Wall-Crossing of D4/D2/D0 on the Conifold ( arXiv: 1007.2731 [hep-th] ) Takahiro Nishinaka ( Osaka U.) (In collaboration with Satoshi Yamaguchi )
Introduction The d=4, N=2 string thoery have a special class of quantum states called 1/2 BPS states , whose “degeneracy” or index is piecewise constant in the moduli space. Ω ( Q ; t ) = − 1 2Tr[( − 1) F F 2 ] : electro-magnetic charge, : vacuum moduli Q t The trace is taken over the Hilbert space of charge , which depends on . t Q
Introduction The d=4, N=2 string thoery have a special class of quantum states called 1/2 BPS states , whose “degeneracy” or index is piecewise constant in the moduli space. Ω ( Q ; t ) = − 1 2Tr[( − 1) F F 2 ] : electro-magnetic charge, : vacuum moduli Q t The trace is taken over the Hilbert space of charge , which depends on . t Q Wall-crossing phenomena moduli space Ω ( Q ; t 1 ) Ω ( Q ; t 2 ) wall of marginal stability
Introduction The d=4, N=2 string thoery have a special class of quantum states called 1/2 BPS states , whose “degeneracy” or index is piecewise constant in the moduli space. Ω ( Q ; t ) = − 1 2Tr[( − 1) F F 2 ] : electro-magnetic charge, : vacuum moduli Q t The trace is taken over the Hilbert space of charge , which depends on . t Q Wall-crossing phenomena discrete change moduli space Ω ( Q ; t 1 ) Ω ( Q ; t 2 ) wall of marginal stability
Introduction The d=4, N=2 string thoery have a special class of quantum states called 1/2 BPS states , whose “degeneracy” or index is piecewise constant in the moduli space. Ω ( Q ; t ) = − 1 2Tr[( − 1) F F 2 ] : electro-magnetic charge, : vacuum moduli Q t The trace is taken over the Hilbert space of charge , which depends on . t Q Wall-crossing phenomena BPS BPS BPS For some decay Q → Q 1 + Q 2 , discrete change moduli space Z ( Q ) = Z ( Q 1 ) + Z ( Q 2 ) | Z ( Q ) | = | Z ( Q 1 ) | + | Z ( Q 2 ) | Ω ( Q ; t 1 ) Ω ( Q ; t 2 ) wall of marginal stability
Introduction The d=4, N=2 string thoery have a special class of quantum states called 1/2 BPS states , whose “degeneracy” or index is piecewise constant in the moduli space. Ω ( Q ; t ) = − 1 2Tr[( − 1) F F 2 ] : electro-magnetic charge, : vacuum moduli Q t The trace is taken over the Hilbert space of charge , which depends on . t Q Wall-crossing phenomena BPS BPS BPS For some decay Q → Q 1 + Q 2 , discrete change moduli space Z ( Q ) = Z ( Q 1 ) + Z ( Q 2 ) Z ( Q 1 ) Z ( Q ) | Z ( Q ) | = | Z ( Q 1 ) | + | Z ( Q 2 ) | Ω ( Q ; t 1 ) namely, Z ( Q 2 ) Ω ( Q ; t 2 ) arg[ Z ( Q )] = arg[ Z ( Q 1 )] wall of marginal stability (= arg[ Z ( Q 2 )])
Introduction Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered) ~ vacuum moduli = Calabi-Yau moduli
Introduction The appearence/disappearence of BPS bound states is related to the existence of multi-centered BPS black holes. [F. Denef] Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered) ~ vacuum moduli = Calabi-Yau moduli
Introduction The appearence/disappearence of BPS bound states is related to the existence of multi-centered BPS black holes. [F. Denef] Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered) ~ vacuum moduli = Calabi-Yau moduli
Introduction The appearence/disappearence of BPS bound states is related to the existence of multi-centered BPS black holes. [F. Denef] Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered) ~ vacuum moduli = Calabi-Yau moduli KS-formula discrete change Recently, Kontsevich and Soibelman have Ω ( Q ; t 1 ) proposed a wall-crossing formula that Ω ( Q ; t 2 ) tells us how the degeneracy changes at the walls of marginal stability. wall of marginal stability
The main topic of this talk Type IIA on Calabi-Yau We study the wall-crossing of one non-compact D4 -brane with arbitrary numbers of D2/D0 on the resolved conifold .
The main topic of this talk Type IIA on Calabi-Yau We study the wall-crossing of one non-compact D4 -brane with arbitrary numbers of D2/D0 on the resolved conifold . The vacuum moduli are the Kahler moduli of the conifold.
The main topic of this talk Type IIA on Calabi-Yau We study the wall-crossing of one non-compact D4 -brane with arbitrary numbers of D2/D0 on the resolved conifold . The vacuum moduli are the Kahler moduli of the conifold. We evaluate the partition function of D4/D2/D0 in various chambers in the moduli space by using the Kontsevich-Soibelman formula (KS-formula).
Resolved conifold Definition ( z 1 , z 2 , z 3 , z 4 ) ; | z 1 | 2 + | z 2 | 2 − | z 3 | 2 − | z 4 | 2 = y C 4 ⊃ M y := � � U(1)-action : → ( e i θ Z 1 , e i θ z 2 , e − i θ z 3 , e − i θ z 4 ) ( z 1 , z 2 , z 3 , z 4 ) −
Resolved conifold Definition ( z 1 , z 2 , z 3 , z 4 ) ; | z 1 | 2 + | z 2 | 2 − | z 3 | 2 − | z 4 | 2 = y C 4 ⊃ M y := � � U(1)-action : → ( e i θ Z 1 , e i θ z 2 , e − i θ z 3 , e − i θ z 4 ) ( z 1 , z 2 , z 3 , z 4 ) − compact 2-cycle x 1 resolved conifold := M y /U (1) compact 4-cycle x 0
Resolved conifold Definition ( z 1 , z 2 , z 3 , z 4 ) ; | z 1 | 2 + | z 2 | 2 − | z 3 | 2 − | z 4 | 2 = y C 4 ⊃ M y := � � U(1)-action : → ( e i θ Z 1 , e i θ z 2 , e − i θ z 3 , e − i θ z 4 ) ( z 1 , z 2 , z 3 , z 4 ) − compact 2-cycle x 1 resolved conifold := M y /U (1) compact 4-cycle x 0 Compact 2-cycle (1) In the case of y > 0 The compact 2-cycle is z 3 = z 4 = 0 ,
Resolved conifold Definition ( z 1 , z 2 , z 3 , z 4 ) ; | z 1 | 2 + | z 2 | 2 − | z 3 | 2 − | z 4 | 2 = y C 4 ⊃ M y := � � U(1)-action : → ( e i θ Z 1 , e i θ z 2 , e − i θ z 3 , e − i θ z 4 ) ( z 1 , z 2 , z 3 , z 4 ) − compact 2-cycle x 1 resolved conifold := M y /U (1) compact 4-cycle x 0 Compact 2-cycle (1) In the case of y > 0 The compact 2-cycle is z 3 = z 4 = 0 , namely, ( z 1 , z 2 ) ; | z 1 | 2 + | z 2 | 2 = y � � /U (1) ≃ P 1
Resolved conifold Definition ( z 1 , z 2 , z 3 , z 4 ) ; | z 1 | 2 + | z 2 | 2 − | z 3 | 2 − | z 4 | 2 = y C 4 ⊃ M y := � � U(1)-action : → ( e i θ Z 1 , e i θ z 2 , e − i θ z 3 , e − i θ z 4 ) ( z 1 , z 2 , z 3 , z 4 ) − compact 2-cycle x 1 resolved conifold := M y /U (1) compact 4-cycle x 0 Compact 2-cycle (2) In the case of (1) In the case of y < 0 y > 0 The compact 2-cycle is The compact 2-cycle is z 1 = z 2 = 0 , z 3 = z 4 = 0 , namely, namely, ( z 1 , z 2 ) ; | z 3 | 2 + | z 4 | 2 = | y | ( z 1 , z 2 ) ; | z 1 | 2 + | z 2 | 2 = y � � /U (1) ≃ P 1 � � /U (1) ≃ P 1
Resolved conifold Definition ( z 1 , z 2 , z 3 , z 4 ) ; | z 1 | 2 + | z 2 | 2 − | z 3 | 2 − | z 4 | 2 = y C 4 ⊃ M y := � � U(1)-action : → ( e i θ Z 1 , e i θ z 2 , e − i θ z 3 , e − i θ z 4 ) ( z 1 , z 2 , z 3 , z 4 ) − compact 2-cycle x 1 resolved conifold := M y /U (1) compact 4-cycle x 0 Compact 2-cycle (2) In the case of (1) In the case of y < 0 y > 0 The compact 2-cycle is The compact 2-cycle is z 1 = z 2 = 0 , z 3 = z 4 = 0 , namely, namely, ( z 1 , z 2 ) ; | z 3 | 2 + | z 4 | 2 = | y | ( z 1 , z 2 ) ; | z 1 | 2 + | z 2 | 2 = y � � /U (1) ≃ P 1 � � /U (1) ≃ P 1 Two limits correspond to large 2-cycle limits. y → ± ∞
Resolved conifold D4-brane and flop We put one D4-brane on a non-compact 4-cycle z 3 = 0
Resolved conifold D4-brane and flop We put one D4-brane on a non-compact 4-cycle z 3 = 0 (1) In the case of y > 0 The compact 2-cycle : z 3 = z 4 = 0 D4 P 1
Resolved conifold D4-brane and flop We put one D4-brane on a non-compact 4-cycle z 3 = 0 (1) In the case of y > 0 The compact 2-cycle : z 3 = z 4 = 0 D4 P 1 The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane
Resolved conifold D4-brane and flop We put one D4-brane on a non-compact 4-cycle z 3 = 0 (1) In the case of (2) In the case of y > 0 y < 0 The compact 2-cycle : The compact 2-cycle : z 1 = z 2 = 0 z 3 = z 4 = 0 D4 D4 P 1 P 1 The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane
Resolved conifold D4-brane and flop We put one D4-brane on a non-compact 4-cycle z 3 = 0 (1) In the case of (2) In the case of y > 0 y < 0 The compact 2-cycle : The compact 2-cycle : z 1 = z 2 = 0 z 3 = z 4 = 0 D4 D4 P 1 P 1 The compact 2-cycle is outside of the The compact 2-cycle is embeded in the 4-cycle wapped by the D4-brane 4-cycle wrapped by the D4-brane
Resolved conifold D4-brane and flop We put one D4-brane on a non-compact 4-cycle z 3 = 0 (1) In the case of (2) In the case of y > 0 y < 0 The compact 2-cycle : The compact 2-cycle : z 1 = z 2 = 0 z 3 = z 4 = 0 D4 D4 flop transition P 1 P 1 The compact 2-cycle is outside of the The compact 2-cycle is embeded in the 4-cycle wapped by the D4-brane 4-cycle wrapped by the D4-brane
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