1. D-branes and KW-dualities in (p,q) minimal superstring theory
Hirotaka Irie
(Kyoto univ.)
Based on
HI, arXiv: [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
= N=1 super-Liouville N=1 (p,q) minimal SCFT
N=1 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 η=±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 η in nonperturvative formulations or in superstring spacetime.
(Order / disorder parameters ? )

5. Plan of the talk D-branes / Cardy states (Review)
Ψ from Modular bootstrap
Boundary states of FZZT branes
Annulus amplitudes of FZZT branes
Summary and future directions

6. 1. D-branes / Cardy states
Superconformally invariant boundary conditions:
The solutions = Ishibashi states (in each Verma module Vi):
( anti-unitary op.)
[Ishibashi ‘89]
Chirality of RR fields
(f: worldsheet fermion #)

7. The Cardy consistency conditions:
∈Z+ =
The solutions = Cardy states:
= fusion number
Bosonic case
Closed-channel labelling
Open-channel labelling

8. Superminimal models (with spin-model GSO)
[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 gravityNS 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 theoryNS 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 ’87’88]
The chirality is flipped by the boundary screening operator

14. The modular matrices S are
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 ∈2Z+1)
: newly obtained in [HI ‘07]

16. The principle FZZT branes
Where
[HI ‘07]
Note that
“Sinh” turns to be “cosh”
The principle η=+1 FZZT brane is not a Cardy state.
The principle η=+1 FZZT brane is not fundamental.
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. η=-1/+1 = order/disorder parameters
Background
= order
= disorder
These principle D-branes are not mutually local in spacetime
(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 η=-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) Kostov’s loop gas models (identify the corresponding supersymmetric integrable lattice models).