From NS-NS D3 to R-R D3

Reference R-R D3 Conclusion and Future

S-Duality for D3-Brane in NS-NS and R-R Backgrounds

Chen-Te Ma Collaborator: Pei-Ming Ho

National Taiwan University

arXiv:1311.3393 [hep-th]

January 17, 2014

Reference R-R D3 Conclusion and Future

Reference

• P. -M. Ho and Y. Matsuo, M5 from M2, JHEP 0806 (2008) 105 [arXiv:0804.3629 [hep-th]].

• P. -M. Ho, Y. Imamura, Y. Matsuo and S. Shiba, M5-brane in three-form flux and multiple M2-branes, JHEP 0808 (2008) 014 [arXiv:0805.2898 [hep-th]].

• P. -M. Ho and C. -H. Yeh, D-brane in R-R Field Background, JHEP 1103 (2011) 143 [arXiv:1101.4054 [hep-th]].

• P. -M. Ho and C. -T. Ma, Effective Action for Dp-Brane in Large RR (p-1)-Form Background, JHEP 1305 (2013) 056 [arXiv:1302.6919 [hep-th]].

• P. -M. Ho and C. -T. Ma, S-Duality for D3-Brane in NS-NS and R-R Backgrounds, arXiv:1311.3393 [hep-th].

Reference R-R D3 Conclusion and Future

From NS-NS D3 to R-R D3

R-R D4 M5

NS-NS D4 D3

Reference R-R D3 Conclusion and Future

From NS-NS D3 to R-R D3

R-R D3 D3

NS-NS D3

Field Redefinition, …

Reference R-R D3 Conclusion and Future

Poisson Limit

L_NS−NS ≡ −1

4F_αβ⁰ F^0αβ− 1

2F_{α ˙}⁰_µF^{0α ˙µ}−1

4F_{µ ˙}⁰_˙νF^{0 ˙µ ˙ν}. F_AB⁰ ≡ F_AB⁰ + g {a⁰_A, a⁰_B}.

b^µ˙ ≡ ^{µ ˙˙ν}a⁰_ν˙. F⁰_{˙1 ˙2} = F⁰_{˙1 ˙2}+ g {a⁰_˙1, a⁰_˙2} = H_{˙1 ˙2}.

F_{α ˙}⁰⁰_µ≡ _αβF^{0β ˙ν}ν ˙˙µ= αβ



∂^βbµ˙− B^{β ˙ν}Vν ˙˙µ

 .

L_NS−NS = −1

4F_αβ⁰ F^0αβ+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

Reference R-R D3 Conclusion and Future

Poisson Limit

L_NS−NS ≡ −1

4F_αβ⁰ F^0αβ− 1

2F_{α ˙}⁰_µF^{0α ˙µ}−1

4F_{µ ˙}⁰_˙νF^{0 ˙µ ˙ν}. F_AB⁰ ≡ F_AB⁰ + g {a⁰_A, a⁰_B}.

b^µ˙ ≡ ^{µ ˙˙ν}a⁰_ν˙. F⁰_{˙1 ˙2}= F_{˙1 ˙2}⁰ + g {a⁰_˙1, a⁰_˙2} = H_{˙1 ˙2}.

F_{α ˙}⁰⁰_µ≡ _αβF^{0β ˙ν}ν ˙˙µ= αβ



∂^βbµ˙− B^{β ˙ν}Vν ˙˙µ

 .

L_NS−NS = −1

4F_αβ⁰ F^0αβ+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

Reference R-R D3 Conclusion and Future

Poisson Limit

L_NS−NS ≡ −1

4F_αβ⁰ F^0αβ− 1

2F_{α ˙}⁰_µF^{0α ˙µ}−1

4F_{µ ˙}⁰_˙νF^{0 ˙µ ˙ν}. F_AB⁰ ≡ F_AB⁰ + g {a⁰_A, a⁰_B}.

b^µ˙ ≡ ^{µ ˙˙ν}a⁰_ν˙. F⁰_{˙1 ˙2}= F_{˙1 ˙2}⁰ + g {a⁰_˙1, a⁰_˙2} = H_{˙1 ˙2}.

F_{α ˙}⁰⁰_µ≡ _αβF^{0β ˙ν}ν ˙˙µ= αβ



∂^βbµ˙− B^{β ˙ν}Vν ˙˙µ

 .

L_NS−NS = −1

4F_αβ⁰ F^0αβ+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

Reference R-R D3 Conclusion and Future

Poisson Limit

L_NS−NS ≡ −1

4F_αβ⁰ F^0αβ− 1

2F_{α ˙}⁰_µF^{0α ˙µ}−1

4F_{µ ˙}⁰_˙νF^{0 ˙µ ˙ν}. F_AB⁰ ≡ F_AB⁰ + g {a⁰_A, a⁰_B}.

b^µ˙ ≡ ^{µ ˙˙ν}a⁰_ν˙. F⁰_{˙1 ˙2}= F_{˙1 ˙2}⁰ + g {a⁰_˙1, a⁰_˙2} = H_{˙1 ˙2}.

F_{α ˙}⁰⁰_µ≡ _αβF^{0β ˙ν}ν ˙˙µ= αβ



∂^βbµ˙− B^{β ˙ν}Vν ˙˙µ

 .

L_NS−NS = −1

4F_αβ⁰ F^0αβ+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

Reference R-R D3 Conclusion and Future

Poisson Limit

L⁽¹⁾_NS−NS ≡ −1 2φ²+1

2^αβF_αβ⁰ φ +1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

φ = F₀₁⁰ .

L⁽²⁾_NS−NS = −1

2F²_{˙1 ˙2}+ 1

2^αβF_αβ⁰ F_{˙1 ˙2}+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}− 1

2H_{˙1 ˙2}H^{˙1 ˙2}.

∂_µ˙



F_{˙1 ˙2}− F₀₁⁰



= 0.

Reference R-R D3 Conclusion and Future

Poisson Limit

L⁽¹⁾_NS−NS ≡ −1 2φ²+1

2^αβF_αβ⁰ φ +1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

φ = F₀₁⁰ .

L⁽²⁾_NS−NS = −1

2F²_{˙1 ˙2}+ 1

2^αβF_αβ⁰ F_{˙1 ˙2}+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}− 1

2H_{˙1 ˙2}H^{˙1 ˙2}.

∂_µ˙



F_{˙1 ˙2}− F₀₁⁰



= 0.

Reference R-R D3 Conclusion and Future

Poisson Limit

L⁽¹⁾_NS−NS ≡ −1 2φ²+1

2^αβF_αβ⁰ φ +1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}−1

2H_{˙1 ˙2}H^{˙1 ˙2}.

φ = F₀₁⁰ .

L⁽²⁾_NS−NS = −1

2F²_{˙1 ˙2}+1

2^αβF_αβ⁰ F_{˙1 ˙2}+1

2F_{α ˙}⁰⁰_µF^{00α ˙µ}− 1

2H_{˙1 ˙2}H^{˙1 ˙2}.

∂_µ˙



F_{˙1 ˙2}− F₀₁⁰



= 0.

Reference R-R D3 Conclusion and Future

Poisson Limit

F₀₁⁰ F_{˙1 ˙2} = F₀₁⁰ F_{˙1 ˙2}+ {a⁰₀, a⁰₁}F_{˙1 ˙2}

= ^αβ∂_βa_µ˙B_α^µ˙ + ^αβF_{˙1 ˙2}B_α^˙1B_β^˙2+ total derivatives.

^αβfβ ˙µ[ ˘Bαµ˙− ^{µ ˙˙ν}∂ν˙a⁰_α].

If we integrate out a_α⁰, we get^{µ ˙˙ν}∂ν˙fβ ˙µ= 0. It implies that locally f_{β ˙µ}= −∂_µ˙a_β for some field a_β.

−^αβ∂µ˙a_βB˘αµ˙.

Reference R-R D3 Conclusion and Future

Poisson Limit

F₀₁⁰ F_{˙1 ˙2} = F₀₁⁰ F_{˙1 ˙2}+ {a⁰₀, a⁰₁}F_{˙1 ˙2}

= ^αβ∂_βa_µ˙B_α^µ˙ + ^αβF_{˙1 ˙2}B_α^˙1B_β^˙2+ total derivatives.

^αβf_{β ˙µ}[ ˘Bαµ˙− ^{µ ˙˙ν}∂ν˙a⁰_α].

If we integrate out a_α⁰, we get^{µ ˙˙ν}∂ν˙fβ ˙µ= 0. It implies that locally f_{β ˙µ}= −∂_µ˙a_β for some field a_β.

−^αβ∂µ˙a_βB˘αµ˙.

Reference R-R D3 Conclusion and Future

Poisson Limit

F₀₁⁰ F_{˙1 ˙2} = F₀₁⁰ F_{˙1 ˙2}+ {a⁰₀, a⁰₁}F_{˙1 ˙2}

= ^αβ∂_βa_µ˙B_α^µ˙ + ^αβF_{˙1 ˙2}B_α^˙1B_β^˙2+ total derivatives.

^αβf_{β ˙µ}[ ˘Bαµ˙− ^{µ ˙˙ν}∂ν˙a⁰_α].

If we integrate outa⁰_α, we get^{µ ˙˙ν}∂ν˙fβ ˙µ= 0. It implies that locally f_{β ˙µ}= −∂_µ˙a_β for some field a_β.

−^αβ∂µ˙a_βB˘αµ˙.

Reference R-R D3 Conclusion and Future

Poisson Limit

F₀₁⁰ F_{˙1 ˙2} = F₀₁⁰ F_{˙1 ˙2}+ {a⁰₀, a⁰₁}F_{˙1 ˙2}

= ^αβ∂_βa_µ˙B_α^µ˙ + ^αβF_{˙1 ˙2}B_α^˙1B_β^˙2+ total derivatives.

^αβf_{β ˙µ}[ ˘Bαµ˙− ^{µ ˙˙ν}∂ν˙a⁰_α].

If we integrate outa⁰_α, we get^{µ ˙˙ν}∂ν˙fβ ˙µ= 0. It implies that locally f_{β ˙µ}= −∂_µ˙a_β for some field a_β.

−^αβ∂µ˙a_βB˘αµ˙.

Reference R-R D3 Conclusion and Future

Poisson Limit

We can now easily integrate out ˘B_α^µ˙. The result is equivalent to replacing ˘B_α^µ˙ by the solution of its equation of motion.

F₀₁⁰ F_{˙1 ˙2} → 1

2g^αβF_αβ F_{α ˙}⁰⁰_µ= F_{α ˙µ}.

L_RR = −1

2H²_{˙1 ˙2}+1

2F_{α ˙µ}F^{α ˙µ}−1

4F_{µ ˙˙ν}F^{µ ˙˙ν}+ 1

2g^αβF_αβ. Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.

Reference R-R D3 Conclusion and Future

Poisson Limit

We can now easily integrate out ˘B_α^µ˙. The result is equivalent to replacing ˘B_α^µ˙ by the solution of its equation of motion.

F₀₁⁰ F_{˙1 ˙2} → 1

2g^αβF_αβ F_{α ˙}⁰⁰_µ= F_{α ˙µ}.

L_RR = −1

2H²_{˙1 ˙2}+1

2F_{α ˙µ}F^{α ˙µ}−1

4F_{µ ˙˙ν}F^{µ ˙˙ν}+ 1

2g^αβF_αβ. Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.

Reference R-R D3 Conclusion and Future

Poisson Limit

We can now easily integrate out ˘B_α^µ˙. The result is equivalent to replacing ˘B_α^µ˙ by the solution of its equation of motion.

F₀₁⁰ F_{˙1 ˙2} → 1

2g^αβF_αβ F_{α ˙}⁰⁰_µ= F_{α ˙µ}.

L_RR = −1

2H²_{˙1 ˙2}+1

2F_{α ˙µ}F^{α ˙µ}−1

4F_{µ ˙˙ν}F^{µ ˙˙ν}+ 1

2g^αβF_αβ.

Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.

Reference R-R D3 Conclusion and Future

Poisson Limit

We can now easily integrate out ˘B_α^µ˙. The result is equivalent to replacing ˘B_α^µ˙ by the solution of its equation of motion.

F₀₁⁰ F_{˙1 ˙2} → 1

2g^αβF_αβ F_{α ˙}⁰⁰_µ= F_{α ˙µ}.

L_RR = −1

2H²_{˙1 ˙2}+1

2F_{α ˙µ}F^{α ˙µ}−1

4F_{µ ˙˙ν}F^{µ ˙˙ν}+ 1

2g^αβF_αβ. Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.

Reference R-R D3 Conclusion and Future

Coupling Constant

L_NS−NS ≡ 1

g_G^{0 2} h

−1

4F_αβ⁰ F^0αβ−1

2F_{α ˙}⁰_µF^{0α ˙µ}−1

4F_{µ ˙}⁰_˙νF^{0 ˙µ ˙ν}i , where g_G⁰ is the gauge coupling.

b^µ˙ →g_G^{0 2}b^µ˙, Bαµ˙ →g_G^{0 2}Bαµ˙, aµ˙ →g_G^{0 2}aµ˙, aα →g_G^{0 2}aα.

Reference R-R D3 Conclusion and Future

Coupling Constant

L_NS−NS ≡ 1

g_G^{0 2} h

−1

4F_αβ⁰ F^0αβ−1

2F_{α ˙}⁰_µF^{0α ˙µ}−1

4F_{µ ˙}⁰_˙νF^{0 ˙µ ˙ν}i , where g_G⁰ is the gauge coupling.

b^µ˙ →g_G^{0 2}b^µ˙, Bαµ˙ →g_G^{0 2}Bαµ˙, aµ˙ →g_G^{0 2}aµ˙, aα →g_G^{0 2}aα.

Reference R-R D3 Conclusion and Future

Coupling Constant

L_RR ≡ 1 g_G²

h−1

2H²_{˙1 ˙2}+1

2F_{α ˙µ}F^{α ˙µ}−1

4F_{µ ˙˙ν}F^{µ ˙˙ν} + 1

2θ^αβF_αβi , but with the gauge coupling gG and noncommutativity parameter θ defined by

g_G ≡ 1 g_G⁰ , θ≡g_G^{0 2}θ⁰,

and with all the coupling constants g replaced by θ.

Reference R-R D3 Conclusion and Future

Coupling Constant

Notice that the need of two independent parameters (g_G⁰ , θ⁰) or (gG, θ) can be seen only if higher order terms are included. At the lowest order, we can scale the gauge fields so that the R-R action has

gG = g_G⁰ and θ = θ⁰.

Reference R-R D3 Conclusion and Future

Double Scaling Limit

The double scaling limit of NS-NS is:

`_s ∼ ^1/4, g_s ∼ ^1/2, B_{µ ˙}⁰_˙ν ∼ 1, g_αβ ∼ 1, g_{µ ˙˙ν} ∼ .

The double scaling limit of R-R is:

`s ∼ ^1/2, gs ∼ ^−1/2, Bµ ˙˙ν ∼ 1, g_αβ ∼ 1, gµ ˙˙ν ∼ .

Despite the fact that the string coupling gs is large for the R-R theory when it is small for the NS-NS theory, the decoupling of the D3-brane from the bulk in the Seiberg-Witten limit ensures that its S-dual picture also has an effective world-volume theory decoupled from the bulk.

Reference R-R D3 Conclusion and Future

Double Scaling Limit

The double scaling limit of NS-NS is:

`_s ∼ ^1/4, g_s ∼ ^1/2, B_{µ ˙}⁰_˙ν ∼ 1, g_αβ ∼ 1, g_{µ ˙˙ν} ∼ .

The double scaling limit of R-R is:

`s ∼ ^1/2, gs ∼ ^−1/2, Bµ ˙˙ν ∼ 1, g_αβ ∼ 1, gµ ˙˙ν ∼ .

Despite the fact that the string coupling gs is large for the R-R theory when it is small for the NS-NS theory, the decoupling of the D3-brane from the bulk in the Seiberg-Witten limit ensures that its S-dual picture also has an effective world-volume theory decoupled from the bulk.

Reference R-R D3 Conclusion and Future

Double Scaling Limit

The double scaling limit of NS-NS is:

`_s ∼ ^1/4, g_s ∼ ^1/2, B_{µ ˙}⁰_˙ν ∼ 1, g_αβ ∼ 1, g_{µ ˙˙ν} ∼ .

The double scaling limit of R-R is:

`s ∼ ^1/2, gs ∼ ^−1/2, Bµ ˙˙ν ∼ 1, g_αβ ∼ 1, gµ ˙˙ν ∼ .

Despite the fact that the string coupling gs is large for the R-R theory when it is small for the NS-NS theory, the decoupling of the D3-brane from the bulk in the Seiberg-Witten limit ensures that its S-dual picture also has an effective world-volume theory decoupled from the bulk.

Reference R-R D3 Conclusion and Future

S-Duality to all Orders

F_AB⁰ ≡ F_AB⁰ + [a⁰_A, a_B⁰ ]∗. [A, B]∗≡ A ∗ B − B ∗ A.

A ∗ B ≡ A expg ^{µ ˙˙ν}←−

∂µ˙

−

→∂ν˙

2

 B.

F⁰_{˙1 ˙2} = H_{˙1 ˙2}, Z

d⁴x 1

2^αβF_αβ⁰ F_{˙1 ˙2} = Z

d⁴x 1

2g^αβF_αβ,

_βαµ ˙˙νF^{0β ˙ν} = F_{α ˙µ}.

Reference R-R D3 Conclusion and Future

S-Duality to all Orders

F_AB⁰ ≡ F_AB⁰ + [a⁰_A, a_B⁰ ]∗. [A, B]∗≡ A ∗ B − B ∗ A.

A ∗ B ≡ A expg ^{µ ˙˙ν}←−

∂µ˙

−

→∂ν˙

2

 B.

F⁰_{˙1 ˙2} = H_{˙1 ˙2}, Z

d⁴x 1

2^αβF_αβ⁰ F_{˙1 ˙2} = Z

d⁴x 1

2g^αβF_αβ,

_βαµ ˙˙νF^{0β ˙ν} = F_{α ˙µ}.

Reference R-R D3 Conclusion and Future

S-Duality to all Orders

H_{˙1 ˙2} = H_{˙1 ˙2}+ [b_˙1, b_˙2]∗, F_{˙1 ˙2} = F_{˙1 ˙2},

_αβF^{β ˙µ} = −



∂αb^µ˙− g {∂_ρ˙X^µ˙, ˆBαρ˙}_∗∗

 ,

F_αβ = F_αβ + g [−Fα ˙µBˆ_β^µ˙ + F_{β ˙µ}Bˆαµ˙] + g²Fµ ˙˙ν{ ˆBαµ˙, ˆB_β^ν˙}_∗∗.

(A ∗ ∗B) ≡ Aexp(^{g }

µ1 ˙ν1˙ ←−

∂µ1˙

−→∂ν1˙

2 ) − 1

g ^µ˙2ν˙2←−

∂_µ˙2−→

∂_ν˙2 B as

{A, B}_∗∗≡ (A ∗ ∗B) + (B ∗ ∗A).

Reference R-R D3 Conclusion and Future

S-Duality to all Orders

H_{˙1 ˙2} = H_{˙1 ˙2}+ [b_˙1, b_˙2]∗, F_{˙1 ˙2} = F_{˙1 ˙2},

_αβF^{β ˙µ} = −



∂αb^µ˙− g {∂_ρ˙X^µ˙, ˆBαρ˙}_∗∗

 ,

F_αβ = F_αβ + g [−Fα ˙µBˆ_β^µ˙ + F_{β ˙µ}Bˆαµ˙] + g²Fµ ˙˙ν{ ˆBαµ˙, ˆB_β^ν˙}_∗∗.

(A ∗ ∗B) ≡ Aexp(^{g }

µ1 ˙ν1˙ ←−

∂µ1˙

−→∂ν1˙

2 ) − 1

g ^µ˙2ν˙2←−

∂_µ˙2−→

∂_ν˙2 B as

{A, B}_∗∗≡ (A ∗ ∗B) + (B ∗ ∗A).

Reference R-R D3 Conclusion and Future

Gauge Algebra and Closedness

Z

{A, {B, C }_∗∗}_∗∗= Z

{{A, B}_∗∗, C }∗∗,

although it is not associative without integration.

g ^{µ ˙˙ν}{∂_µ˙f , ∂_ν˙g }∗∗= [f , g ]∗.

Reference R-R D3 Conclusion and Future

Gauge Algebra and Closedness

Z

{A, {B, C }_∗∗}_∗∗= Z

{{A, B}_∗∗, C }∗∗,

although it is not associative without integration.

g ^{µ ˙˙ν}{∂_µ˙f , ∂_ν˙g }∗∗= [f , g ]∗.

Reference R-R D3 Conclusion and Future

Conclusion and Future

• We extend S-duality toinfinite order. We also find

non-commutative R-R D3-brane beyond Poisson. The gauge algebra of this theory is also extended to all orders.

• DBI ?

Reference R-R D3 Conclusion and Future

Conclusion and Future

• We extend S-duality toinfinite order. We also find

non-commutative R-R D3-brane beyond Poisson. The gauge algebra of this theory is also extended to all orders.

• DBI ?