The final purpose in this thesis is to find effective field theory of all extended objects in all possible background fields. However, we are still far from this purpose. One way to extend our recent work is to study the D3-brane in R-R 4-form background from T-duality. Firstly, we do dimensional reduction on x1, then we view the gauge field a1 as the transverse direction of D3-brane ( ˜X1). Finally we get the effective D3-brane theory in large R-R 4-form background. The R-R 4-form field D(4):
D(4) = V(1)∧ C(3). (5.1)
The C(3) is original 3-form background in D4-brane theory, and the V(1) is the one-form on the transverse direction ˜X1 of D3-brane. Extending this conclusion to Dp-branes, we claim that for a Dp-brane in R-R (p + 1)-form potential background
D(p+1) = V(1)∧ C(p), (5.2)
where V(1) is transverse and C(p)is parallel to the Dp-brane. The VPD, corresponding to the volume-form C(p), shares the same gauge field degrees of freedom with the component of the momentum p along the direction of V(1).
The other brane systems in large field background are also interesting. For example, we want to know the behavior of NS5-branes in large R-R field background, the effective action of KK monopole in large field background, etc.
There are some interesting papers which describe the possible effective action of Dp-brane in R-R field background [37–39, 51]. However, they only focus on the D3-Dp-brane in R-R 2-form background case and the supersymmetry is N = 12. This topic is called by non-anticommutative field theory, which is motivated from the extension of original anticommutative field theory (supersymmetry theory). How to understand the relation between their work and our new methods is an important problem.
Furthermore, we know the original AdS/CFT correspondence which describe the physics of D-branes in R-R field background. How to apply the correspondence into large R-R field background is another important application.
Appendix A
Conventions and Notations
In this article, we use these indices to label the 6 worldvolume directions:
M, N, R = µ, ν, ρ, ˙µ, ˙ν, ˙ρ , (A.1)
µ, ν, ρ = 0, 1, 2 , (A.2)
˙µ, ˙ν, ˙ρ = ˙1, ˙2, ˙3 . (A.3)
The metric and Levi-Civita tensor are:
gM N = ηµν 0 0 η˙µ ˙ν
!
, (A.4)
ηµν =
−1 0 0
0 1 0
0 0 1
, (A.5)
η˙µ ˙ν =
1 0 0
0 1 0
0 0 1
, (A.6)
ǫ012 =−ǫ012= 1, (A.7)
ǫ˙1˙2˙3 = ǫ˙1˙2˙3= 1. (A.8)
The conventions of Gamma matrix are:
{ΓM, ΓN} = 2gM N, (A.9)
(Γ0)† = −Γ0, (A.10)
(ΓM 6=0)† = ΓM 6=0, (A.11)
Γ7 ≡ Γ0Γ1Γ2Γ˙1Γ˙2Γ˙3, (A.12)
ΓµνρΓ˙1˙2˙3 = ǫµνρΓ7, (A.13)
Γµνρ = ǫµνρΓ˙1˙2˙3Γ7, (A.14)
(Γ7)2 = 1. (A.15)
We use these conventions to label the fields and directions in D4-brane theory:
α, β, γ, δ = 0, 1 , (A.16)
A, B, C = 0, 1, ˙1, ˙2, ˙3 , (A.17)
b˙µ2≡ a˙µ, (A.18)
Λ2 ≡ λ, (A.19)
FAB ≡ ∂AaB− ∂BaA, (A.20)
ǫαβ2 ≡ ǫαβ. (A.21)
On the other hand, we also use the index I to label the transverse directions of brane.
In this article, we use I = 6, 7, 8, 9, 11 in M5-brane or D4-brane cases.
Appendix B
Some Useful Identities
In order to check supersymmetry, we need to use some identities. Here is the summary.
• Chirarity condition
Γ7Ψ = Ψ, (B.1)
Γ7ǫ = −ǫ. (B.2)
Hence, we can get:
ΓµνρΓ˙1˙2˙3ψ = ǫµνρΓ7ψ = ǫµνρψ, (B.3)
ΓαβΓ2Γ˙1˙2˙3ψ = ǫαβψ, (B.4)
ΓαβΓ˙1˙2˙3ψ = ǫαβΓ2ψ, (B.5)
ΓαΓ˙1˙2˙3ψ = −ǫαβΓβΓ2ψ, (B.6)
Γ˙1˙2˙3ψ = −1
2ǫαβΓαΓβΓ2ψ, (B.7) Γ˙µΓ˙1˙2˙3ψ = 1
2ǫαβΓαΓβΓ2Γ˙µψ. (B.8)
• Gamma matrix
There are some useful identities of gamma matrix:
ΓAΓB = ΓAB+ ηAB, (B.9)
Γ˙µΓ˙µ = 3, (B.10)
Γ˙µΓ˙µ ˙ν = 2Γ˙ν, (B.11)
Γ˙µΓ˙µ ˙ν ˙ρ = Γ˙ν ˙ρ, (B.12)
ǫ˙µ ˙ν ˙ρ = −Γ˙µ ˙ν ˙ρΓ˙1˙2˙3, (B.13)
Γ˙µǫ˙µ ˙ν ˙ρ = −Γ˙ν ˙ρΓ˙1˙2˙3, (B.14)
Γ˙µ ˙νǫ˙µ ˙ν ˙ρ = 2Γρ˙Γ˙1˙2˙3, (B.15)
Γ˙µΓ˙1˙2˙3 = Γ˙1˙2˙3Γ˙µ, (B.16)
ΓαΓα = 2, (B.17)
ΓαΓβΓα = 0. (B.18)
• Levi-Civita tensor
ǫαβǫγδ = −ηγαηδβ + ηγβηδα, (B.19) ǫ˙µ ˙ν ˙ρǫ˙σ ˙λ ˙δ = η˙σ˙µη˙λ˙νη˙δρ˙+ η˙σ˙νη˙λρ˙η˙δ˙µ+ η˙σρ˙η˙λ˙µη˙δ˙ν
−η˙σ˙µη˙λρ˙η˙δ˙ν− η˙σ˙νη˙λ˙µη˙δρ˙− η˙σρ˙η˙λ˙νη˙δ˙µ, (B.20) ǫαβηγδ = −ηαγǫβδ+ ηβγǫαδ, (B.21) ǫ˙µ ˙ν ˙ρη˙σ ˙δ = ǫ˙µ ˙ν ˙σηρ ˙δ˙ + ǫ˙ν ˙ρ ˙ση˙µ ˙δ+ ǫρ ˙µ ˙σ˙ η˙ν ˙δ. (B.22)
Appendix C
Suitable Scaling Limit in Different Cases
When we describe the effective field theory of Dp-brane, we need to choose some limit of original exact theory. The effective field theory of open string ending on Dp-brane should be described by some limit. For example, the DBI action of Dp-brane is given by slowly varying limit( ∂F ≪ 1) of original string scattering amplitude analysis. Hence, the DBI action is a effective description of string theory without higher derivative term.
Scaling limit(zero slope limit α′ → 0) is a low energy limit, which make theory be more easy for analysis. For example, the zero slope limit of DBI action is Yang-Mill action. In this section, we want to describe the suitable low energy limit of theory in different fields background.
C.1 Scaling Limit of Dp-brane in B-field Background
The low energy limit means the theory without string behavior, the first example is Yang-Mill theory. People [12] find the commutative Yang-Mill theory can relate to non-commutative Yang-Mill theory by using Seiberg-Witten map and taking scaling limit.
This scaling limit is called Seiberg-Witten limit:
α′ ∼ √
ǫ→ 0, (C.1)
gij ∼ ǫ → 0, (C.2)
here i, j is the non-vanish component of B field. This limit is understood the low energy limit of Dp-brane in NS-NS B-field background.
C.2 Scaling Limit of M5 in Large C-field Background
Following the logic in previous section, the NP M5 theory is some special limit of M5 in large C-field background. The reason is the kinetic terms of gauge field which is quadratic (H2) as Yang-Mill theory case. So, we should ask what is the scale limit of this NP M5 theory. The NP M5 theory can relate to the action of Dp-brane in NS-NS B-field background after DDR, so we can get the clue of scaling limit from this relation.
Following the calculations in paper [47], we summary it by below equations:
ℓP ∼ √3
here the ℓP is Plank length and C is background 3-form. The scaling limit will match the scaling limit of D4 in B-field background after DDR on y˙3. To understand this result. we introduce the radius of compact direction(y˙3) Rphys˙3 , the radius can be calculated by this way: TD4 = 2πRphys˙3 TM 5= 2πR
phys 3˙
(2π)5ℓ6p. This is the knowledge of M-IIA which D4 is given by M5 after DDR. From the relation TD4 = 2πRphys˙3 TM 5, it is consistent with the results:
Rphys˙3 = gsℓs, ℓP =√3
gsℓs. (C.8)
On the other hand, the C field can relate to B field in D4 by this way:
C˙1˙2˙3= B˙1˙2 2πRcoord˙3 =
√g˙3˙3B˙1˙2
2πRphys˙3 . (C.9)
From these relation we can find the scaling limit after DDR on y˙3: α′ ∼ √
here the scaling of gs can get from the constraints of finite Yang-Mills coupling. These
C.3 Scaling Limit of D4 in Large C-field Background
To carry out the double dimensional reduction (DDR) for the M5-brane along the x2 -direction, we set
x2 ∼ x2+ 2πR, (C.15)
and let all other fields to be independent of x2. As a result we can set ∂2 to zero when it acts on any field. Here R is the radius of the circle of compactification and we should take R ≪ 1 such that the 6 dimensional field theory on M5 reduces to a 5 dimensional field theory for D4. Since the NP M5-brane action is a good low energy effective theory in the limit in previous section, the 5 dimensional field theory is a good low energy effective description of a D4-brane in the limit ǫ→ 0 for
ℓs∼ ǫ1/2, gs∼ ǫ−1/2, gαβ ∼ 1, g˙µ ˙ν ∼ ǫ, C˙µ ˙ν ˙λ ∼ 1, (C.16)
with
gsℓs≪ 1, (C.17)
from the perspective of the type IIA theory. The indices α, β = 0, 1 are used to distinguish from the M5-brane indices µ, ν = 0, 1, 2.
Note that in the scaling limit of NP M5 theory, another three C-field component C012 ∼ ǫ−1 look like divergence. As a result the B-field component B01 ∼ ǫ−1 and the noncommutative parameter θ01 ∼ B−1 ∼ ǫ vanishes in the limit ǫ → 0. However, the combination 2πα′B is finite in the limit, and thus the D4-brane is not only in a C-field background but also in the B-field background. Using the nonlinear self-dual relation derived in [23,24], we can express C012 in terms of C˙1˙2˙3, and then the B-field background is given by
2πα′B01= C˙1˙2˙3
2π . (C.18)
In the convention (normalization of the worldvolume coordinates) of [2], we have C˙1˙2˙3 = 1
g2 ⇒ 2πα′B01 = 1
2πg2. (C.19)
References
[1] P. M. Ho and Y. Matsuo, “M5 from M2,” JHEP 0806, 105 (2008) [arXiv:0804.3629 [hep-th]].
[2] P. M. Ho, Y. Imamura, Y. Matsuo and S. Shiba, “M5-brane in three-form flux and multiple M2-branes,” JHEP 0808, 014 (2008) [arXiv:0805.2898 [hep-th]].
[3] P. M. Ho and C. H. Yeh, “D-brane in R-R Field Background,” JHEP 1103, 143 (2011) [arXiv:1101.4054 [hep-th]].
[4] R. G. Leigh, “Dirac-Born-Infeld Action from Dirichlet Sigma Model,” Mod. Phys.
Lett. A 4, 2767 (1989).
[5] E. Witten, “Bound states of strings and p-branes,” Nucl. Phys. B 460, 335 (1996) [arXiv:hep-th/9510135].
[6] J. Polchinski, “Tasi lectures on D-branes,” [hep-th/9611050].
[7] S. Forste, “Strings, branes and extra dimensions,” Fortsch. Phys. 50, 221 (2002) [arXiv:hep-th/0110055].
[8] C. V. Johnson, “D-branes,” Cambridge, USA: Univ. Pr. (2003) 548 p.
[9] C. S. Chu and P. M. Ho, “Noncommutative open string and D-brane,” Nucl. Phys.
B 550, 151 (1999) [arXiv:hep-th/9812219].
[10] C. S. Chu and P. M. Ho, “Constrained quantization of open string in background B field and noncommutative D-brane,” Nucl. Phys. B 568, 447 (2000) [arXiv:hep-th/9906192].
[11] V. Schomerus, “D-branes and deformation quantization,” JHEP 9906, 030 (1999) [arXiv:hep-th/9903205].
[12] N. Seiberg and E. Witten, “String theory and noncommutative geometry,” JHEP 9909, 032 (1999) [arXiv:hep-th/9908142].
[13] M. R. Douglas, “Branes within branes,” [hep-th/9512077].
[14] M. B. Green, J. A. Harvey, G. W. Moore, “I-brane inflow and anomalous couplings on d-branes,” Class. Quant. Grav. 14, 47-52 (1997). [hep-th/9605033].
[15] A. A. Tseytlin, “Born-Infeld action, supersymmetry and string theory,” In *Shifman, M.A. (ed.): The many faces of the superworld* 417-452. [hep-th/9908105].
[16] J. Bogaerts, A. Sevrin, J. Troost, W. Troost, S. van der Loo, “D-branes and constant electromagnetic backgrounds,” Fortsch. Phys. 49, 641-647 (2001). [hep-th/0101018].
[17] Y. Nambu, “Generalized Hamiltonian dynamics,” Phys. Rev. D 7, 2405 (1973).
[18] F. Bayen, M. Flato, “Remarks concerning Nambu’s generalized mechanics,” Phys.
Rev. D 11, 3049 (1975).
[19] N. Mukunda, E. Sudarshan, “Relations between Nambu and Hamiltonian mechan-ics,” Phys. Rev. D 13, 2846 (1976).
[20] L. Takhtajan, “On Foundation Of The Generalized Nambu Mechanics (Second Ver-sion),” Commun. Math. Phys. 160, 295 (1994) [arXiv:hep-th/9301111].
[21] For a review of the Poisson bracket, see: I. Vaisman, “A survey on Nambu-Poisson brackets,” Acta. Math. Univ. Comenianae 2 (1999), 213.
[22] P. K. Townsend, “M-theory from its superalgebra,” arXiv:hep-th/9712004.
[23] P. S. Howe and E. Sezgin, “D = 11, p = 5,” Phys. Lett. B 394, 62 (1997) [arXiv:hep-th/9611008].
[24] P. S. Howe, E. Sezgin and P. C. West, “Covariant field equations of the M-theory five-brane,” Phys. Lett. B 399, 49 (1997) [arXiv:hep-th/9702008].
[25] P. Pasti, D. P. Sorokin and M. Tonin, “Covariant action for a D = 11 five-brane with the chiral field,” Phys. Lett. B 398, 41 (1997) [arXiv:hep-th/9701037].
[26] I. A. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. P. Sorokin and M. Tonin,
“Covariant action for the super-five-brane of M-theory,” Phys. Rev. Lett. 78, 4332 (1997) [arXiv:hep-th/9701149].
[27] M. Aganagic, J. Park, C. Popescu and J. H. Schwarz, “World-volume action of the M-theory five-brane,” Nucl. Phys. B 496, 191 (1997) [arXiv:hep-th/9701166].
[28] I. A. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D. P. Sorokin and M. Tonin,
“On the equivalence of different formulations of the M theory five-brane,” Phys. Lett.
B 408, 135 (1997) [arXiv:hep-th/9703127].
[29] W. -M. Chen and P. -M. Ho, “Lagrangian Formulations of Self-dual Gauge Theories in Diverse Dimensions,” Nucl. Phys. B 837, 1 (2010) [arXiv:1001.3608 [hep-th]].
[30] P. -M. Ho, K. -W. Huang and Y. Matsuo, “A Non-Abelian Self-Dual Gauge Theory in 5+1 Dimensions,” JHEP 1107, 021 (2011) [arXiv:1104.4040 [hep-th]].
[31] S. Kawamoto and N. Sasakura, “Open membranes in a constant C-field back-ground and noncommutative boundary strings,” JHEP 0007, 014 (2000) [arXiv:hep-th/0005123].
[32] P. M. Ho and Y. Matsuo, “A toy model of open membrane field theory in constant 3-form flux,” Gen. Rel. Grav. 39, 913 (2007) [arXiv:hep-th/0701130].
[33] J. Bagger and N. Lambert, “Modeling multiple M2’s, Phys. Rev. D 75, 045020 (2007) [arXiv:hep-th/0611108].
[34] J. Bagger and N. Lambert, “Gauge Symmetry and Supersymmetry of Multiple M2-Branes,” Phys. Rev. D 77, 065008 (2008) [arXiv:0711.0955 [hep-th]].
[35] J. Bagger and N. Lambert, “Comments On Multiple M2-branes,” JHEP 0802, 105 (2008) [arXiv:0712.3738 [hep-th]].
[36] A. Gustavsson, “Algebraic structures on parallel M2-branes,” Nucl. Phys. B 811, 66 (2009) [arXiv:0709.1260 [hep-th]].
[37] L. Cornalba, M. S. Costa and R. Schiappa, “D-brane dynamics in constant Ramond-Ramond potentials and noncommutative geometry,” Adv. Theor. Math. Phys. 9, 355 (2005) [arXiv:hep-th/0209164].
[38] H. Ooguri and C. Vafa, “The C-deformation of gluino and non-planar diagrams,”
Adv. Theor. Math. Phys. 7, 53 (2003) [arXiv:hep-th/0302109].
[39] J. de Boer, P. A. Grassi and P. van Nieuwenhuizen, “Non-commutative superspace from string theory,” Phys. Lett. B 574, 98 (2003) [arXiv:hep-th/0302078].
[40] K. Furuuchi and T. Takimi, “String solitons in the M5-brane worldvolume action with Nambu-Poisson structure and Seiberg-Witten map,” JHEP 0908, 050 (2009) [arXiv:0906.3172 [hep-th]].
[41] P. M. Ho, “A Concise Review on M5-brane in Large C-Field Background,”
arXiv:0912.0445 [hep-th].
[42] C. S. Chu and D. J. Smith, “Towards the Quantum Geometry of the M5-brane in a Constant C-Field from Multiple MemM5-branes,” JHEP 0904, 097 (2009) [arXiv:0901.1847 [hep-th]].
[43] J. Huddleston, “Relations between M-brane and D-brane quantum geometries,”
arXiv:1006.5375 [hep-th].
[44] P. Pasti, I. Samsonov, D. Sorokin and M. Tonin, “BLG-motivated Lagrangian for-mulation for the chiral two-form gauge field in D=6 and M5-branes,” Phys. Rev. D 80, 086008 (2009) [arXiv:0907.4596 [hep-th]].
[45] K. Furuuchi, “Non-Linearly Extended Self-Dual Relations From The Nambu-Bracket Description Of M5-Brane In A Constant C-Field Background,” JHEP 1003, 127 (2010) [arXiv:1001.2300 [hep-th]].
[46] C. H. Chen, P. M. Ho and T. Takimi, “A No-Go Theorem for M5-brane Theory,”
JHEP 1003, 104 (2010) [arXiv:1001.3244 [hep-th]];
[47] C. H. Chen, K. Furuuchi, P. M. Ho and T. Takimi, “More on the Nambu-Poisson M5-brane Theory: Scaling limit, background independence and an all order solution to the Seiberg-Witten map,” arXiv:1006.5291 [hep-th].
[48] X. Bekaert, M. Henneaux and A. Sevrin, “Deformations of chiral two-forms in six dimensions,” Phys. Lett. B 468, 228 (1999) [arXiv:hep-th/9909094].
[49] A. M. Low, “Worldvolume Superalgebra Of BLG Theory With Nambu-Poisson Structure,” JHEP 1004, 089 (2010). [arXiv:0909.1941 [hep-th]].
[50] N. Nekrasov and A. S. Schwarz, “Instantons on noncommutative R**4 and (2,0) superconformal six-dimensional theory,” Commun. Math. Phys. 198, 689 (1998) [hep-th/9802068].
[51] M. Billo, M. Frau, I. Pesando, A. Lerda, “N = 1/2 gauge theory and its instanton moduli space from open strings in RR background,” JHEP 0405, 023 (2004). [hep-th/0402160].