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Superspace Sigma Models

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### Superspace Sigma Models

IST String Fest

Volker Schomerus

based on work w. C. Candu, C.Creutzig, V. Mitev,

T Quella,H. Saleur; 2 papers in preparation

Superspace Sigma Models

Aim: Study non-linear sigma models with

target space supersymmetry not world-sheet

Strings in AdS backgrounds [pure spinor]

c = 0 CFTs; symmetry e.g. PSU(2,2|4) ...

Focus on scale invariant QFT, i.e. 2D CFT

Properties:Weird: logarithmic Conformal Field Theories!

Remarkable: Many families with cont. varying exponents

Plan of 2nd Lecture

M

O

D

E

L

S

- SM on CP0|1 - Symplectic Fermions
- Sigma Model on Superspace CP1|2
- Properties of the Chiral Field on CP1|2
- Conclusion & Some Open Problems

Bulk and twisted Neumann boundary conditions

M

E

T

H

O

D

S

Exact boundary partition functions

Quasi-abelian evolution; Lattice models;

Symplectic fermions as cohomology

1.1 Symplectic Fermions - Bulk

Only global u(1)

sym. is manifest

Has affine psu(1|1) sym:

Action rewritten in terms of currents:

U(1) x U(1)

Possible modification: θ-angle

trivial in bulk

1.1 Symplectic Fermions - Bulk

Has affine psu(1|1) sym:

Action rewritten in terms of currents:

U(1) x U(1)

Possible modification: θ-angle

trivial in bulk

ΘΘ

x

1.2 SF – Boundary Conditionsboundary

term

Implies twisted Neumann boundary conditions:

A

with currents:

Results:

4 ground states

In P0 of u(1|1)

Θ1Θ2

-1

Ground states: λ(Θ1,Θ2) = λ(tr(A1A2 ))

x

twist fields

[Creutzig,Quella,VS] [Creutzig,Roenne]

2.1 The Sigma Model on CPN-1|N

zα, ηα→

ωzα, ωηα

=

→ 2 parameter family of 2D CFTs;c = -2

a - non-dynamical

gauge field

D = ∂ - ia

θ term non-trivial; θ=θ+2π

Non-abelian extension of symplectic fermion

2.2 CPN-1|N – Boundary Conditions

Boundary condition in σ-Model on target X is

hypersurface Y + bundle with connection A

Dirichlet BC ┴ to Y; Neumann BC || to Y;

U(N|N) symmetric boundary cond. for CPN-1|N

line bundles on Y = CPN-1|N with monopole Aμ

Spectrum of sections e.g. N=2; μ = 0

μ integer

~ (1+14+1) + 48 + 80 + ... + (2n+1) x 16 +

superspherical

harmonics

atypicals

typicals of U(2|2)

2.3 Spectra/Partition Function

Count boundary condition changing operators

at R = ∞ (free field theory)

Built from Zα,Zα,∂xZα, ...

fermionic contr.

monopole numbers

Euler fct

bosonic contr.

u(2|2) characters

for N=2

and μ = ν

~ χ(1+14+1) + χ48 + χ80 + ... q χad + ...

Result: Spectrum at finite R

universal

indep of Λ,N

fμν|μ-ν| ~ λ

Characters

of u(2|2) reps

Branching fcts

from R = ∞

λ = λ(Θ1,Θ2) from

symplectic Fermions

Casimir

of U(2|2)

3.1 Quasi-abelian Evolution

Free Boson:

In boundary theory

bulk more involved

Prop.: Boundary spectra of CP1|2 chiral field :

quadratic

Casimir

Deformation of conf. weights is `quasi-abelian’

[Bershadsky et al] [Quella,VS,Creutzig] [Candu, Saleur]

e.g. (1+14+1) remains at Δ=0; 48, 80, .... are lifted

3.1 Quasi-abelian Evolution

Free Boson:

In boundary theory

bulk more involved

at R=R0 universal U(1) charge

Prop.: Boundary spectra of CP1|2 chiral field :

quadratic

Casimir

Deformation of conf. weights is `quasi-abelian’

[Bershadsky et al] [Quella,VS,Creutzig] [Candu, Saleur]

e.g. (1+14+1) remains at Δ=0; 48, 80, .... are lifted

3.2 Lattice Models and Numerics

boundary term

f00

acts on states space:

level 1

level 2

level 3

Extract fνμ(R) from

Δhk= fνμ(R)(Clk-Cl0)

l = μ - ν

no sign of

instanton effects

w=w(R)

3.3 SFermions as Cohomology

U(2|2):

U(1)

x

U(1|2)

in 3

of U(1|2)

Result: [Candu,Creutzig,Mitev,VS]

Cohomology of Q arises from states in atypical

modules with sdim ≠ 0

all multiplets of ground states

contribute → λ(Θ1,Θ2) from SF

Conclusions and Open Problems

- Exact Boundary Partition Functions for CP1|2
- For S2S+1|S there is WZ-point at radius R = 1
- Is there WZ-point in moduli space of CPN-1|N?
- Much generalizes to PCM on PSU(1,1|2)!

Techniques: QA evolution; Lattice; Q-Cohomology

[Candu,H.Saleur]

[Mitev,Quella,VS]

osp(2S+2|2s) at level k=1

e.g. psu(N|N) at level k=1

CP0|1 = PSU(1|1)k=1

AdS3 x S3

QA evolution, simple subsector

Examples: Super-Cosets

[Candu]

[thesis]

Familiesv w. compact form, w.o. H-flux:

cpct symmetric superspaces

volume

G/GZ

2

OSP(2S+2|2S)

OSP(2S+1|2S)

→ S2S+1|2S

OSP(2S+2|2S)

OSP(2S+2-n|2S) x SO(n)

c = 1

U(N|N)

U(N-1|N)xU(1)

→ CPN-1|N

U(N|N)

U(N-n|N) x U(n)

c = -2

.

.

- note: cv (GL(N|N)) = 0 = cv (OSP(2S+2|2S))

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