Towards the construction of local Logarithmic Conformal Field Theories

Transcription

1 Towards the construction of local Logarithmic Conformal Field Theories Anne-Ly Do Max-Planck-Institut für Physik komplexer Systeme Dresden July 26, 2007

2 Outline Fundamentals of two-dimensional conformal field theories (CFTs)

3 Outline Fundamentals of two-dimensional conformal field theories (CFTs) Logarithmic conformal field theories (LCFTs)

4 Outline Fundamentals of two-dimensional conformal field theories (CFTs) Logarithmic conformal field theories (LCFTs) The problem of constructing local LCFTs

5 Outline Fundamentals of two-dimensional conformal field theories (CFTs) Logarithmic conformal field theories (LCFTs) The problem of constructing local LCFTs Two methods to construct the space of states

6 Outline Fundamentals of two-dimensional conformal field theories (CFTs) Logarithmic conformal field theories (LCFTs) The problem of constructing local LCFTs Two methods to construct the space of states Consulting a specific model

7 The Power of Symmetry The concept of symmetry is the core of quantum field theory (QFT).

8 The Power of Symmetry The concept of symmetry is the core of quantum field theory (QFT). Symmetry rules the construction: Two of three key points of a QFT are governed by its symmetry:

9 The Power of Symmetry The concept of symmetry is the core of quantum field theory (QFT). Symmetry rules the construction: Two of three key points of a QFT are governed by its symmetry: Which space? Which fields? Which interactions?

10 The Power of Symmetry The concept of symmetry is the core of quantum field theory (QFT). Symmetry rules the construction: Two of three key points of a QFT are governed by its symmetry: Which space? Which fields? Which interactions? Symmetry allows the solution: Powerful tools are can be derived from symmetry:

11 The Power of Symmetry The concept of symmetry is the core of quantum field theory (QFT). Symmetry rules the construction: Two of three key points of a QFT are governed by its symmetry: Which space? Which fields? Which interactions? Symmetry allows the solution: Powerful tools are can be derived from symmetry: Noether s theorem, selection rules, conserved charges...

12 Conformal Symmetry A prime example is the conformal symmetry in two dimensions.

13 Conformal Symmetry A prime example is the conformal symmetry in two dimensions. Definition: Conformal transformations x x g µν x g µν x λ x g µν x

14 Conformal Symmetry A prime example is the conformal symmetry in two dimensions. Definition: Conformal transformations x x g µν x g µν x λ x g µν x d 2 Coordinates z x iy and z x iy, Every holomorphic map f : z z f z, z f z 0 is conformal, every anti-holomorphic too!

15 Conformal Symmetry A prime example is the conformal symmetry in two dimensions. Definition: Conformal transformations x x g µν x g µν x λ x g µν x d 2 Coordinates z x iy and z x iy, Every holomorphic map f : z z f z, z f z 0 is conformal, every anti-holomorphic too! CFTs possess an infinite number of local symmetries.

16 The Virasoro algebra The generators of the local conformal transformations are: l n z n 1 z and l n z n 1 z n Z

17 The Virasoro algebra The generators of the local conformal transformations are: l n z n 1 z and l n z n 1 z n Z Quantization: l n L n and l n L n L n,l m n m L n m c 12 L n, L m n m L n m c 12 L n, L m 0 ( n 3 n ) δ n, m ( n 3 n ) δ n, m

18 The Virasoro algebra The generators of the local conformal transformations are: l n z n 1 z and l n z n 1 z n Z Quantization: l n L n and l n L n L n,l m n m L n m c 12 L n, L m n m L n m c 12 L n, L m 0 ( n 3 n ) δ n, m ( n 3 n ) δ n, m Properties of the Virasoro algebra: Class of algebras, parametrized by the central charge c The chiral and the anti-chiral part decouple

19 Virasoro representation theory I Highest weight representations of the Virasoro algebra

20 Virasoro representation theory I Highest weight representations of the Virasoro algebra Fundamental fields

21 Virasoro representation theory I Highest weight representations of the Virasoro algebra Fundamental fields Ground states

22 Virasoro representation theory I Highest weight representations of the Virasoro algebra Fundamental fields Ground states In the simplest case, these representations are irreducible: Virasoro SU 2 L 0 c,h h c,h L z l,m l m l,m l L n c,h 0 n > 0 L l,m l 0 L nk L n1 c,h c,h;{n} L n l,m l l,m l n

23 Virasoro representation theory II Irreducible heighest weight representations: L 0 h h h, L n h 0 n > 0.

24 Virasoro representation theory II Irreducible heighest weight representations: L 0 h h h, L n h 0 n > 0. Reducible but indecomposable heighest weight representations:

25 Virasoro representation theory II Irreducible heighest weight representations: L 0 h h h, L n h 0 n > 0. Reducible but indecomposable heighest weight representations: r heighest weight states h,k with k {0, 1,...,r 1}, L n h,k 0 k n > 0

26 Virasoro representation theory II Irreducible heighest weight representations: L 0 h h h, L n h 0 n > 0. Reducible but indecomposable heighest weight representations: r heighest weight states h,k with k {0, 1,...,r 1}, L n h,k 0 k n > 0 Connected through the non-diagonal action of L 0 : L 0 h,k h h,k 1 δ k,0 h,k 1.

27 Virasoro representation theory II Irreducible heighest weight representations: L 0 h h h, L n h 0 n > 0. Reducible but indecomposable heighest weight representations: r heighest weight states h,k with k {0, 1,...,r 1}, L n h,k 0 k n > 0 Connected through the non-diagonal action of L 0 : L 0 h,k h h,k 1 δ k,0 h,k 1. Naming covention: δ h h,k 1 δ 0,k h,k 1.

28 Virasoro representation theory II Irreducible heighest weight representations: L 0 h h h, L n h 0 n > 0. Reducible but indecomposable heighest weight representations: r heighest weight states h,k with k {0, 1,...,r 1}, L n h,k 0 k n > 0 Connected through the non-diagonal action of L 0 : L 0 h,k h h,k 1 δ k,0 h,k 1. Naming covention: δ h h,k 1 δ 0,k h,k 1. k: Jordan level, number of different k s: Jordan rank r

29 Solving CFT problems A CFT is characterized by three parameters:

30 Solving CFT problems A CFT is characterized by three parameters: symmetry algebra representation dim representation central charge c conformal weights h i Jordan rank r

31 Solving CFT problems A CFT is characterized by three parameters: symmetry algebra representation dim representation central charge c conformal weights h i Jordan rank r General strategy: Treating many theories at the same time.

32 Solving CFT problems A CFT is characterized by three parameters: symmetry algebra representation dim representation central charge c conformal weights h i Jordan rank r General strategy: Treating many theories at the same time. Theories with different r cannot be treated simultaneously

33 Solving CFT problems A CFT is characterized by three parameters: symmetry algebra representation dim representation central charge c conformal weights h i Jordan rank r General strategy: Treating many theories at the same time. Theories with different r cannot be treated simultaneously For fixed r, generic solutions with defined dependency on the parameters h, c

34 Logarithmic CFTs r 1 versus r 2 The generic form of correlation functions changes crucially, when r is changed.

35 Logarithmic CFTs r 1 versus r 2 The generic form of correlation functions changes crucially, when r is changed. (Gurarie, 1993) Virasoro representations with r 2 Logarithmic divergences in correlation functions

36 Logarithmic CFTs r 1 versus r 2 The generic form of correlation functions changes crucially, when r is changed. The second difference concerns locality:

37 Logarithmic CFTs r 1 versus r 2 The generic form of correlation functions changes crucially, when r is changed. The second difference concerns locality: Theories with r 1 factorize, i. e. the local theory can be obtained as product of the chiral and anti-chiral halves.

38 Logarithmic CFTs r 1 versus r 2 The generic form of correlation functions changes crucially, when r is changed. The second difference concerns locality: Theories with r 1 factorize, i. e. the local theory can be obtained as product of the chiral and anti-chiral halves. This is not true for r 2.

39 The state of affairs The problem: Well-elaborated: exact generic solutions for the chiral halves Unknown: how to assemble two halves to a local theory

40 The state of affairs The problem: Well-elaborated: exact generic solutions for the chiral halves Unknown: how to assemble two halves to a local theory Only approach so far: (Gaberdiel and Kausch, 1999) Constructed a local LCFT at c 2 Based on model specific data cannot be generalized

41 The state of affairs The problem: Well-elaborated: exact generic solutions for the chiral halves Unknown: how to assemble two halves to a local theory Only approach so far: (Gaberdiel and Kausch, 1999) Constructed a local LCFT at c 2 Based on model specific data cannot be generalized Our contribution: We developed methods how to assemble generic local correlation functions and local states from the chiral halves.

42 The local space of states R h { h, 0, h, 1 }

43 The local space of states R h { h, 0, h, 1 } R h { h, 0, h, 1 }

44 The local space of states R h { h, 0, h, 1 } R h { h, 0, h, 1 } R h R h { 1 1, 0 1, 1 0, 0 0 },

45 The local space of states R h { h, 0, h, 1 } R h { h, 0, h, 1 } R h R h { 1 1, 0 1, 1 0, 0 0 }, k i k j : h,k i h, k j

46 The local space of states R h { h, 0, h, 1 } R h { h, 0, h, 1 } R h R h { 1 1, 0 1, 1 0, 0 0 }, k i k j : h,k i h, k j Locality constraint: δ h δ h L h h 0,

47 The local space of states R h { h, 0, h, 1 } R h { h, 0, h, 1 } R h R h { 1 1, 0 1, 1 0, 0 0 }, k i k j : h,k i h, k j Locality constraint: δ h δ h L h h 0, δ h h,k 1 δ 0,k h,k 1

48 The local space of states R h { h, 0, h, 1 } R h { h, 0, h, 1 } R h R h { 1 1, 0 1, 1 0, 0 0 }, k i k j : h,k i h, k j Locality constraint: δ h δ h L h h 0, δ h h,k 1 δ 0,k h,k 1 level ւ δ h level level ց δ h

49 Quotient space construction Locality constraint: δ h δ h L h h 0 level level level L h h R h R h

50 Quotient space construction Locality constraint: δ h δ h L h h 0 level level level L h h R h R h/s h h Approach Nº 1: (Gaberdiel and Kausch, 1999) Identification of 0 1 and 0 1 Dividing out a subrepresentation

51 Quotient space construction Locality constraint: δ h δ h L h h 0 level level level L h h R h R h/s h h Approach Nº 1: (Gaberdiel and Kausch, 1999) Identification of 0 1 and 0 1 Dividing out a subrepresentation S h h {( ), 0 0 } image of R h R h under δ h δ h

52 Quotient space construction Locality constraint: δ h δ h L h h 0 level level level L h h R h R h/s h h Approach Nº 1: (Gaberdiel and Kausch, 1999) Identification of 0 1 and 0 1 Dividing out a subrepresentation S h h {( ), 0 0 } image of R h R h under δ h δ h

53 Generalization For arbitrary Jordan rank L h h R h R h/s h h possible level r level r 1 level r 2.. level 0 level 1 level (Do and Flohr, 2007)

54 Generalization For arbitrary Jordan rank L h h R h R h/s h h possible Minimal choice for S h h image of R h R h under δ h δ h level r level r 1 level r 2.. level 0 level 1 level (Do and Flohr, 2007)

55 Generalization For arbitrary Jordan rank L h h R h R h/s h h possible Minimal choice for S h h image of R h R h under δ h δ h L h h { equivalence classes with level 0 } level r level r 1 level r 2.. level 0 level 1 level (Do and Flohr, 2007)

56 Kernel construction Approach Nº 2: (Do and Flohr, 2007) δ h δ h L h h 0 L h h is the kernel of δ h δ h.

57 Kernel construction Approach Nº 2: (Do and Flohr, 2007) δ h δ h L h h 0 L h h is the kernel of δ h δ h. r r 1 r

58 Kernel construction Approach Nº 2: (Do and Flohr, 2007) δ h δ h L h h 0 L h h is the kernel of δ h δ h. r r 1 r 2 δ h δ h acting on the sum over all states of same level: telescoping series

59 Kernel construction Approach Nº 2: (Do and Flohr, 2007) δ h δ h L h h 0 L h h is the kernel of δ h δ h. r r 1 r 2 δ h δ h acting on the sum over all states of same level: telescoping series for l 0 the first and the last term vanish

60 Kernel construction Approach Nº 2: (Do and Flohr, 2007) δ h δ h L h h 0 L h h is the kernel of δ h δ h. r r 1 r 2 δ h δ h acting on the sum over all states of same level: telescoping series for l 0 the first and the last term vanish Kernel { level l states l 0}

61 Discussion Comparison of both methods:

62 Discussion Comparison of both methods: With both methods we find: rank L h h rank R h rank R h

63 Discussion Comparison of both methods: With both methods we find: rank L h h rank R h rank R h Quotient space method: L h h { level n equivalence classes n 0 } r 3

64 Discussion Comparison of both methods: With both methods we find: rank L h h rank R h rank R h Quotient space method: L h h { level n equivalence classes n 0 } Kernel construction: L h h { level n states n 0} r 3

65 Discussion Comparison of both methods: With both methods we find: rank L h h rank R h rank R h Quotient space method: L h h { level n equivalence classes n 0 } Kernel construction: L h h { level n states n 0} r 3 Which method is to choose?

66 Help from a specific model I rank-2 theory at c 2, h { 0, 1, 1 8, 3 8} free symplectic fermion

67 Help from a specific model I rank-2 theory at c 2, h { 0, 1, 1 8, 3 8} free symplectic fermion The stress energy tensor of this theory is given by: T z 1 2 ǫαβ : θ α θ β : z.

68 Help from a specific model I rank-2 theory at c 2, h { 0, 1, 1 8, 3 8} free symplectic fermion The stress energy tensor of this theory is given by: T z 1 2 ǫαβ : θ α θ β : z. The mode expansion of the component fields reads: θ α n 0θ α,n z n θ α,0 log z ξ α.

69 Help from a specific model I rank-2 theory at c 2, h { 0, 1, 1 8, 3 8} free symplectic fermion The stress energy tensor of this theory is given by: T z 1 2 ǫαβ : θ α θ β : z. The mode expansion of the component fields reads: θ α n 0θ α,n z n θ α,0 log z ξ α. The ξ s are Grassmann numbers.

70 Help from a specific model II ξ s act as creation operator: h 0,k ǫαβ ξ α ξ β h 0,k 0 : ξ α ξ β 0

71 Help from a specific model II ξ s act as creation operator: h 0,k ǫαβ ξ α ξ β h 0,k 0 : ξ α ξ β 0 Of course an analogue identity holds for the anti-chiral half.

72 Help from a specific model II ξ s act as creation operator: h 0,k ǫαβ ξ α ξ β h 0,k 0 : ξ α ξ β 0 Of course an analogue identity holds for the anti-chiral half. r 2 ξ α ξ β 0 ξ α ξ β 0 ξ α ξ β ξα ξβ 0 0 ξ α ξ β ξ α ξ β 0 ( ξ α ξ β ξ α ξ ) β

73 Help from a specific model II ξ s act as creation operator: h 0,k ǫαβ ξ α ξ β h 0,k 0 : ξ α ξ β 0 Of course an analogue identity holds for the anti-chiral half. r 2 ξ α ξ β 0 ξ α ξ β 0 ξ α ξ β ξα ξβ 0 0 ξ α ξ β ξ α ξ β 0 ( ξ α ξ β ξ α ξ ) β It is always possible to choose a basis such that ξ α ξ α.

74 Help from a specific model II ξ s act as creation operator: h 0,k ǫαβ ξ α ξ β h 0,k 0 : ξ α ξ β 0 Of course an analogue identity holds for the anti-chiral half. r 2 ξ α ξ β 0 ξ α ξ β 0 ξ α ξ β ξα ξβ 0 0 ξ α ξ β ξ α ξ β 0 ( ξ α ξ β ξ α ξ ) β It is always possible to choose a basis such that ξ α ξ α.

75 Conclusions L h h R h R h

76 Conclusions L h h R h R h The local subspace L h h can be sorted out by two methods: the quotient space construction and the kernel construction.

77 Conclusions L h h R h R h The local subspace L h h can be sorted out by two methods: the quotient space construction and the kernel construction. All computable data of theories constructed in the one or the other manner is identical.

78 Conclusions L h h R h R h The local subspace L h h can be sorted out by two methods: the quotient space construction and the kernel construction. All computable data of theories constructed in the one or the other manner is identical. The Symplectic Fermion model suggests the kernel construction to be the appropriate method.

79 References A.-L. Do and M. A. I. Flohr: Towards the construction of local Logarithmic Conformal Field Theories. In preparation (2007). V. Gurarie: Logarithmic Operators in Conformal Field Theory. Nucl. Phys. B 410, (1993). M. R. Gaberdiel and H. G. Kausch: A local Logarithmic Conformal Field Theory. Nucl. Phys. B 538, (1999). H. G. Kausch: Symplectic Fermions. Nucl. Phys. B 583, (2000). M. A. I. Flohr : Operator Product Expansion in Logarithmic Conformal Field Theory Nucl. Phys. B 634, (2002).