D-branes and KW-dualities in (p,q) minimal superstring theory

1
D-branes and KW-dualities in (p,q) minimal
superstring theory

• Hirotaka Irie
• (Kyoto univ.)

Based on HI, arXiv0706.4471
hep-th Notes on D-branes and
dualities in (p,q) minimal superstring theory
2
Minimal superstring theory
Worldsheet description
FY-SFT description
Matrix-model description
WS description is also useful to see its basic
properties (Our intuition is mainly based on
this description)
3
The worldsheet SCFT
(p,q) minimal superstring theory N1
super-Liouville N1 (p,q) minimal SCFT
N1 super-ghost
type 0 GSO
Type 0 GSO diagonal GSO
Left/right WS fermion
(0B, -0A)
4
Problems to solve / Motivation

• Independent D-brane degrees of freedom
• (principle ?1 FZZT branes)
• in matrix models / its FY-SFT description
  Fukuma-HI 06
• The complete form of Boundary states and
• annulus amplitudes of the D-branes in any
  (p,q)
• (p,q)(2,4) pure-SUGRA case has been
• investigated in Seiberg-Shih 03,04
  Okuyama 05
• What is the meaning of ? in nonperturvative
  formulations or in superstring spacetime.
• (Order / disorder parameters ? )

5
Plan of the talk

1. D-branes / Cardy states (Review)
2. ? from Modular bootstrap
3. Boundary states of FZZT branes
4. Annulus amplitudes of FZZT branes
5. Summary and future directions

6
1. D-branes / Cardy states
Superconformally invariant boundary conditions
The solutions Ishibashi states (in each Verma
module Vi)
Ishibashi 89
Chirality of RR fields
(f worldsheet fermion )
7
The Cardy consistency conditions
Cardy 89
?Z

fusion number
The solutions Cardy states
Bosonic case
Closed-channel labelling
Open-channel labelling
8
Superminimal models
Nepomechie 01
NS
NS
R
R
spin-model GSO
X
N.B.
Superminimal models (with spin-model GSO) ?
minimal models with accidental symmetry
9
Superminimal matters coupled to super-Liouville
gravity
NS
NS
R
R
We need this amplitude for type 0B GSO projection

Klebanov-Maldacena-Seiberg 03 Seiberg-Shih
03
N.B.
Superminimal matters (without spin-model GSO)
? minimal models with accidental symmetry
10
Super-Liouville field theory
NS
NS
R
R
Obtained from the conformal bootstrap method
Fukuda-Hosomichi 02
First we derive this result from the modular
bootstrap method
11
2. from modular bootstrap
open-string fusion number
Consider the OPE with (1,2) deg. op.
HI 07
The chirality is flipped by the fermion in the
boundary action
Fukuda-Hosomichi 02
12
The corresponding Cardy equations
HI 07
This correctly reproduces the results of
Fukuda-Hosomichi
13
from modular bootstrap
Consider the OPE with (1,2) deg. op.
HI 07
OPE ? Super-Coulomb gas
Bershadski-Knizhnik-Teitelman 85
Mussardo-Sotkov-Sanishkov 8788
The chirality is flipped by the boundary
screening operator
14
The first result
HI 07
cf.)
The modular matrices S
are
Matsuo-Yahikozawa 86
15
3. FZZT-brane boundary states (0B)
Seiberg-Shih 03 HI 07
(k-s ?2Z)
(k-s ?2Z1)
newly obtained in HI 07
16
The principle FZZT branes
HI 07
Where
Sinh turns to be cosh
Note that

1. The principle ?1 FZZT brane is not a Cardy
   state.
2. The principle ?1 FZZT brane is not fundamental.
3. This is needed to construct all the spectrum of
   D-branes.

17
4. Amplitudes of FZZT branes in (p,q)
HI 07
(p,q) even model
(p,q) odd model
From the previous technique of Martinec 03
Kutasov-Okuyama-Park-Seiberg-Shih 04 (bosinic)
Okuyama 05 (fermionic (p,q)(2,4)) This
generalizes the results of Okuyama in (p,q)(2,4)
Okuyama 05
18
?-1/1 order/disorder parameters
Background
order
disorder
(This comes from the vanishing property of )
19
5. Summary

• We completely determine the Cardy states of (p,q)
  minimal SCFT and minimal superstring theory.
• We identify principle FZZT branes in this system.
• We evaluate the annulus amplitudes of the
  principle FZZT branes.
• The principle ?-1/1 FZZT branes are reminiscent
  of order / disorder parameters in spacetime KW
  duality.
• We also evaluate the boundary states of 0A theory
  in HI 07 .
• (e.g. We can see that only odd 0A theories
  can be identified with orbifolding of 0B
  theories.)

20
Future direction

• Other consistency checks of our Cardy states.
• Complete classification of boundary states in
  SCFT and other possible minimal superstring
  theories.
• Construction of Supersymmetric (type 0) Kostovs
  loop gas models (identify the corresponding
  supersymmetric integrable lattice models).