In order to understand that the NP M5 theory describes the M5-brane in large C field background. One can study the relative superstring theory. This relation between M theory and superstring theory can be done by dimensional reduction. There are several ways of dimensional reductions. One way is just to compactify one target space dimension on circle, then M2-brane and M5-brane in eleven dimensions will relate to D2-brane and NS5-brane in ten dimensions. Another way is to compactify one target space dimension and one worldvolume space dimension on a circle at the same time. It is called Double Dimensional Reduction (DDR). After DDR, the M2-brane and M5-brane in eleven di-mensions will relate to F1 string and D4-brane in ten didi-mensions. These objects (F1, D2, D4, and NS5) are the main elements in IIA superstring theory in ten dimensions.
Similarly, if we compactify one target space dimension on S1/Z2, it will relate to the E8× E8 heterotic superstring theory. In this section, we will focus on the DDR method, then we can study the relative D4-brane action of NP M5 theory.
2.4.1 Poisson D4 Description From Nambu-Poisson M5 Theory
In this subsection, we will re-derive the D4 in large NS-NS B field background from the NP M5 theory. Firstly, we know theory D4-brane theory can be obtained from M5-brane theory after double dimensional reduction on a circle. The double dimensional reduction (DDR) means that we do the dimensional reduction on worldvolume and target space at the same time. In original, people [1, 2] want to show the evidence of the NP M5 theory is the effective description of M5-brane in large C field background. Hence, they expect to get the D4 in large NS-NS B field background after compactification the circle, which live in the direction y˙3 and has radius R. There are several reasons for this choice. The
first thing is the C field background field C˙1˙2˙3 will be explained as B˙1˙2 after DDR on y˙3. The relation between C˙1˙2˙3 and B˙1˙2 is written by:
Z y3˙=2πR y3˙=0
C˙1˙2˙3dy˙1dy˙2dy˙3 ≡ B˙1˙2dy˙1dy˙2. (2.33)
The second thing is the Nambu-Poisson bracket will relate to Poisson bracket by this way: {f, g, y˙3} = ǫα ˙˙β ˙3∂˙µf ∂˙νg ≡ {f, g}p.b.. Here the indices ˙α are {˙1, ˙2}.
After integrating out the auxiliary field (bµ ˙α) and renaming some fields, we get4:
SD4inB = are understood as the one form gauge field in D4-brane theory after DDR. The gauge field ˆaa:= ba ˙3 can be used to define the covariant derivative and field strength:
δΛˆaa = ∂aΛ− g{Λ, ˆaa}p.b., Λ≡ Λ˙3, (2.35) Fˆab = ∂aˆab − ∂bˆaa+ g{ˆaa, ˆab}p.b., (2.36) DˆaΦ = ∂aΦ + g{ˆaa, Φ}p.b.. (2.37) This theory describes the D4-brane in large NS-NS B field background(B˙1˙2).
In this chapter, we show the main characters of NP M5 theory. We give several evi-dences of the M5-brane in large C field background. For example, the constant term exists in action, the supersymmetry law is nonlinear, the two form gauge field has non-abelian structure, and it reproduces D4-brane in NS-NS B field background, etc. However, we find another possible D4-brane formalism, which can describe the D4-brane in large C field background. It can be achieved by DDR on another circle x2. We will deal with it in next chapter.
4Here the indices ‘a’ are{µ; ˙α}.
Chapter 3
D4 in R-R Three Form Background
In this chapter, we will start to consider the effective action of D4-brane in the large three-form background. This is motivated from NP M5 theory, which describes the single M5-brane in large C field background. If we do double dimensional reduction along the codimension of C field. We will get the effective description of D4-brane in large C field background.
3.1 D4-Brane in C Field Background via DDR
To carry out the double dimensional reduction (DDR) for the M5-brane along the x2 -direction, we set
x2 ∼ x2+ 2πR, (3.1)
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. To keep zero mode of fields in x2 direction, we need to explain the meaning of field with component 2. For example, the bµ ˙µ → {b2 ˙µ, bα ˙µ}, where α = 0, 1 and the field b2 ˙µ is understood by one form field on D4-brane theory. Hence, we define
b˙µ2 ≡ a˙µ. (3.2)
On the other hand, the Gamma matrix Γ2 is understood by ten dimensional chirality matrix. It is used to define the chirality condition of fermion (gaugino) in D4-brane theory.
3.1.1 Gauge Transformation of Fields
We expect that the U (1) gauge symmetry on the D4-brane has its origin in the gauge transformations (2.16), (2.17) on the M5-brane. The gauge transformation parameter Λ2 shall be identified with the U (1) gauge transformation parameter. This is consistent with the identification of a˙µ with b˙µ2. The gauge symmetry parametrized by Λ˙µ, i.e., the VPD, is also still present on the D4-brane. Hence, we can have the gauge transformation of a˙µ:
δΛa˙µ = ∂˙µλ + g(κ˙ν∂˙νa˙µ+ a˙ν∂˙µκ˙ν). (3.7) The gauge symmetry combines U(1) gauge symmetry and volume-preserving diffeomor-phism symmetry. This is the first new character of the new D4 theory. The 3-dimensional volume-preserving diffeomorphism is the evidence of D4 in large C-field background. We want to ask how to find the other DOF of one form fields (aα), and we also want to know how to find the gauge transformation law of aα. We will deal with it in next section.
3.1.2 Action
After keeping the zero mode of fields in x2 direction, we get the effective description of five dimensions worldvolume theory. The action is what we expect for the new D4-brane action, which describe the effective action of D4-brane in large C-field background. The complete action form can be represented in different parts. The result of DDR on Sgauge
is
where we use the definition of ǫαβ2 ≡ ǫαβ. The result of DDR on SX is
In this chapter, we will focus on the gauge field part. To understand if the gauge part has a well description of D4 in large C-field background will teach us how to deal with matter fields part. After turning off the mater fields, we only need to consider the equation (3.8).
Focus on the action of gauge fields after DDR, we identify a˙µ as components of the one-form potential on the D4-brane. In terms of the field strength
F˙µ ˙ν ≡ ∂˙µa˙ν − ∂˙νa˙µ, (3.11)
we can rewrite H2 ˙µ ˙ν as
H2 ˙µ ˙ν = F˙µ ˙ν +g
2ǫ˙µ ˙ν ˙λǫ˙σ ˙ρ ˙τ∂˙σb˙λFρ ˙τ˙ . (3.12)
In the above we see that part of the two-form potential b on the M5-brane transforms into part of the one-form potential a on D4. However, in order to interpret this action as a D4-brane action, we still need to identify the rest of the components aα of the one-form gauge potential, and to re-interpret bα ˙µ and b˙µ ˙ν from the D4-brane viewpoint. We also need to find all components of field strength or find all covariant variables in this theory. On the other hand, we also need to understand the new D4 action in usually D4 viewpoint. We will deal with these problems in different sections.