D-branes and KW-dualities in (p,q) minimal superstring theory Hirotaka Irie (Kyoto univ.) Based on HI, arXiv:0706.4471 [hep-th] “Notes on D-branes and dualities in (p,q) minimal superstring theory”

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)

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)

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 ? )

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

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 #)

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

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

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

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

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]

The corresponding Cardy equations [HI ’07] This correctly reproduces the results of Fukuda-Hosomichi

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

The modular matrices S are The first result [HI ’07] cf.) The modular matrices S are [Matsuo-Yahikozawa ‘86]

3. FZZT-brane boundary states (0B) [Seiberg-Shih ’03] [HI ‘07] (k-s ∈2Z) (k-s ∈2Z+1) : newly obtained in [HI ‘07]

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.

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]

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 )

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.)

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).