z2)2h

〈B(z1)B(z2)〉 = −2Kln(z1 − z2)

(z1 − z2)2h


Why holography?

• Topologically Massive, New Massive and Generalized MassiveGravities are good candidates for duality

• Application of AdS3/(L)CFT2 correspondence[]Grumiller, Riedler, Rosseel’13]


Partition function of TMG= Partition function of LCFT

One loop graviton partition function ZTMG

[Gaberdiel, Grumiller, Vassilevich’10]

ZTMG(q, q) =∞∏n=2

1

|1− qn|2∞∏

m=2

∞∏m=0

1

|1− qmqm|2.

ZLCFT(q, q) = Z0LCFT(q, q) +

∑h,h

Nh,hqhqh

∞∏n=1

1

|1− qn|2,

with

Z0LCFT(q, q) = ZΩ + Zt =

∞∏n=2

1

|1− qn|2

(1 +

q2

|1− q|2

),

Ω: the Virasoro vacuumt: log. partner of energy momentum tensor T .


But partition function of TMG...... in terms of Bell Polynomials

ZTMG = A(q, q)B(q, q),

with:

A(q, q) =∞∏n=2

1

|1− qn|2; B(q, q) =

∞∏m=0

∞∏m=0

1

|1− q2qmqm|2

B(q, q) =∑∞

n=0Ynn!

(q2)n

(Generating function of Bell polynomials)

Yn(g1, g2, . . . , gn) =∑k`n

n!

k1! · · · kn!

(g1

1!

)k1(g2

2!

)k2

· · ·(gnn!

)kngn = (n − 1)!

∑m≥0,m≥0

qnmqnm.


...= Partition function of LCFT...... in an ordered way.

[]Grumiller, Riedler, Rosseel’13]


Additional resultsHeisenberg-Weyl action

Raising/Lowering operators:

X = g1 +n∑

k=1

gk+1∂

∂gk, D =

∂

∂g1

Heisenberg-Weyl Algebra:[X , D

]= 1

Action on Bell Polynomials:XYn = Yn+1

DYn = nYn−1

X DYn = nYn


Additional resultssl2 action

Raising/Lowering operators:

f =1

2R2, h = RL +

1

2, e =

1

2L2 (1)

sl2 Algebra: f = 12 X

2, h = X D + 12 , e = 1

2 D2

Action on Bell Polynomials:eYn = 1

2n(n − 1)Yn−2

fYn = 12Yn+2

hYn =(n + 1

2

)Yn


Additional resultsPictorially

Ladder operators acting on Y (n odd)

Ladder operators acting on Y (n even)


GoalsImmediate goal: Topologically Massive Higher Spin Gravity...

Partition function of Topologically Massive Higher Spin Gravity:[Bagchi, Lal, Saha, Sahoo’11]


Goals... starting with addition of a spin 3 field

Partition function of Topologically Massive Higher Spin Gravity:


GoalsOther goals

• Quantum Group action on our holographic LCFT model viaKazhdan-Lusztig correspondence.

• Novel relation between Hopf Algebras and functions in severalvariables [Lentner’17]

• Appearance of a differential operator acting on objects


Some literature

1. M.R. Gaberdiel, D. Grumiller, and D. Vassilevich, Graviton1-loop partition function for 3-dimensional massive gravity,JHEP 1011 (2010) 094, arxiv:1007.5189 .

2. A. Bagchi, S. Lal, A. Saha, B. Sahoo, One loop partitionfunction for Topologically Massive Higher Spin Gravity, arXiv:1107.2063v2.

3. D. Grumiller, W. Riedler, J. Rosseel, Holographic applicationsof logarithmic conformal field theories, arXiv:1302.0280.

4. S. D. Lentner, Quantum groups and Nichols algebras actingon conformal field theories, arXiv preprint arXiv:1702.06431(2017).


Acknowledgement

• Mandelstam Institute of Theoretical Physics

• NITheP

• University of Pretoria

• Professor Konstantinos Zoubos

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