Dimensional Reduction of the Generalized DBI

We study the generalized Dirac-Born-Infeld (DBI) action, which describes a $q$-brane ending on a $p$-brane with a ($q$+1)-form background. This action has the equivalent descriptions in commutative and non-commutative settings, which can be shown from the generalized metric and Nambu-Sigma model. We mainly discuss the dimensional reduction of the generalized DBI at the massless level on the flat spacetime and constant antisymmetric background in the case of flat spacetime, constant antisymmetric background and the gauge potential vanishes for all time-like components. In the case of $q=2$, we can do the dimensional reduction to get the DBI theory. We also try to extend this theory by including a one-form gauge potential.


Introduction
In string theory, T-duality shows the equivalence of two theories that seem different by exchanging radius R and radius α ′ R . For closed string, T-duality exchanges winding and momentum modes. In the case of open string, it exchanges the Dirichlet and Neumann boundary conditions. The low energy effective description of an open string ending on a single D-brane can be described by the DBI model [1]. We can observe T-duality in the DBI model. T-duality can also be seen in string field theory [2].
One interesting problem of brane theory is to construct action of a single M5-brane. An important observation is the equivalence of commutative and non-commutative description [3,4] by studying the DBI, non-abelian DBI and Nambu-Poisson M5 [5] model. Hopefully, the equivalence of commutative and non-commutative description gives a strong constraint on the final form of a single M5-brane action. From this equivalence, we can deduce the suitable form of brane theory. The related closed-open string relations can also be obtained from the Nambu-Sigma model, which is the generalization of the Poisson-Sigma model. The description of this theory is a q-brane ending on a p-brane. This theory is called the generalized DBI which can be consistently reduced to DBI by considering 2-form background. One nice thing of the generalized DBI is that we can find the same form for the M5-brane as given in [6] up to the second order when considering the 5-brane theory in 3-form background. This implies that the DBI-part of the M5-brane theory can be obtained from the generalized DBI [7]. More evidences on the validity of the generalized DBI can be found by looking into the generalized metric as done in [8].
On the other hand, the recent interesting developments of T-duality are double field theory [9] and generalized geometry [10]. We can see manifest O(D, D) by doubling coordinates in the formulation of the double field theory. The meaning of manifest O(D, D) is to embed Busher's rule in the O(D, D) structure. Then we can exchange coordinates to obtain Busher's rule. The current stage of double field theory is established on closed strings. Currently, we only have few understanding on open strings. The most important thing is the "stringy geometry" [11,12] constructed in double field theory in understanding the non-geometric flux [13]. We already understood how to generalize the standard 10-dimensional supergravity (NS-NS) to the new 10-dimensional supergravity as shown in [12,14]. We also expect that this different structure can give inspirations to other fields in analogy to the famous example that the quantum correction of string theory inspires new gravity model [15]. Under the strong constraint (that is a constraint for removing the additional coordinate), double field theory will reduce to the generalized geometry. The study of relaxing constraint is in [16]. Other interesting features of double field theory are α ′ geometry [17], exceptional field theory [18], D-brane [19] and others [20]. The recent reviews are in [21]. The related discussions of generalized geometry are curvature, torsion [22], Courant algebra [23], reduction [24], exceptional generalized geometry [25] and supergravity [26].
The main task of this paper is to carry out the dimensional reduction of the generalized DBI. We perform dimensional reduction from a (q + 1)-brane ending on a (p + 1)brane to a q-brane ending on a p-brane. In this work, we only consider flat spacetime, constant background and (q +1)-form gauge field only exists in (q +1)-dimensional worldvolume directions (no time direction) in q-p system. The non-trivial result is that the appearance of the 2(q + 1)-th root can be shown by the equivalence of commutative and non-commutative description, which is inherited under the consistent dimensional reduction. The most interesting case is a 2-brane ending on a 5-brane. It can reduce to a 1-brane ending on a 4-brane. It shows that the non-trivial 2-brane ending on a 5-brane can reduce to DBI theory in our simple consideration, and gives an interesting interpretation to our calculations. The 2-5 system possibly inspires to the M2-M5 system, which can go to F1-D4 system by dimensional reduction. We also explore the possibility of extending the generalized DBI. We show that this theory is possible to include one-form gauge field based on the consistency of dimensional reduction.
The plan of this paper is to first review the generalized DBI in Sec. 2. Then we show the discussion of dimensional reduction without scalar fields in Sec. 3. Dimensional reduction with scalar fields is in Sec. 4. Finally, we conclude in Sec. 5.

Review of the Generalized DBI
In this section, we review the generalized DBI. At first, we show the closed-open string relations from string sigma model. Secondly, we generalize Poisson-Sigma model to Nambu-Sigma model. We can also obtain generalized closed-open relations from the Nambu-Sigma model. Thirdly, we introduce membrane action. We find that the action is equivalent to Nambu sigma model under the gauge fixing. We also introduce worldsheet metric in this part. Finally, we use the generalized closed-open relations to construct an action.
We define our notations as follow. We denote the index A to be worldvolume direction. While a, b=1, 2, · · · , p are reserved for the spatial components of worldvolume coordinates. µ, ν=0, 1, · · · , D − 1 denote target space indices and w=0, 1 denote worldsheet indices. In addition, we use I to denote transverse direction and i, j to denote antisymmetric indices, i = (i 1 , i 2 , · · · , i r ) with 0 ≤ i 1 < i 2 < · · · < i r ≤ (r + 1), where r is the dimension of i.

Closed-Open Relations
We first introduce action of the Poisson-Sigma model [27] where X : Σ → M, Σ is the two dimensional world-sheet and M is the target space manifold. The one-form field A(σ) is on Σ and Π is an antisymmetric tensor. From the equations of motion we can show that the bi-vector Π satisfy the Jacobi identity. The first equation is the equation of motion for A µ . The other one is the equation of motion for X µ . We can add a metric term in the Poisson-Sigma model to obtain the non-topological generalized Poisson-Sigma model where * A ν is the Hodge dual of A ν . The signature of world-sheet is (−, +) and volume form d 2 σ ≡ dσ 0 ∧ dσ 1 . The A µ ≡ A µw (σ)dσ w are auxiliary fields. By using the equation of motion of A µ , the action (3) can be rewritten as the string sigma model action, where the g and B are defined by the closed-open string relations [3] 1 The action (3) can also be rewritten in terms of the components of η µ ≡ −A µ1 (σ) and η ν ≡ A ν0 (σ), the action is We can use matrix notation to rewrite it. We define The action is where the superscript T indicates transpose of matrix. By using the matrix notation, it is easier to generalize Poisson-Sigma model.

Generalized Closed-Open Relations
We introduce Nambu-Sigma model at first. It is a generalized Poisson-Sigma model. The action is π is a permutation and the antisymmetric product of partial derivatives where 0 ≤ i 1 < · · · < i q ≤ (q + 1). There are two types of metrics G andG, auxiliary fields η andη, and an antisymmetric (q + 1)-form tensor Π. We can integrate out the fields η andη. Then the resulting action is where We identify g by For the special case q = 1, these relations reduce to the closed-open string relations (5) [3]. We can also rewrite the action after using Wick rotation (σ 0 → −iσ 0 ) by the compact matrix form Let G denote the matrix We can find the inverse matrix by using the analytic inversion formula We also consider Interestingly, we can get the generalized closed-open relations by setting G −1 = H, the results are These relations imply that we can use to write the action in terms of G, Φ and Π. If q = 1, we can get We can use G = H −1 to get another form of the generalized closed-open relations as well.
We determine g explicitly by this way Before we give the explicit answer, we show the trick for the third and fourth terms. The third term is The fourth term is By using the same method, the first and second term are We can see explicit answer by combining all terms.
Then C is also easy to obtain Explicit expression forg −1 can be shown below The first term can be rewritten by The above formula can be derived from Binomial Inverse Theorem. The first term is If we combine the first term and second term, we obtainG −1 . Now we can combine all terms to see explicit answer.
However, the (21) and (22) can be a possibility of the generalization of the closed-open relations. We call these relations "generalized closed-open relations". We can also use the generalized metric to see the generalized closed-open relations. The generalized metric is exactly the matrix in the Hamiltonian. We start from Then we show the Hamiltonian where P is the canonical momenta corresponding to the fields X (P = g∂ 0 X − iC ∂X).
If we consider q = 1, the matrix in Hamiltonian has the same structure as the familiar generalized metric.
We can use another way to write the generalized metric Then we can get the generalized closed-open relations from the generalized metric.

Membrane Action
We start from the action to introduce membrane action. We can introduce an auxiliary field h AB and write the classically equivalent action We used equations of motion of h AB to derive the equivalence. For q = 1, we also have Then we can get h AB = ∂ A X µ ∂ B X ν g µν . Even for q = 1, we can also show the equivalence as q = 1. After we gauge fix (by reparametrization invariance) the components h a0 , h 0b and h 00 by choosing h a0 = h 0b = 0 and h 00 = − det(h ab ), and use the equations of motion of h ab we get a classically equivalent action with gauge fixing The action (44) can also be rewritten as We can also add a (q + 1)-form background field term, If we combine S gf with S C , we can get the same action as the Nambu-Sigma model.

Generalized DBI
Before we generalize the DBI action, we first review the well-known theory, DBI theory, which is an effective action for an open string ending on a D-brane. The action is where g s , g and B are closed string coupling constant, metric and background. F is the usual abelian field strength (F = dA). Before we show the equivalence of commutative and non-commutative description, we discuss the relations between the closed and open string parameters. These are The meaning of the above relations is to determine the open string variables from closed string variables by choosing Π. We can also rewrite G s as Now we include the gauge field in the generalized metric Now we add one new block matrix N to factorize of the generalized metric. Later we will combine them to see the equivalence of non-commutative and commutative description where Φ ′ =Φ + F ′ . From we can obtain We can also find useful formula from Then we find The DBI action can be rewritten from the closed string parameters to the open string parameters by the above relations Then we perform Seiberg-Witten map to get where the superscriptˆmeans the fields evaluated at covariant coordinates. When we change the coordinates, x → ρ * A (x) =x = x + ΠÂ induced by a map Π → Π ′ = (1 + Π · F ) −1 Π. The coordinatex µ is called covariant coordinate. We used to show the equivalence of the non-commutative and commutative description in the DBI theory.
We expect that we can use the similar method to generalize DBI theory. In other words, we use the equivalence of non-commutative and commutative description to construct the generalized DBI theory. In the generalized DBI theory, the background field is (q + 1)-form. When q=1, we can obtain the usual DBI theory.
The generalization of DBI can also be done by a similar decomposition of matrix We can also add square matrix M and N by the similar way we can obtain We can also find useful formula from Then we find Thus, we have From the down right block of we can obtain We also have We can get ( ∂x ∂x ) 2(q+1) in the action. From this term, we postulate the action can be because the term ( ∂x ∂x ) 2(q+1) cancel with the Jacobian which arise from coordinate transformation, such that the Lagrangian is an integral density. The coupling constant g b is called closed brane coupling constant. We can also rewrite open brane coupling constant G b as We used in the last equality. The action of the generalized DBI can be rewritten from the closed brane parameters to the open brane parameters.
. (72) This action is based on the equivalence of non-commutative and commutative gauge theory. The closed-open relations can be generalized from the generalized metric. On the other hand, it can also be derived from the Nambu-Sigma model. This generalized DBI theory can also be viewed as a generalization of the DBI. If we consider 2-form background, it goes back to the usual DBI theory. If we choose 3-form background and p=5, the action is where k µ ν = (H +C) µρσ (H +C) νρσ . This action is consistent with the [6] up to the second order. This action up to the second order can be understood from the κ-symmetry and equivalence of non-commutative and commutative gauge description. The understanding of full order comes from the equivalence of non-commutative and commutative gauge description. The supersymmetric extension and other formulation of the membrane theory are in [28].

Consistency of Dimensional Reduction
In this section, we discuss dimensional reduction of the action (69) without scalar fields. At first, we show dimensional reduction from (q + 1) − (p + 1) to q − p. We only consider flat spacetime, constant background, and (q + 1)-form gauge field exists in (q + 1)dimensional worldvolume directions (without time direction) in q-p system. In other words, we will have two types worldvolume directions. We denote α is the worldvolume direction without background andα is the other one direction with background. For a consistent notation, we define (1,2, · · · ,q) ≡ (p − q, p − q + 1,· · · , p − 1). The generalized DBI theory (69) gives We used in the above action. Then we calculate H Aig ij H Cj g CB The only non-zero components in ( Substituting the result and taking trace in the action (74), we get If we compactify one direction, the final expression (77) simply becomes In conclusion, we start from a system of (q + 1)-(p + 1), we can get an effective action for q-p system by dimensional reduction. We want to emphasize that this is not a trivial check because the 2(q + 1) root in the action is so far predicted based on the equivalence of non-commutative and commutative gauge theory. Thus, the calculation of this simple example gives us a confidence to show that this theory can also be consistent with dimensional reduction.

Comments on Pull-Back
If we also require that the generalized DBI can also go from (q + 1) − (p + 1) to q − p with scalar fields (by pull-back). Generalized DBI (69) needs to include a one-form gauge potential for a U(1) gauge symmetry. For the non-commutative gauge theory, we also have these similar systems [29]. We wish to explore the possibility by dimensional reduction.

Scalar Fields and Gauge Potential
When a worldvolume direction is compactified, the component of the compactified direction of a gauge potential A I give a scalar field X I , The scalar fields X I are actually the positions of a brane in transverse directions. The way we introduce scalar field is simply to set the metric by pull-back. In static gauge and the case of flat spacetime The inverse of this metric is which indeed satisfy the condition g AB g BC = δ A C . We define for convenience and notice that it is symmetric, i.e. ω AB = ω BA .

Conclusion
The generalized DBI is aimed for describing a q-brane ending on a p-brane. The most non-trivial feature of this action is that it contains a 2(q+1) root, which is predicted by the existence of the equivalence of the commutative and non-commutative description of the q-p system. In this paper, we showed that the generalized DBI action is also consistent with dimensional reduction to all orders perturbatively in the absence of scalar fields for (q+1)-(p+1)→ q-p in flat spacetime, constant background, and (q+1)-form gauge field which only exists in (q+1)-dimensional worldvolume (without time direction). It gives more understanding on the relation between 2-5 with M2-M5 system. In addition, we also find the possibility of the extension of including one-form gauge field in the presence of scalar fields based on dimensional reduction. The full understanding of dimensional reduction for (q+1)-(p+1)→ q-p leave it to the future.
In this paper, we focus on the dimensional reduction. However, the most interesting problem should be T-duality rule. Of course, we still have familiar Buscher's rule for q=1 with different p. Exploring T-duality rule is a challenging and interesting problem. It should give more interesting understanding to q-p system.
One related interesting problem is to explore double field theory of the DBI. By now, we do not get any insight to put one-form gauge fields. It is still an open problem about how to consider gauge fields in the double field theory. The starting direction is to find the gauge transformation which can be related to Courant bracket. It can offer the unique structure to constrain the DBI theory in double field theory. One more interesting thing related to open string of double field theory is to understand string sigma model with manifest Buscher's rule. It is a well-known fact that DBI model is equivalent to the calculation of the one-loop β function of the string sigma model. If we can include strong constraints in double field theory of the string sigma model, the one-loop β function would be an interesting thing. We also point out that one-loop β function of the Nambu-Sigma model is an important problem. So far, we only use the generalized metric and equivalence of the commutative and non-commutative description to understand the generalized DBI. We expect that one-loop β function of the Nambu-Sigma model should give the generalized DBI.