### D-brane in R-R Field Background

Chi-Hsien Yeh

NTU department of Physics

March 25, 2011

Based on Arxiv:1101.4054 with Professor Pei-Ming Ho

### Outline

Introduction

Effective Theory and Large Field Background Noncommutative and NS-NS B Field Background C Field Background Extension

Single M5-brane in Three-Form Background Field Contents and Gauge Symmetry D4 in B Field Background

Another Possible D4 Brane Theory Dp brane in R-R p-1 Field Background

Dual Transformation Covariant Variables Order Expansion Analysis Multi-Dp branes generalization Conclusion

Effective Theory and Large field background

### Overview

System in large field background will have different description with original case.

For example, when we study the charged particles in large magnetic filed background, we will find the geometry is noncommutative.

The low energy effective theory of D-brane in constant NS-NS B field background can be described by

noncommutative field theory.

M5 brane in large C field background is described by Nambu-Poisson bracket.

In this talk, we try to figure out the low energy effective theory of D-brane in large R-R field background.

Noncommutative and NS-NS B field background

### Moyal Product and Poisson Bracket

Noncomutative gauge field theory is governed by Moyal product:

f(x) ∗ g (x) = e^{2}^{i}^{θ}^{ij ∂}^{∂ξi} ^{∂ζj}^{∂} f(x + ξ)g (x + ζ)

ξ=ζ=0

= f (x)g (x) + i

2θ^{ij}∂if(x)∂jg(x) + O(θ^{2}).

Hence, the geometry is noncommutative [x^{i}, x^{j}] = iθ^{ij}.
Noncommutative effects depend on the reciprocal of B
field. θ^{ij} = (_{B}^{1})^{ij}.

In suitable truncation and limit, the effective theory can relate to original effective theory(DBI) with

Seiberg-Witten map.

Noncommutative and NS-NS B field background

### Area Preserving Diffeomorphism and Generalization

In first θ order, noncommutative theory describe by
Poisson bracket:{f , g }p.b. = ǫ^{ij}∂if(x)∂jg(x).

Poisson Bracket is used to generate the 2D area preserving diffeomorphism: δαΦ = {α, Φ}.

Following the same logic, Poisson bracket can generalize to Nambu-Poisson bracket:

{f , g , h} = ǫ^{˙µ ˙ν ˙}^{ρ}∂_{˙µ}f ∂_{˙ν}g ∂ρ_{˙}h,
to generate the 3D volume preserving
diffeomorphism(VPD): δα,βΦ = {α, β, Φ}.

VPD expect to describe the behavior of theory in C-field background as B-field case.

We have practical example of M5-brane in C-field background.

C field background Extension

### Nambu Poisson Bracket and C Field Background

Recently, people apply the Nambu-Poisson bracket into internal 3-manifold of BLG theory, as a realization of Lie-3 algebra. They find the new description of single M5 brane theory in large C field background. It is called by NP M5 theory.

There are many evidences to show the NP M5 theory is M5 in C field background.

For example, the constant term exists in action, the supersymmetry law is nonlinear, the two form gauge field has nonabelian structure, and it reproduces D4 brane in NS-NS B field background, etc.

Field Contents and Gauge Symmetry

The NP M5 theory has self-dual 2-form(bµ˙ν,b˙µ ˙ν), 5 scalar
(X^{i}) and dimensional reduction 11 dimensional Majorana
fermion(Ψ) with chirality.

In this theory, we use the worldvolume coordinate

(x^{µ}, y^{˙µ}) = (x^{0}, x^{1}, x^{2}, y^{˙1}, y^{˙2}, y^{˙3}), here, the y^{˙µ} coordinate
describe the direction of C-field → C_{˙1˙2˙3}.

The fundamental fields transform under the gauge transformation as

δ_{Λ}Φ = g κ^{ρ}^{˙}∂ρ_{˙}Φ (Φ = X^{i},Ψ),
δ_{Λ}b_{˙κ ˙λ} = ∂˙κΛ_{˙λ}− ∂_{˙λ}Λ˙κ+ g κ^{ρ}^{˙}∂ρ_{˙}b_{˙κ ˙λ},

δ_{Λ}b_{λ}_{˙σ} = ∂λΛ˙σ − ∂˙σΛλ+ g κ^{˙τ}∂_{˙τ}b_{λ}_{˙σ}+ g (∂˙σκ^{˙τ})bλ˙τ,
where κ^{˙λ} ≡ ǫ^{˙λ ˙µ ˙ν}∂˙µΛ˙ν(x, y ).

Gauge transformation law of two form fields can be concisely expressed in terms of new variables.

Gauge Symmetry and VPD

Let’s the two useful variables: b^{˙µ}, Bµ˙µ

b^{˙µ} ≡ 1

2ǫ^{˙µ ˙ν ˙λ}b_{˙ν ˙λ},
B_{µ}^{˙µ} ≡ ǫ^{˙µ ˙ν ˙λ}∂˙νb_{µ ˙λ}

Hence, the gauge transformations of the new variables are:

δ_{Λ}b^{˙µ} = κ^{˙µ}+ g κ^{˙ν}∂_{˙ν}b^{˙µ},

δΛB_{µ}^{˙µ} = ∂µκ^{˙µ}+ g κ^{˙ν}∂˙νB_{µ}^{˙µ}− g (∂˙νκ^{˙µ})Bµ˙ν.
Physics of NP M5 theory can describe by (Bµ˙µ,b^{˙µ},X^{i},Ψ),
their gauge transformation parameter is only κ^{˙λ}.

Gauge transformation can understand as
(VPD):δy^{˙µ} = g κ^{˙µ}, with ∂˙µκ^{˙µ} = 0.

b^{˙µ} is the gauge potential for VPD in 3 dimension space
which is the direction of C-field background.

Action

S = SX + SΨ+ Sgauge,Sgauge = SH^{2}+ SCS,where

S_{X} =
Z

d^{3}xd^{3}y

−1

2(DµX^{i})^{2}− 1

2(D_{˙λ}X^{i})^{2}

− 1

2g^{2} −g^{4}

4 {X^{˙µ}, X^{i}, X^{j}}^{2}− g^{4}

12{X^{i}, X^{j}, X^{k}}^{2}

,
S_{Ψ} =

Z

d^{3}xd^{3}y i

2ΨΓ^{µ}DµΨ + i

2ΨΓ^{ρ}^{˙}Dρ˙Ψ
+ig^{2}

2 ΨΓ˙µi{X^{˙µ}, X^{i},Ψ} − ig^{2}

4 ΨΓijΓ_{˙1˙2˙3}{X^{i}, X^{j},Ψ}

,
SH^{2} =

Z

d^{3}xd^{3}y

− 1

12H^{2}_{˙µ ˙ν ˙}_{ρ}− 1
4H^{2}_{λ}_{˙µ ˙ν}

,
S_{CS} =

Z

d^{3}xd^{3}y ǫ^{µνλ}ǫ^{˙µ ˙ν ˙λ}

−1

2∂˙µb_{µ˙ν}∂νb_{λ ˙λ}
+g

6∂_{˙µ}b_{ν}_{˙ν}ǫ^{ρ}^{˙}^{˙σ ˙τ}∂_{˙σ}b_{λ}_{ρ}_{˙}(∂_{˙λ}b_{µ˙τ} − ∂˙τb_{µ ˙λ})i
.

Covariant Derivatives And Field Strength

Covariant derivatives are defined by (Φ = X^{i},Ψ):

DµΦ = ∂µΦ − gBµ˙µ∂_{˙µ}Φ,
D˙µΦ = g^{2}

2 ǫ˙µ ˙ν ˙ρ{X^{˙ν}, X^{ρ}^{˙},Φ},
where X^{˙µ} = ^{y}_{g}^{µ}^{˙} + b^{˙µ}.

Field strengths are defined by:

Hλ˙µ ˙ν = ǫ_{˙µ ˙ν ˙λ}DλX^{˙λ}

= Hλ˙µ ˙ν − g ǫ^{˙σ ˙τ ˙}^{ρ}(∂˙σb_{λ}_{˙τ})∂ρ˙b_{˙µ ˙ν},
H_{˙1˙2˙3} = g^{2}{X^{˙1}, X^{˙2}, X^{˙3}} − 1

g

= H_{˙1˙2˙3}+ g

2(∂˙µb^{˙µ}∂˙νb^{˙ν} − ∂˙µb^{˙ν}∂˙νb^{˙µ}) + g^{2}{b^{˙1}, b^{˙2}, b^{˙3}},
where, H is the normal abelian 2-form field strength.

D4 in B field Background

M5 brane theory can get D4 brane theory after double dimension reduction.

Double Dimension Reduction(DDR) means that we do the dimension reduction on worldvolume and target space at the same time.

We can compact the worldvolume y^{˙3} and keep only zero
mode, then relate {f , g , y^{˙3}} = ǫ^{˙µ ˙ν ˙3}∂˙µf ∂˙νg ≡ {f , g }p.b.

This case is expected to be D4 brane in B-field

background, because the background field B_{˙1˙2} after DDR.

After integrating out the auxiliary field and filed rename,
we get S_{boson}^{D4} =

Z

d^{3}xd^{2}y

−1

2(DaX^{i})^{2}− 1

4(Fab)^{2}− g^{2}

4 {X^{i}, X^{j}}^{2} − 1
2g^{2}

Another possible D4 brane theory

If we do DDR in another dimension (x^{2}) to keep C field
background in D4 theory. We get the new D4 theory:

S[b^{˙µ}, a_{˙µ}, B_{α}^{˙µ}]

= Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}− 1

4H^{2}_{2 ˙µ ˙ν} −1

2(∂αb^{˙µ}− V_{˙σ}^{˙µ}B_{α}^{˙σ})^{2}
+ǫ^{αβ}∂βa_{˙µ}B_{α}^{˙µ}+ g

2ǫ^{αβ}F_{˙µ ˙ν}B_{α}^{˙µ}B_{β}^{˙ν}o
,
where

a_{˙µ} ≡ b˙µ2, B_{α}^{˙µ} ≡ ǫ^{˙µ ˙ν ˙}^{ρ}∂_{˙ν}b_{α}_{ρ}_{˙}, V_{˙σ}^{˙µ} ≡ (δ˙σ˙µ+g ∂˙σb^{˙µ}), α = {0, 1}.

For simplify, we only show the gauge field terms.

### D4 brane in R-R 3-Form Background

If we want to identify our theory to be D4 brane theory, we need to answer several questions:

D4 brane should have one-form gauge field aA, where
A= {α, ˙µ} = {0, 1, ˙1, ˙2, ˙3}. Where are the gauge fields
a_{0}, a1?

Even if we find all one form fields, we still need to deal
with these additional two-form gauge fields(b^{˙µ}, Bα˙µ) after
DDR.

Gauge transformation of aA include U(1) and VPD, how to define the field strength or covariant variables?

After solving these questions, then we can analyze what can we learn from this new D4 brane action.

Dual Transformation

To find missing one form aα=0,1, we use the dual transformation.

Rewrite the action in another equivalent
form:S^{(1)}[b^{˙µ}, a_{˙µ}, ˘B_{α}^{˙µ}, fα˙µ, bα˙µ] =

S[b^{˙µ}, a_{˙µ}, ˘B_{α}^{˙µ}] − ǫ^{αβ}f_{β}_{˙µ}[ ˘B_{α}^{˙µ}− ǫ^{˙µ ˙ν ˙}^{ρ}∂_{˙ν}b_{α}_{ρ}_{˙}] by auxiliary field
f_{α}_{˙µ}.

This new action is same with original action after integrating out fα˙µ.

On the other hand, we can integrate out bα˙µ and ˘B_{α}^{˙µ} to
get another dual description.

Firstly, we integrate out bα˙µ, then we find
ǫ^{˙µ ˙ν ˙λ}∂˙µf_{α}_{˙ν} = 0 → fα˙µ = ∂˙µa_{α}.

Gauge transformation of aα is easy to find:

δ_{Λ}a_{α} = ∂αλ+ g (κ^{˙σ}∂_{˙σ}a_{α}+ a˙σ∂ακ^{˙σ}), this is same form of
δ_{Λ}a_{˙µ}.

It is original U(1) with additional VPD gauge symmetry.

Final Action Form

After integrating out bα˙µ, it is equivalent to replace
action S[b^{˙µ}, a˙µ, ˘B_{α}^{˙µ}] by this way

:ǫ^{αβ}∂βa_{˙µ}B˘_{α}^{˙µ} → ǫ^{αβ}F_{β}_{˙µ}B˘_{α}^{˙µ}, where FAB ≡ ∂Aa_{B} − ∂Ba_{A}.
To integrate out ˘B_{α}^{˙µ}, we solve the equation of motion of
B˘_{α}^{˙µ}.

After integrating out ˘B_{α}^{˙µ}, we get S^{(2)}[b^{˙µ}, a_{˙µ}, aα] =
Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}−1

4(H2 ˙ν ˙ρ)^{2}−1

2∂αb^{˙µ}∂^{α}b_{˙µ}
+1

2(ǫ^{αγ}F_{γ}_{˙µ}+ V_{˙µ}^{˙σ}∂^{α}b_{˙σ})(M^{−}^{1})αβ˙µ ˙ν(ǫ^{βδ}F_{δ}_{˙ν}+ V˙ν˙λ∂^{β}b_{˙λ})

,
where, ˆB_{α}^{˙µ} ≡ (M^{−}^{1})αβ˙µ ˙ν(V_{˙ν}^{˙σ}∂^{β}b_{˙σ}+ ǫ^{βγ}F_{γ}_{˙ν}),

and M_{˙µ ˙ν}^{αβ} ≡ V_{˙µ ˙}ρV_{˙ν}^{ρ}^{˙}δ^{αβ} − g ǫ^{αβ}F_{˙µ ˙ν}

Covariant Variables

To find the missing field strength, we start to search covariant
variables δΛΦ = g κ^{˙µ}∂_{˙µ}Φ, after some complex calculations, we
get:

H_{˙1˙2˙3} = ∂˙µb^{˙µ}+1

2g(∂˙νb^{˙ν}∂ρ_{˙}b^{ρ}^{˙}− ∂˙νb^{ρ}^{˙}∂ρ_{˙}b^{˙ν}) + g^{2}{b^{˙1}, b^{˙2}, b^{˙3}},
F˙µ ˙ν ≡ H2 ˙µ ˙ν = F˙µ ˙ν + g [∂˙σb^{˙σ}F_{˙µ ˙ν} − ∂˙µb^{˙σ}F_{˙σ ˙ν} − ∂˙νb^{˙σ}F_{˙µ ˙σ}],
Fα˙µ = V^{−}^{1}_{˙µ}^{˙ν}(Fα˙ν + gF_{˙ν ˙σ}Bˆ_{α}^{˙σ}),

Fαβ = Fαβ + g [−Fα˙µBˆ_{β}^{˙µ}− F_{˙µβ}Bˆ_{α}^{˙µ}+ gF_{˙µ ˙ν}Bˆ_{α}^{˙µ}Bˆ_{β}^{˙ν}].

To construct these covariant variables, we need the
combination of b^{˙µ}.

This situation is not same with original D4 case, because the variable FAB is not covariant.

Hence, how to deal with b^{˙µ} field in final action is a
important problem.

Simplify Action by Covariant Variables

After using covariant variables, we can get more simply action
form:S_{gauge}^{′} [b^{˙µ}, aA] =

Z

d^{2}xd^{3}y

−1

2H_{˙1˙2˙3}H^{˙1˙2˙3}− 1

4F˙ν ˙ρF^{˙ν ˙}^{ρ}+ 1

2Fβ˙µF^{β}^{˙µ}+ 1

2gǫ^{αβ}Fαβ

.

Although, this action does not look like familiar form of original case.

We can see the interesting structure in the action:

1

2gǫ^{αβ}Fαβ is Wess-Zumino term for the C-field.

Order expansion is a good way to more understand the property of action.

It also can help us to understand the meaning of b^{˙µ} in
the action.

Order Expansion Analysis

### Zeroth Order Analysis

To the 0-th order of g , the action S^{′(0)}gauge[b^{˙µ}, aA] can now be
expressed as

Z

d^{2}xd^{3}y

−1

2H^{2}_{˙1˙2˙3}−1

2ǫ^{αβ}F_{αβ}H_{˙1˙2˙3}− 1

4F_{˙µ ˙ν}F^{˙µ ˙ν} −1

2F_{α}_{˙µ}F^{α}^{˙µ}

= Z

d^{2}xd^{3}y

−1

2(H_{˙1˙2˙3}+ F01)^{2}− 1

4F_{AB}F^{AB}

,
where H_{˙1˙2˙3}= ∂˙µb^{˙µ} and A, B = ( ˙µ, α).

Because b^{˙µ} has no kinetic term, we can integrate it.

Finally we get the Maxwell action −^{1}_{4}F_{AB}F^{AB}.
To get the Maxwell action, we use the relation

:H_{˙1˙2˙3} = −F01, the degree of freedom of b^{˙µ} is transformed

into d.o.f of aα.

Hence, the new D4 action only includes the one-form degree of freedom in this sense.

Order Expansion Analysis

### First Order Analysis

The first order corrective action S^{′(1)}gauge[b^{˙µ}, aA] is
g

Z

d^{2}xd^{3}y

(−1

2H^{2}_{˙1˙2˙3}+1

2∂˙µb^{˙ν}∂˙νb^{˙µ})(H_{˙1˙2˙3}+ F01)
+H_{˙1˙2˙3}

−1

2F_{˙µ ˙ν}F^{˙µ ˙ν} + ǫ^{αβ}F_{α}_{˙µ}∂βb^{˙µ}

− 1

2ǫαβF_{˙µ ˙ν}F^{α}^{˙µ}F^{β}^{˙ν}
+F^{˙µ ˙ν}F_{˙λ ˙ν}∂_{˙µ}b^{˙λ} + Fα˙µF^{α}_{˙ν}∂^{˙µ}b^{˙ν} − Fα˙µ∂^{α}b_{˙ν}F^{˙µ ˙ν}o

.

We can gauge fix the degree of freedom of b^{˙µ}. Hence, we
chose ǫ^{˙µ ˙ν ˙}^{ρ}∂_{˙ν}b_{ρ}_{˙} = 0 to keep divergence part of b^{˙µ}.

The divergence part of b^{˙µ} can be represented by

b^{˙µ} = ∂^{˙µ}∂˙^{−}^{2}H_{˙1˙2˙3}, where the notation ˙∂^{−}^{2} is the inverse of
Laplacian ˙∂^{2} ≡ ∂˙µ∂^{˙µ}.

Nonlocal Phenomena

After these calculations, we get an action as a functional of
H_{˙1˙2˙3} and aA. To integrate out H_{˙1˙2˙3}, we can use

H_{˙1˙2˙3} = −F_{01}+ O(g ) to find final action S_{gauge}^{′′} [aA] up to first
order:

Z

d^{2}xd^{3}y

−1

4F_{AB}F^{AB} + g 1

2F_{01}F_{˙µ ˙ν}F^{˙µ ˙ν}
+ǫ^{αβ}F_{01}F_{α}_{˙µ}∂β∂^{˙µ}∂˙^{−}^{2}F_{01}− 1

2ǫαβF_{˙µ ˙ν}F^{α}^{˙µ}F^{β}^{˙ν}

−F^{˙µ ˙ν}F_{˙λ ˙ν}∂˙µ∂^{˙λ}∂˙^{−}^{2}F_{01}− Fα˙µF^{α}_{˙ν}∂^{˙µ}∂^{˙ν}∂˙^{−}^{2}F_{01}
+Fα˙µF^{˙µ ˙ν}∂^{α}∂˙ν∂˙^{−}^{2}F_{01}io

,

The nonlocal terms appear in this action if we want to describe action only by aA. It should be interesting physics phenomena of D4 brane in C-field background.

Multi-Dp branes generalization

### Generalize VPD for (p-1)-form

To generalize D4 to Dp brane, we introduce the (p-1)-bracket which is the generator of VPD in p-1 dimension:

{f1, f2,· · · , fp−1} ≡ ǫ^{˙µ}^{1}^{˙µ}^{2}^{···}^{˙µ}^{p−1}∂˙µ1f_{1}∂˙µ2f_{2}· · · ∂˙µ_{p−1}f_{p−1}.
The (p-1) form field strength:

H˙µ1˙µ2···˙µ_{p−1} ≡ g^{p−2}{X^{˙µ}^{1}, X^{˙µ}^{2},· · · , X^{˙µ}^{p−1}}−1

g = ∂˙µb^{˙µ}+O(g ),
Field strengths are defined by similar VPD gauge

potential: b^{˙µ}^{1} = _{(p−2)!}^{1} ǫ^{˙µ}^{1}^{˙µ}^{2}^{···}^{˙µ}^{p−1}b_{˙µ}_{2}···˙µ_{p−1}, X^{˙µ}^{i} ≡ ^{y}_{g}^{i} + b^{˙µ}^{i}.

Multi-Dp branes generalization

### Gauge Symmetry and Covariant Variables

To generalize the multi Dp-branes cases, we need to
promote gauge fields to matrix, however, it is difficult to
modify the gauge transformation of b^{˙µ}. One way is that
we try to keep this field as U(1) case and promote aA to
be matrix.

Hence, gauge transformation of aA is defined by
:δaA = [DA, λ] + g (κ^{˙µ}∂˙µa_{A}+ a˙µ∂Aκ^{˙µ}), where
D_{A} = ∂A + aA.

We choose λ is N × N matrix but κ^{˙µ} is 1 × 1 parameters.

Following these definitions, we can also define these covariant variables as same way as before.

The gauge transformation laws are:

δFAB = [FAB, λ− g κ^{˙µ}∂_{˙µ}]

Dp brane in R-R Field Background

Following the results of D4-brane action with covariant variables, we can try to generalize the action for multiple Dp-branes in R-R (p − 1)-form field background:

Z

d^{2}xd^{p−1}y

−1 2

1

(p − 1)!H˙µ1···˙µ_{p−1}H^{˙µ}^{1}^{···}^{˙µ}^{p−1} + 1

2gǫ^{αβ}F_{αβ}^{U(1)}

−1

4F^{U(1)}_{˙ν ˙}_{ρ} F_{U(1)}^{˙ν ˙}^{ρ} + 1

2F_{β}^{U(1)}_{˙µ} F_{U(1)}^{β}^{˙µ} − 1
4tr

F_{AB}^{SU(N)}F_{SU(N)}^{AB}
.
From this action, we can see the F_{01}^{U(1)} is still dual with the
VPD field strength H. It is easy to read from the 0-th order
action:

Z

d^{2}xd^{3}y

−1

2(H23···p+ F_{01}^{U(1)})^{2}− 1

4F_{AB}^{U(1)}F_{U(1)}^{AB}

−1 4tr

F_{AB}^{SU(N)}F_{SU(N)}^{AB}
,

### Summary

In this lecture, we construct D4 in R-R three form background from NP M5 theory.

We solve how to introduce another missing d.o.f of one form, and explain how to deal with the addition two form fields.

Theory include VPD gauge symmetry.

The order expansion analysis can reproduce Maxwell action in zeroth order, and we find some nonlocal effects in first order calculation.

We also generalize the formalism to Dp brane and multi Dp branes in R-R (p-1) form background.

### Furthermore Questions

In this talk, we try to understand the low energy effective action of D brane in R-R field background.

This formalism has still unknown parts about nonabelian extension of VPD, supersymmetry transformation of dual field, more simple action of high order g expansion with matter fields, etc.

To focus on supersymmetry’s effect, BPS states of the new theory may be different with previous cases and may be non trivial.

How to relate our formalism with the research of D brane in R-R field background(non-anticommutative theory) is interesting.

The story of AdS/CFT are also the application of R-R field background research. What can we relate it to our case?