Conformal field theories with supergroup symmetry From supergeometry to logarithmic CFT
SIS, Isaac Newton Institute, Cambridge
Thomas Quella (University of Amsterdam)
Based on the articles: [ hep-th/0610070 ]: “Supergroup PSU(1,1|2)”, with Gerhard Gotz¨ and Volker Schomerus [ arXiv:0706.0744 ]: “Type I supergroups”, with Volker Schomerus [ arXiv:0708.0583 ]: “D-branes on GL(1|1)”, with Thomas Creutzig and Volker Schomerus
The project “SuperCFT” is funded by the European FP6 (contract MEIF-CT-2007-041765)
Conformal field theories with supergroup symmetry – p. 1/41 Theories with supersymmetry I
String theory
Frequently one desires target space supersymmetry because of Stability Finiteness Phenomenological reasons
Maximally supersymmetric AdS-spaces play a prominent role in the AdS/CFT correspondence [Maldacena] [...] String theory on AdS-space ⇔ Gauge theory on boundary
The pure spinor formalism provides the most promising way of quantizing flux backgrounds [Berkovits] [...] [Hoogeveen,Skenderis]
Conformal field theories with supergroup symmetry – p. 2/41 Making the supersymmetry manifest...
All AdS backgrounds preserve a superconformal symmetry which is realized geometrically (as an isometry) [Nahm] In the supergroup formulation one has
P SU(1, 1|2) AdS × S2 : 2 U(1) × U(1) 3 AdS3 × S : PSU(1, 1|2)
3 3 AdS3 × S × S : D(2, 1; α)
P SU(2, 2|4) AdS × S5 : 5 SO(4, 1) × SO(5)
[Metsaev,Tseytlin] [Berkovits,Bershadsky,Hauer,Zhukov,Zwiebach] [Berkovits,Vafa,Witten]
Conformal field theories with supergroup symmetry – p. 3/41 Strings on (super)manifolds
The propagation of strings in some target space M is described by (conformal) 2D non-linear σ-models
Σ M
Remarks: One studies maps X : Σ → M The coordinates Xµ are promoted to fields Xµ(z, z¯) The isometries act on the fields Xµ(z, z¯)
Conformal field theories with supergroup symmetry – p. 4/41 Strings on (super)manifolds
2D non-linear σ-models Dynamics: Action functional S[Xµ] The physics depends on background parameters: Metric Gauge fields (fluxes): NS and RR fields Dilaton (Super)symmetry: Invariance of S[Xµ] under isometries Conformal invariance ⇔ Fields solve (super)gravity equations
Conformal field theories with supergroup symmetry – p. 4/41 Strings on (super)manifolds
2D non-linear σ-models Dynamics: Action functional S[Xµ] The physics depends on background parameters: Metric Gauge fields (fluxes): NS and RR fields Dilaton (Super)symmetry: Invariance of S[Xµ] under isometries Conformal invariance ⇔ Fields solve (super)gravity equations
String theory: 1. Solve quantum non-linear σ-model 2. Impose constraints
Conformal field theories with supergroup symmetry – p. 4/41 Theories with supersymmetry II
Condensed matter, disordered systems and statistical physics
Disordered systems [Efetov]
Spin quantum Hall effect [Gruzberg,Ludwig,Read]
Integer quantum Hall effect [Pruisken] [Zirnbauer] [Tsvelik] [LeClair]
Polymers and percolation [Parisi,Sourlas] [Saleur,Read] [Jacobsen,Read,Saleur]
Conformal field theories with supergroup symmetry – p. 5/41 Some technical observations
Standard CFT descriptions do not apply in many cases...
The chiral algebra is unknown or rather small [Bershadsky,Vaintrob,Zhukov] Theories with target space supersymmetry are frequently logarithmic conformal field theories [Rozansky,Saleur] [Maassarani,Serban] [...]
Some models admit new kinds of moduli [Berkovits,Vafa,Witten] [Read,Saleur]
Conformal field theories with supergroup symmetry – p. 6/41 Plan of the talk
Envisaged lesson: Why do logarithmic CFTs arise in connection with supergeometry?
1. Reminder: Specific features of 2D logarithmic CFTs 2. Discussion of (type I) supergroup WZW models
GL(1|1) ∼= SUSY(R1,1)
3 P SU(1, 1|2) ∼= SUSY(AdS3 × S )
3. A few comments on other AdS backgrounds...
Conformal field theories with supergroup symmetry – p. 7/41 2D conformal field theories in a nutshell
The essential ingredients of every CFT are
Chiral symmetry algebra A⊕ A¯
Free boson: J(z) = −i∂X(z), J¯(¯z) = i∂X¯ (¯z)
Fields (vertex operators) ΦR(z, z¯)
: eikX(w,w¯) : J(z) : eikX(w,w¯) : = k z − w
Space of states (fields) H = HR
L ∗ H = dk H(k,−k) = dk Hk ⊗Hk R R Z Z
Conformal field theories with supergroup symmetry – p. 8/41 Characteristics of logarithmic CFTs...
A typical operator product has the form
Φ(z, z¯) Φ∗(0) = (zz¯)−hΦ ω(0) + ln(zz¯) Ω(0) + · · · This is only consistent if L0 and L¯0 are not diagonalizable
L0ω = Ω , L0Ω=0
The theory is non-unitary There are some issues with factorization
Rotations L0 − L¯0 have to be diagonalizable The dilatation operator L0 + L¯0 is not diagonalizable
Conformal field theories with supergroup symmetry – p. 9/41 ...and their consequence
(Certain) Representations of A⊕ A¯ have to be non-chiral
HR =6 Hµ ⊗ H¯ν
Which representations contribute precisely?
Logarithmic triplet model: Brute force calculation [Gaberdiel,Kausch]
Supergroup WZW models: Supergeometry [TQ,Schomerus]
General proposal: Constraints from D-branes [Gaberdiel,Runkel]
Conformal field theories with supergroup symmetry – p. 10/41 Supergroups and Lie superalgebras
Lie supergroups arise through exponentiation of Lie superalgebras
Procedure: Introduce Lie superalgebras Discuss their representation theory Do harmonic analysis on supergroups Explain the relation to WZW models
First GL(1|1), then more general supergroups
Conformal field theories with supergroup symmetry – p. 11/41 Lie superalgebras
A vector space g with multiplication [·, ·] is a Lie superalgebra iff
1. g is Z2-graded: g = g0¯ ⊕ g1¯ 2. For the bracket one has
Consistency with the grading: [gi, gj] ⊂ gi+j Graded antisymmetry Graded Jacobi identity
“Lie algebra with bosonic and fermionic generators”
Conformal field theories with supergroup symmetry – p. 12/41 Supergroups of type I
We will only be concerned with type I Lie superalgebras: [Kac] Non-degenerate invariant bilinear form
Particular Z-grading: g = g1 ⊕ g0 ⊕ g−1
Examples: gl(M|N), sl(M|N), psl(N|N), ...
bosonic M × M fermionic M × N gl(M|N) : A = fermionic N × M bosonic N × N !
Conformal field theories with supergroup symmetry – p. 13/41 Supergroups of type I
We will only be concerned with type I Lie superalgebras: [Kac] Non-degenerate invariant bilinear form
Particular Z-grading: g = g1 ⊕ g0 ⊕ g−1
Gauss-like decomposition for supergroup elements... θ·F1 θ¯·F−1 g = e gB e with gB ∈ GB
Conformal field theories with supergroup symmetry – p. 13/41 Isometries of supergroups
On a supergroup G one has the supersymmetry G × G:
Left regular action: g 7→ hg Right regular action: g 7→ gk−1
These transformation induce an action of G × G on the (linear) space F(G) of functions on the supergroup:
(h × k) · f : g 7→ f(h−1gk)
Remarks:
Inf. SUSY → differential operators → Lie superalgebra Metric ↔ Laplacian ↔ quadratic Casimir
Conformal field theories with supergroup symmetry – p. 14/41 Wess-Zumino-Witten models
The WZW action depends only on the metric
WZW WZ S [g] = Skin[g] + S [g]
The action is invariant under
g 7→ h(z)gk(¯z)−1
This leads to affine Lie superalgebras: A⊕ A¯ = gˆ ⊕ gˆ (Certain) vertex operators can be considered as normal ordered matrix elements
(R) VR(z, z¯) = : D g(z, z¯) :
Conformal field theories with supergroup symmetry – p. 15/41 A useful dictionary
WZW model Geometry
State space H ↔ Algebra of functions F(G)
Chiral algebra gˆ ⊕ gˆ ↔ Left/right regular action of g ⊕ g,
J(z), J¯(¯z) differential operators J0, J¯0 Dilatation operators ↔ Laplacian, Casimir operator ¯ 2 ¯2 L0, L0 ∆ ∼ trJ0 = trJ0 Vertex operators ↔ Eigenfunctions of the Laplacian
Conformal dimension ↔ Eigenvalue
Geometry is a good approximation at large volume!
Conformal field theories with supergroup symmetry – p. 16/41 An example: GL(1|1)
The superalgebra gl(1|1) is defined by the commutation relations
[N,ψ±] = ±ψ± [ψ+,ψ−] = E
Supergroup elements are given by [Rozansky,Saleur] [Schomerus,Saleur]
+ − g = eiη+ψ ei(xE+yN) eiη−ψ
Conformal field theories with supergroup symmetry – p. 17/41 An example: GL(1|1)
The superalgebra gl(1|1) is defined by the commutation relations
[N,ψ±] = ±ψ± [ψ+,ψ−] = E
Supergroup elements are given by [Rozansky,Saleur] [Schomerus,Saleur]
+ − g = eiη+ψ ei(xE+yN) eiη−ψ
The superalgebra gl(1|1) in disguise: Supersymmetric quantum mechanics (forget about N) deRham complex and Hodge theory
Conformal field theories with supergroup symmetry – p. 17/41 The Kac modules of gl(1|1)
The standard representations are 2D Kac modules he, ni.
N n − 1 n
They are defined by the highest weight condition
E|e, ni = e|e, ni N|e, ni = n|e, ni ψ+|e, ni = 0
Conformal field theories with supergroup symmetry – p. 18/41 The Kac modules of gl(1|1)
The standard representations are 2D Kac modules he, ni.
The Kac modules degenerate for e = 0: They exhibit a 1-dimensional invariant subspace hni
ψ+ ψ−|e, ni = [ψ+,ψ−]|e, ni = e |e, ni
One has to distinguish Kac modules and anti Kac modules
Conformal field theories with supergroup symmetry – p. 18/41 Summary on irreducible modules of gl(1|1)
Module he, ni hni
Value of E e =06 e =0
Qualifier typical atypical
Dimension 2 1
Projective? yes no
Extendable? no yes
Important concept: Composition series
h0, ni : hni → hn − 1i
Conformal field theories with supergroup symmetry – p. 19/41 Projective covers of gl(1|1)
The projective cover Pn of hni is four-dimensional
The quadratic Casimir is not diagonalizable Projective covers arise as tensor products of typical modules
he, mi⊗h−e, ni = Pm+n−1
Tensor products close on projective modules
Conformal field theories with supergroup symmetry – p. 20/41 Projective covers of gl(1|1)
Projective covers have various invariant subspaces and quotients...
Atypical irreducible representation
Pn : hni → hn − 1i ⊕ hn +1i → hni
Conformal field theories with supergroup symmetry – p. 20/41 Projective covers of gl(1|1)
Projective covers have various invariant subspaces and quotients...
Degenerate Kac module
Pn : hni → hn − 1i⊕ hn +1i → hni
Conformal field theories with supergroup symmetry – p. 20/41 Projective covers of gl(1|1)
Projective covers have various invariant subspaces and quotients...
Degenerate Kac module
Pn : h0, n +1i → h0, ni
Conformal field theories with supergroup symmetry – p. 20/41 Projective covers of gl(1|1)
Projective covers have various invariant subspaces and quotients...
Degenerate anti Kac module
Pn : hni → hn − 1i ⊕ hn +1i → hni
Conformal field theories with supergroup symmetry – p. 20/41 Projective covers of gl(1|1)
Projective covers have various invariant subspaces and quotients...
Zigzag module
Pn : hni → hn +1i ⊕ hn +1i → hni
Conformal field theories with supergroup symmetry – p. 20/41 Projective covers of gl(1|1)
Projective covers have various invariant subspaces and quotients...
Dual zigzag module
Pn : hni → hn +1i ⊕ hn +1i → hni
Conformal field theories with supergroup symmetry – p. 20/41 Harmonic analysis on GL(1|1)
The algebra of functions on GL(1|1) has the following basis
i(ex+ny) i(ex+ny) i(ex+ny) e , e η± , e η+η−
In a weight diagram wrt to N and N¯ one has N¯ N¯
N N
all values of n n ∈ Z
Conformal field theories with supergroup symmetry – p. 21/41 Harmonic analysis on GL(1|1)
In the typical sectors one finds the spectrum
left action right action both actions
de 2 · he, ni e6=0 Z Z Mn∈
Conformal field theories with supergroup symmetry – p. 21/41 Harmonic analysis on GL(1|1)
In the typical sectors one finds the spectrum
left action right action both actions
de 2 · he, ni → de dn he, ni ⊗ he, ni∗ e6=0 Z e6=0 Z Z Mn∈ Z Mn∈
Conformal field theories with supergroup symmetry – p. 21/41 Harmonic analysis on GL(1|1)
In the atypical sectors of the WZW model, however, one obtains
left action right action both actions
1 · Pn Z Mn∈
Conformal field theories with supergroup symmetry – p. 21/41 Harmonic analysis on GL(1|1)
In the atypical sectors of the WZW model, however, one obtains
left action right action both actions
1 · Pn → In Z Mn∈
Conformal field theories with supergroup symmetry – p. 21/41 Solving supergroup WZW models
Free fermion resolution I: Semi-classical analysis Rewriting the Lagrangian Symmetry and spectrum: Minisuperspace approximation Point-particle limit ∼= Harmonic analysis on G
Conformal field theories with supergroup symmetry – p. 22/41 Solving supergroup WZW models
Free fermion resolution I: Semi-classical analysis Rewriting the Lagrangian Symmetry and spectrum: Minisuperspace approximation Point-particle limit ∼= Harmonic analysis on G Free fermion resolution II: Quantum description Symmetry and spectrum: Affine Lie superalgebras and reps Correlation functions
Conformal field theories with supergroup symmetry – p. 22/41 Solving supergroup WZW models
Free fermion resolution I: Semi-classical analysis Rewriting the Lagrangian Symmetry and spectrum: Minisuperspace approximation Point-particle limit ∼= Harmonic analysis on G Free fermion resolution II: Quantum description Symmetry and spectrum: Affine Lie superalgebras and reps Correlation functions
Special features: [Rozansky,Saleur] [Serban,Maassarani] [Schomerus,Saleur] Non-diagonalizability: Logarithmic correlation functions Non-chirality: Representations of the chiral algebra do not factorize into holomorphic times antiholomorphic Non-unitarity
Conformal field theories with supergroup symmetry – p. 22/41 Rewriting the WZW Lagrangian
We start with the Gauss decomposition
θ·F1 θ¯·F−1 g = e gB e with gB ∈ GB
Using the Polyakov-Wiegmann identity and introducing auxiliary fermionic fields p and p¯, the Lagrangian can be written as
WZW ¯ S = Sren [gB] + SF[θ,p] + SF[θ, p¯] + Sint[g] 1 S [g] = d2z pR(g )¯p int 2π B Z
Achievement: Reduction to bosonic WZW model
Conformal field theories with supergroup symmetry – p. 23/41 The strategy for a solution
Assume that the WZW model for gB is solved [Knizhnik,Zam] [Gepner,Witten] The free fermion theory is under complete control
Treat Sint ∼ pR(gB)¯p as a perturbation
R −Sint hΦ1 · · · ΦniWZW = hφ1 · · · φn e idecoupled
Conformal field theories with supergroup symmetry – p. 24/41 The strategy for a solution
Assume that the WZW model for gB is solved [Knizhnik,Zam] [Gepner,Witten] The free fermion theory is under complete control
Treat Sint ∼ pR(gB)¯p as a perturbation
R −Sint hΦ1 · · · ΦniWZW = hφ1 · · · φn e idecoupled
Still to determine: The spectrum, i.e. the range of fields Φ The map Φ(z, z¯) 7→ φ gB(z, z¯), θ(z), θ¯(¯z)
Conformal field theories with supergroup symmetry – p. 24/41 The strategy for a solution
Assume that the WZW model for gB is solved [Knizhnik,Zam] [Gepner,Witten] The free fermion theory is under complete control
Treat Sint ∼ pR(gB)¯p as a perturbation
R −Sint hΦ1 · · · ΦniWZW = hφ1 · · · φn e idecoupled
Still to determine: The spectrum, i.e. the range of fields Φ The map Φ(z, z¯) 7→ φ gB(z, z¯), θ(z), θ¯(¯z)
Tools: Point particle limit: Only keep zero-modes Representation theory of g
Harmonic analysis [Gepner,Witten]
Conformal field theories with supergroup symmetry – p. 24/41 Decoupled versus non-decoupled
Both Langrangians have the same supersymmetry gˆ ⊕ gˆ...... but it is realized differently! Interacting theory: g ⊕ g ∼= isometries of G Decoupled theory: g ⊕ g not related to G
Conformal field theories with supergroup symmetry – p. 25/41 Realization of supersymmetry
Supersymmetry g ⊕ g LL rrr LL rr LLL rrr LL rr LLL rx rr L& J0, J¯0 Differential operators J0, J¯0
∆0 Laplacian (Casimir) ∆
F0(G) Representation space F(G)
Decoupled Interacting
Conformal field theories with supergroup symmetry – p. 26/41 The auxiliary problem
The auxiliary algebra of functions is
F0(G) = F(GB) ⊗ (θ, θ¯) ^ Solve the eigenvalue problem for the Laplacian ∆0
Since ∆0 is the quadratic Casimir (with respect to the generators J) the problem is again reduced to harmonic analysis
We only need to know how F0(G) decomposes wrt g ⊕ g
The interaction term Sint induces a map
Ξ : F0(G) → F(G) mapping eigenfunctions of ∆0 to (generalized) eigenfunctions of ∆
Conformal field theories with supergroup symmetry – p. 27/41 Restriction to the bosonic subsymmetry
The Peter-Weyl theorem (or related knowledge) implies
F(G ) = V ⊗ V ∗ B g0⊕g0 µ µ irreps µ M
The Grassmann algebra transforms in
(θ, θ¯) = F ⊗ F ∗ g0⊕g0 ^ The bosonic representation content is thus given by
∗ F (G) = V ⊗ F ⊗ V ⊗ F 0 g0⊕g0 µ µ irreps µ M h i h i
Conformal field theories with supergroup symmetry – p. 28/41 Supersymmetrization of the bosonic multiplets
The relevant functions without fermions are matrix elements α (µ) “vertex operators of bosonic WZW model” D (gB) β
When the fermions are included, the bosonic modules are promoted to Kac modules K = V ⊗ F µ g0 µ The total (auxiliary) representation space may then be written as
∗ F0(G) g⊕g = Kµ ⊗ Kµ irreps µ (of GB ) M
The eigenvalues of the Laplacian ∆0 are the Casimirs
Conformal field theories with supergroup symmetry – p. 29/41 Typical versus atypical representations
There are two different types of Kac modules Irreducible → “typical” (non-BPS) Reducible (but not fully) → “atypical” (BPS)
∗ ∗ F0(G) g⊕g = Kµ ⊗ Kµ ⊕ Kσ ⊗ Kσ µ typ σ atyp M M
Conformal field theories with supergroup symmetry – p. 30/41 Typical versus atypical representations
There are two different types of Kac modules Irreducible → “typical” (non-BPS) Reducible (but not fully) → “atypical” (BPS)
∗ ∗ F0(G) g⊕g = Kµ ⊗ Kµ ⊕ Kσ ⊗ Kσ µ typ σ atyp M M
With Kac modules one can associate Irreducible representations Lµ via quotients Projective covers Pµ of Lµ
The trivial representation L0 is always atypical!
Conformal field theories with supergroup symmetry – p. 30/41 Projective covers and a duality
Projective covers have a Kac composition series There exists a nice duality between irreducible modules, Kac modules and projective covers [Zou] [Brundan]
[Pµ : Kν] = [Kν : Lµ]
Conformal field theories with supergroup symmetry – p. 31/41 Peter-Weyl theorem for supergroups
Theorem: ∗ F(G) g⊕g = Kµ ⊗ Kµ ⊕ I[σ] µ typ [σ] atyp M M
∗ The map Ξ entangles the modules Kσ ⊗ Kσ into non-chiral indecomposable representations I[σ] in the atypical subspace Restriction to either left or right action yields
∗ ∗ F(G) g = dim(Kµ) Kµ ⊕ dim(Lσ) Pσ µ typ σ atyp M M
∆ is not diagonalizable on the spaces I[σ]
Conformal field theories with supergroup symmetry – p. 32/41 The full quantum theory
The generalization of the point particle results to the full quantum theory is relatively straightforward:
Symmetry:
Left and right regular actions are promoted to affine current superalgebra symmetries
There exist expressions for the currents in terms of free fermions and renormalized bosonic currents They closely resemble the differential operators
Conformal field theories with supergroup symmetry – p. 33/41 The full quantum theory
The generalization of the point particle results to the full quantum theory is relatively straightforward:
Representation theory:
The metric and thus the Casimir eigenvalues are renormalized
The energy operator L0 is not diagonalizable Finite volume leads to a truncation of the spectrum
Affine representations still factorize in the typical sector
There might be twisted and spectrally flowed sectors
Conformal field theories with supergroup symmetry – p. 33/41 The full quantum theory
The generalization of the point particle results to the full quantum theory is relatively straightforward:
Correlation functions:
They may be calculated algorithmically The relevant formulas are
−Sint hΦ1 · · · ΦniWZW = hφ1 · · · φn e idecoupled
Sint ∼ pR(gB)¯p Z The perturbation series truncates after a finite number of terms Certain correlation functions will be logarithmic
Conformal field theories with supergroup symmetry – p. 33/41 D-branes in GL(1|1)
There is an analogue of the Cardy case Trivial gluing conditions for the currents D-branes are labelled by irreducible representations
The spectrum is given by fusion [Creutzig,TQ,Schomerus]
There are logarithmic boundary sectors, e.g.
∗ µ ⊗ µ = P0 + · · ·
Semi-classical methods confirm the indecomposability One has a phenomenon of localized atypical branes
Structurally similar results exist for the triplet model [Gaberdiel,Runkel]
Conformal field theories with supergroup symmetry – p. 34/41 Quotients of (super)groups
A (right) supercoset G/H is a supergroup with identification
g ∼ gh with h ∈ H ⊂ G
The (super)symmetry is again realized geometrically:
Left regular action: g 7→ hg
This transformation induces an action on the (linear) space F(G/H) = InvH F(G) of functions on the supercoset:
h · f : g 7→ f(h−1g)
Remark: The identification above is not the one which leads to gauged WZW models! → difficult to quantize
Conformal field theories with supergroup symmetry – p. 35/41 Conformal invariance?
One sided (super)cosets G/H n Generalized symmetric spaces, e.g. AdSn × S H is fixed point set under an automorphism of G Conformally invariant examples:
G has vanishing Killing form [Kagan,Young] → AdS-spaces, flat superspace, superspheres, ... C + Non-compact cosets H /H, e.g. H3 [Gawedzki] No general algebraic construction known Chiral symmetry? Chiral splitting? Relation to massive perturbations of gauged WZW models?
[Grigoriev,Tseytlin] [Mikhailov,Schafer-Nameki]¨
Conformal field theories with supergroup symmetry – p. 36/41 Are coset theories G/H logarithmic?
Recall the Peter-Weyl theorem for supergroups
F(G) = K ⊗ K∗ ⊕ P ⊗ L∗ g⊕g0 µ µ g0 σ σ g0 µ typ σ atyp M M
H bosonic ⇒ gauging the right action gives rise to atypical projective covers and hence logarithmic behaviour This applies to All AdS-spaces Flat superspace = super-Poincare/Lorentz´ It does, however, not (necessarily) apply to Superspheres such as S3|2 = OSP (4|2)/OSP (3|2) Projective superspaces
Conformal field theories with supergroup symmetry – p. 37/41 Summary and conclusions
There is an overwhelming number of “new” conformal field theories with supergroup symmetry Supersymmetry(!) implies the existence of an atypical sector which is manifestly non-chiral → logarithmic CFT A large class of supergroup WZW models can be solved using a free fermion resolution → reduction to bosonic subgroup The geometric interpretation of these models might teach us a lot about the structure of general logarithmic CFTs [TQ,Schomerus]
Conformal field theories with supergroup symmetry – p. 38/41 Outlook and speculations
Generalization to type II supergroups [Hikida,Schomerus] Solve gauged WZW theories and other coset models ⇒ get a handle on AdSn theories
Might indecomposability at the end even help? [Read,Saleur]
Implications for the pure spinor approach? [Berkovits] [Hoogeveen,Skenderis] Applications to disordered systems, percolation, ...
Conformal field theories with supergroup symmetry – p. 39/41 3 String theory on AdS3 × S × M4
λ
Deformed WZW model: mixture of NS and RR flux
WZW model: pure NS flux
Principal chiral model: pure RR flux
[Berkovits,Vafa,Witten] k [Bershadsky,Vaintrob,Zhukov]
P SU(1,1|2) WZ S = λ Skin + k S
Conformal field theories with supergroup symmetry – p. 40/41 Non-standard CFTs
Supergroups with vanishing Killing form → New marginal deformations of WZW models
Metric perturbation
J(z)Adg J¯(¯z) = J(z)Φad(z, z¯)J¯(¯z)
Marginality ⇒ vanishing Killing form Full global G × G symmetry preserved
W-algebra symmetry [Bershadsky,Vaintrob,Zhukov]
W (n)(z) = tr(Jn)
Mixture of NS and RR fluxes [Berkovits,Vafa,Witten]
Conformal field theories with supergroup symmetry – p. 41/41