Wall-Crossing of D4/D2/D0 on the Conifold ( arXiv: 1007.2731 - - PowerPoint PPT Presentation
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
( Osaka U.) (In collaboration with Satoshi Yamaguchi ) ( arXiv: 1007.2731 [hep-th] )
Q Q
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.
: electro-magnetic charge, : vacuum moduli
The trace is taken over the Hilbert space of charge , which depends on .
Q Q
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. Wall-crossing phenomena
: electro-magnetic charge, : vacuum moduli
The trace is taken over the Hilbert space of charge , which depends on . moduli space wall of marginal stability
Q Q
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. Wall-crossing phenomena
: electro-magnetic charge, : vacuum moduli
The trace is taken over the Hilbert space of charge , which depends on . moduli space wall of marginal stability discrete change
Q Q
Z(Q) = Z(Q1) + Z(Q2)
|Z(Q)| = |Z(Q1)| + |Z(Q2)|
Q → Q1 + Q2,
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. Wall-crossing phenomena
: electro-magnetic charge, : vacuum moduli
The trace is taken over the Hilbert space of charge , which depends on . moduli space wall of marginal stability discrete change For some decay
BPS BPS BPS
Q Q
Z(Q) = Z(Q1) + Z(Q2)
|Z(Q)| = |Z(Q1)| + |Z(Q2)|
Q → Q1 + Q2,
Z(Q)
Z(Q1) Z(Q2)
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. Wall-crossing phenomena
: electro-magnetic charge, : vacuum moduli
The trace is taken over the Hilbert space of charge , which depends on . moduli space wall of marginal stability discrete change namely, For some decay
BPS BPS BPS
Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered)
vacuum moduli = Calabi-Yau moduli
Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered)
The appearence/disappearence of BPS bound states is related to the existence of multi-centered BPS black holes. [F. Denef] vacuum moduli = Calabi-Yau moduli
Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered)
The appearence/disappearence of BPS bound states is related to the existence of multi-centered BPS black holes. [F. Denef] vacuum moduli = Calabi-Yau moduli
KS-formula Recently, Kontsevich and Soibelman have proposed a wall-crossing formula that tells us how the degeneracy changes at the walls of marginal stability. Branes vs Black holes wrapped D-branes in CY3 BPS black holes in 4dim (single- or multi-centered)
The appearence/disappearence of BPS bound states is related to the existence of multi-centered BPS black holes. [F. Denef] vacuum moduli = Calabi-Yau moduli
wall of marginal stability discrete change
Type IIA on Calabi-Yau
Type IIA on Calabi-Yau
Type IIA on Calabi-Yau
→ (eiθZ1, eiθz2, e−iθz3, e−iθz4)
Definition U(1)-action :
→ (eiθZ1, eiθz2, e−iθz3, e−iθz4)
Definition U(1)-action :
resolved conifold compact 2-cycle x 1 compact 4-cycle x 0
→ (eiθZ1, eiθz2, e−iθz3, e−iθz4) z3 = z4 = 0,
Definition U(1)-action :
resolved conifold Compact 2-cycle compact 2-cycle x 1 compact 4-cycle x 0
(1) In the case of The compact 2-cycle is
→ (eiθZ1, eiθz2, e−iθz3, e−iθz4) z3 = z4 = 0,
Definition U(1)-action :
resolved conifold Compact 2-cycle compact 2-cycle x 1 compact 4-cycle x 0
(1) In the case of The compact 2-cycle is namely,
→ (eiθZ1, eiθz2, e−iθz3, e−iθz4) z3 = z4 = 0, z1 = z2 = 0,
Definition U(1)-action :
resolved conifold Compact 2-cycle compact 2-cycle x 1 compact 4-cycle x 0
(1) In the case of
(2) In the case of The compact 2-cycle is namely, The compact 2-cycle is namely,
→ (eiθZ1, eiθz2, e−iθz3, e−iθz4) z3 = z4 = 0, z1 = z2 = 0,
Definition U(1)-action :
resolved conifold Compact 2-cycle compact 2-cycle x 1 compact 4-cycle x 0
(1) In the case of
(2) In the case of The compact 2-cycle is namely, The compact 2-cycle is namely,
y → ±∞
Two limits correspond to large 2-cycle limits.
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
(1) In the case of The compact 2-cycle :
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
(1) In the case of The compact 2-cycle :
The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
(1) In the case of
(2) In the case of The compact 2-cycle : The compact 2-cycle :
The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
(1) In the case of
(2) In the case of The compact 2-cycle : The compact 2-cycle :
The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane The compact 2-cycle is outside of the 4-cycle wapped by the D4-brane
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
(1) In the case of
(2) In the case of flop transition The compact 2-cycle : The compact 2-cycle :
The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane The compact 2-cycle is outside of the 4-cycle wapped by the D4-brane
D4-brane and flop We put one D4-brane on a non-compact 4-cycle
(1) In the case of
(2) In the case of flop transition The compact 2-cycle : The compact 2-cycle :
The compact 2-cycle is embeded in the 4-cycle wrapped by the D4-brane The compact 2-cycle is outside of the 4-cycle wapped by the D4-brane
The point is geometrically singular but the spectrum has no singularity if we tune the B-field for the compact 2-cycle.
(P, P ′ ∈ H2(X))
In the final result, the local limit should be taken.
Λ → ∞
Kahler moduli : Kahler parameter for the compact cycle
( : B-field, : size ) : Kahler parameter for other non-compact cycles
(P, P ′ ∈ H2(X))
Q → Q1 + Q2
arg[Z(Q)] = arg[Z(Q1)] = arg[Z(Q2)]
In the final result, the local limit should be taken.
Λ → ∞
Kahler moduli : Kahler parameter for the compact cycle
( : B-field, : size ) : Kahler parameter for other non-compact cycles Walls of marginal stability For a decay channel , the walls are defined by
(P, P ′ ∈ H2(X))
D4 + kD2 + lD0
Q → Q1 + Q2
arg[Z(Q)] = arg[Z(Q1)] = arg[Z(Q2)] D4 + (k ∓ 1)D2 + (l − n)D0 (±1)D2 + nD0
In the final result, the local limit should be taken.
Λ → ∞
Kahler moduli : Kahler parameter for the compact cycle
( : B-field, : size ) : Kahler parameter for other non-compact cycles Walls of marginal stability For a decay channel , the walls are defined by The relevant walls are
(P, P ′ ∈ H2(X))
D4 + kD2 + lD0
Q → Q1 + Q2
arg[Z(Q)] = arg[Z(Q1)] = arg[Z(Q2)] D4 + (k ∓ 1)D2 + (l − n)D0 (±1)D2 + nD0
Z((±1)D2 + nD0) = ±z + n
In the final result, the local limit should be taken.
Λ → ∞
Kahler moduli : Kahler parameter for the compact cycle
( : B-field, : size ) : Kahler parameter for other non-compact cycles Walls of marginal stability For a decay channel , the walls are defined by The relevant walls are
Z(D4 + kD2 + lD0) ∼ −1 2Λ2e2iϕ
central charges
Z((±1)D2 + nD0) = ±z + n
Z(D4 + kD2 + lD0) ∼ −1 2Λ2e2iϕ
Walls of marginal stability are the subspace in the moduli space where these two central charges are aligned,
Z((±1)D2 + nD0) = ±z + n
Z(D4 + kD2 + lD0) ∼ −1 2Λ2e2iϕ
ϕ = 1 2 arg[∓z − n]
Walls of marginal stability are the subspace in the moduli space where these two central charges are aligned, namely,
Z((±1)D2 + nD0) = ±z + n
Z(D4 + kD2 + lD0) ∼ −1 2Λ2e2iϕ
ϕ = 1 2 arg[∓z − n]
Walls of marginal stability are the subspace in the moduli space where these two central charges are aligned, (±1, n) W ±1
n
These walls are labeled by . So we denote them . namely,
2ϕ
2
−1 −2
Rez
W −1
3
W −1
2
W −1
1
W −1
W −1
−1
W +1
−2
W +1
−1
W +1 W +1
1
W +1
2
Z((±1)D2 + nD0) = ±z + n
Z(D4 + kD2 + lD0) ∼ −1 2Λ2e2iϕ
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation
A =
− →
KΩ(QBPS;t)
QBPS
∞
1 n2 enQ]
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product
A =
− →
KΩ(QBPS;t)
QBPS
∞
1 n2 enQ]
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product The product is taken in the decreasing order of arg Z(QBPS)
A =
− →
KΩ(QBPS;t)
QBPS
∞
1 n2 enQ]
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product The product is taken in the decreasing order of arg Z(QBPS)
The product depends on the moduli in two ways, the degeneracy and the order in the product depend on .
A =
− →
KΩ(QBPS;t)
QBPS
∞
1 n2 enQ]
Z(Q2) Z(Q1 + Q2)
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product The product is taken in the decreasing order of arg Z(QBPS)
The product depends on the moduli in two ways, the degeneracy and the order in the product depend on .
A =
− →
KΩ(QBPS;t)
QBPS
∞
1 n2 enQ]
Z(Q2) Z(Q1 + Q2)
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product The product is taken in the decreasing order of arg Z(QBPS)
The product depends on the moduli in two ways, the degeneracy and the order in the product depend on .
Z(Q1) Z(Q2) Z(Q1 + Q2)
wall-crossing
A =
− →
KΩ(QBPS;t)
QBPS
∞
1 n2 enQ]
Z(Q2) Z(Q1 + Q2)
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product The product is taken in the decreasing order of arg Z(QBPS)
The product depends on the moduli in two ways, the degeneracy and the order in the product depend on . KS-formula says that “nevertheless, the product is independent of ! ”
Z(Q1) Z(Q2) Z(Q1 + Q2)
wall-crossing
A =
− →
KΩ(QBPS;t)
QBPS
Ω(QBPS; t)
∞
1 n2 enQ]
Z(Q2) Z(Q1 + Q2)
[eQ1, eQ2] = (−1)Q1,Q2 Q1, Q2 eQ1+Q2 Consider an infinite dimensional Lie algebra with a commutation relation and the following product The product is taken in the decreasing order of arg Z(QBPS)
The product depends on the moduli in two ways, the degeneracy and the order in the product depend on . KS-formula says that “nevertheless, the product is independent of ! ” We can read off the change in from the invariance of .
Z(Q1) Z(Q2) Z(Q1 + Q2)
wall-crossing
Partition function
Partition function
Im z = ∞ Im z = −∞
Suppose that we move the moduli from to as above.
2ϕ
1 2
−1
−2
Rez W −1
3
W −1
2
W −1
1
W −1 W −1
−1
W +1
−2
W +1
−1
W +1 W +1
1
W +1
2
Partition function
Im z = ∞ Im z = −∞
Suppose that we move the moduli from to as above.
2ϕ
1 2
−1
−2
Rez W −1
3
W −1
2
W −1
1
W −1 W −1
−1
W +1
−2
W +1
−1
W +1 W +1
1
W +1
2
Im z > 0
{W −1
∞ , · · · , W −1 2
, W −1
1
} (1) For , there are the walls of .
Z(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1)
Partition function
Im z = ∞ Im z = −∞
Suppose that we move the moduli from to as above.
2ϕ
1 2
−1
−2
Rez W −1
3
W −1
2
W −1
1
W −1 W −1
−1
W +1
−2
W +1
−1
W +1 W +1
1
W +1
2
Im z > 0
{W −1
∞ , · · · , W −1 2
, W −1
1
} (1) For , there are the walls of .
W −1
n
W −1
n−1
When the moduli is in the chamber between and ,
Z(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1)
Partition function
Im z = ∞ Im z = −∞
Suppose that we move the moduli from to as above.
2ϕ
1 2
−1
−2
Rez W −1
3
W −1
2
W −1
1
W −1 W −1
−1
W +1
−2
W +1
−1
W +1 W +1
1
W +1
2
Im z > 0
{W −1
∞ , · · · , W −1 2
, W −1
1
} (1) For , there are the walls of .
Im z < 0
{W +1 , W +1
1
, · · · , W +1
∞ }
(2) For , there are the walls of .
W −1
n
W −1
n−1
When the moduli is in the chamber between and ,
Z(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1)
Z(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
n
1 (1 − urv)
Partition function
Im z = ∞ Im z = −∞
Suppose that we move the moduli from to as above.
2ϕ
1 2
−1
−2
Rez W −1
3
W −1
2
W −1
1
W −1 W −1
−1
W +1
−2
W +1
−1
W +1 W +1
1
W +1
2
Im z > 0
{W −1
∞ , · · · , W −1 2
, W −1
1
} (1) For , there are the walls of .
Im z < 0
{W +1 , W +1
1
, · · · , W +1
∞ }
(2) For , there are the walls of .
W −1
n
W −1
n−1
When the moduli is in the chamber between and ,
W +1
n
W +1
n+1
When the moduli is in the chamber between and ,
Im z = ±∞
Z±∞(u, v)
In particular, we obtain where denotes the partition function in the limit .
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
Im z = ±∞
Z±∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture:
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
flop transition
D4 D4
Im z = ±∞
Z±∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
flop transition
D4 D4
[Aganagic-Ooguri-Saulina-Vafa ’04]
Im z = ±∞
Z±∞(u, v)
Z+∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Actually, was already evaluated in a literature: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
Z+∞(u, v) = f(u)(1 − v)
∞
(1 − ur)(1 − urv)(1 − urv−1)
flop transition
D4 D4
[Aganagic-Ooguri-Saulina-Vafa ’04]
Im z = ±∞
Z±∞(u, v)
Z+∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Actually, was already evaluated in a literature: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
Z+∞(u, v) = f(u)(1 − v)
∞
(1 − ur)(1 − urv)(1 − urv−1)
flop transition
D4 D4
[Aganagic-Ooguri-Saulina-Vafa ’04]
Im z = ±∞
Z±∞(u, v)
Z+∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Actually, was already evaluated in a literature: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
independent of the chemical potential for D2-branes !! = f(u)
∞
(1 − ur)
Z+∞(u, v) = f(u)(1 − v)
∞
(1 − ur)(1 − urv)(1 − urv−1)
flop transition
D4 D4
[Aganagic-Ooguri-Saulina-Vafa ’04]
Im z = ±∞
Z±∞(u, v)
Z+∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Actually, was already evaluated in a literature: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
independent of the chemical potential for D2-branes !! = f(u)
∞
(1 − ur)
Z+∞(u, v) = f(u)(1 − v)
∞
(1 − ur)(1 − urv)(1 − urv−1)
flop transition
D4 D4
[Aganagic-Ooguri-Saulina-Vafa ’04]
Im z = ±∞
Z±∞(u, v)
Z+∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Actually, was already evaluated in a literature: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
independent of the chemical potential for D2-branes !! = f(u)
∞
(1 − ur)
f(u) ∼
∞
(1 − ur)−χ(C4) Z+∞(u, v) = f(u)(1 − v)
∞
(1 − ur)(1 − urv)(1 − urv−1)
flop transition
D4 D4
[Aganagic-Ooguri-Saulina-Vafa ’04]
Im z = ±∞
Z±∞(u, v)
Z+∞(u, v)
In particular, we obtain where denotes the partition function in the limit . These two limits correspond to the large limit in left and right hand side of the following picture: Actually, was already evaluated in a literature: Moreover, these two limits coinside with the attractor moduli of the MSW black holes, where BPS microstates are counted in the field theory on D4-brane.
Z−∞(u, v) = Z+∞(u, v) ×
∞
1 (1 − urv−1) ×
∞
1 (1 − urv)
independent of the chemical potential for D2-branes !! = f(u)
∞
(1 − ur)
f(u) ∼
∞
(1 − ur)−χ(C4) Z+∞(u, v) = f(u)(1 − v)
∞
(1 − ur)(1 − urv)(1 − urv−1)
flop transition
D4 D4
This is consistent with the fact that decreases by one through the flop transition. χ(C4)
conifold through which the topology of the conifold is changed.
Soibelman wall-crossing formula.