# Canonical approach to the closed string non-commutativity

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## Abstract

We consider the closed string moving in a weakly curved background and its totally T-dualized background. Using T-duality transformation laws, we find the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. From this structure we see that the commutative original theory is equivalent to the non-commutative T-dual theory, whose Poisson brackets are proportional to the background fluxes times winding and momentum numbers. The non-commutative theory of the present article is more nongeometrical than T-folds and in the case of three space-time dimensions corresponds to the nongeometric space-time with \(R\)-flux.

## Keywords

Poisson Bracket Closed String Bosonic String Flat Space Twisted Torus## 1 Introduction

It is well known that the open string endpoints, attached to a \(Dp\)-brane, are non-commutative [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The non-commutativity is implied by the fact that for the solution of the boundary conditions the initial coordinate is given as a linear combination of the effective coordinate and the effective momentum, which have a nonzero Poisson bracket (PB). In the constant background case, the coefficient in front of the momenta is proportional to the Kalb–Ramond field \(B_{\mu \nu }\), whose presence is crucial in gaining the non-commutativity.

The closed string does not have endpoints and in the flat space the boundary conditions are satisfied automatically. But, to understand the closed string non-commutativity, we are going to use a explanation similar to the open string case. We will express the closed string coordinates in terms of the coordinates and momenta of some other space. The relation between different spaces will be established using the T-duality transformations.

The T-dualization along isometry directions, and the construction of T-dual theory was first realized through a Buscher procedure [13, 14]. The procedure is in fact a localization of the translation invariance symmetry, in which beside the covariantization of derivatives one adds the Lagrangian multiplier term to the action, which ensures the physical equivalence of the initial and the T-dual theory.

In flat space, T-duality relates \(\sigma \)-derivatives of the coordinates of the original theory with the momenta of its T-dual theory, and vice versa. As the momenta of the original theory are taken to be commutative, it follows that the coordinates commute as well. So, in flat space there is no non-commutativity of the closed string T-dual coordinates. This is in agreement with the fact that T-duality is a canonical transformation in the flat space, and with the fact that PB’s are invariant under such transformations.

The closed string non-commutativity was first observed in the papers [15], and investigated further in [16, 17, 18, 19, 20], where it was found that the commutators of the coordinates are proportional to the flux and the winding number.

We considered the bosonic string moving in a background with constant metric \(G_{\mu \nu }=\mathrm{const}\) and the linear Kalb–Ramond field \(B_{\mu \nu }=b_{\mu \nu }+\frac{1}{3}B_{\mu \nu \rho }x^\rho \), where the field strength of the Kalb–Ramond field \(B_{\mu \nu \rho }\) is infinitesimally small (for more details see the introductory part of Sect. 2). The T-dual theory obtained is of the same form as the initial theory, so that the T-dual string moves in the T-dual background, but in the doubled space given by the coordinates \(y_\mu ,\, \tilde{y}_\mu \). The dual coordinates satisfy the following conditions: \(\dot{y}_\mu =\tilde{y}^\prime _\mu ,\,{y}^\prime _\mu =\dot{\tilde{y}}_\mu \). The improvement, in comparison to the standard Buscher procedure, is the covariantization of the coordinates \(x^\mu \). In fact, because \(x^\mu \) is gauge dependent, it is replaced by the gauge invariant expression \(\Delta x^\mu _{inv}=\int \mathrm{d}\xi ^\alpha D_{\alpha }x^\mu \). As pointed out in [21, 22], the T-dual background of the present paper is of the ‘new class that is even more nongeometrical than \(T\)-folds’. Unlike the T-folds, this background is not a standard manifold even locally. In our formulation, this stems from the fact that the argument of the background fields \(\Delta x^\mu _{inv}\) is the line integral. Some authors argued that such a spaces (for \(D =3\) known as R-flux background) involve nonassociative geometries [24].

In the canonical formalism, the T-dual variables can be expressed in terms of the original ones in the simple form \(y^\prime _\mu \cong \frac{1}{\kappa }\pi _\mu -\beta ^{0}_\mu \left[ x\right] \) and \( ^\star \pi ^\mu \cong \kappa x^{\prime \mu }+\kappa ^{2}\theta _{0}^{\mu \nu }\beta ^{0}_\nu \left[ x\right] . \) The infinitesimal expression \(\beta ^{0}_\mu \) is an improvement in comparison to the flat background case. Because the coordinates and momenta of the original theory do not commute, \(\beta ^{0}_\mu \) is the source of the closed string non-commutativity.

We will follow the main idea of Ref. [16], using the T-duality transformation laws between the T-dual backgrounds in order to study the non-commutativity of the coordinates. In the paper [16], the \(T_{2}\)-duality connects coordinates \(Z^{a}=Z^{a}(Y^{a})\) of the nongeometric background (\(Z^a\) with \(Q\)-flux) and the geometric background (twisted torus with \(Y^a\) and \(f\)-flux). We performed the T-dualization procedure along all the coordinates, and we obtained the T-duality transformation \(y_\mu = y_\mu (x^\mu )\) of the locally nongeometric background (the end of the chain (1.2) with \(y_\mu \) and \(f_D\)-flux) and the geometric background (torus with \(H\)-flux in the beginning of the chain (1.2)). In both approaches it was assumed that the geometric backgrounds (described by \(Y^{a}\) in [16] and by \(X^{a}\) in our paper) have the standard commutation relations. The PB between the \(y_\mu \) is proportional to the flux \(B_{\mu \nu \rho }\) and the winding number \(N^\mu \) of the initial theory. In addition, we obtain the complete algebra of the T-dual coordinates and momenta in terms of the fluxes.

For \(D=3\), the case of the present article corresponds to T-duality, \(T=T_1\circ T_2\circ T_{3}\), which connects the coordinates \(W^{a}=W^{a}(X^{a})\) of the nongeometric background (\(W^a\) with \(R\)-flux) and the geometric background (torus with \(X^a\) and \(H\)-flux). In comparison to Ref. [16], this procedure contains one \(T\)-dualization more, \(T_3\)-dualization along the coordinate \(X^{3}=Y^{3}=Z^{3}\), which cannot be done using the standard Buscher prescription because the Kalb–Ramond field \(B_{ab}\) depends on \(Z^{3}\). Thus, in terms of Ref. [16], we obtained the non-commutativity of the nongeometric background, with R-flux configuration. This background does not look like the conventional space even locally.

At the end we give three appendices. In the first one we derive in detail the expression for the dual momentum \({}^\star \pi ^\mu \), while in the second one we present a list of the fluxes used in the paper. The third appendix contains the mathematical details regarding the transition from PB \(\{\Delta X,\Delta Y\}\) to PB \(\{X,Y\}\).

## 2 Bosonic string in the weakly curved background and its T-dual picture

### 2.1 T-dual bosonic string

The T-dualization of closed string theory in a weakly curved background was the subject of investigation in [23]. There we presented the T-dualization procedure performed along all the coordinates, in a background which depends on these coordinates. Here we will give a short overview of the most important results.

### 2.2 Transformation laws

## 3 Non-commutativity relations between canonical variables

In flat space the coordinate dependent part of the Kalb–Ramond field is absent, \(h_{\mu \nu }=0\), and consequently \(\beta ^{0}_\mu =0\). Thus, from Eqs. (2.22a) and (2.22b) follows \(y^\prime _\mu \cong \frac{1}{\kappa }\pi _\mu \) and \(^\star \pi ^\mu \cong \kappa x^{\prime \mu }\). Therefore, the PB of the canonical variables of the T-dual theory remain the standard ones, the same as in the original theory. So, the nontrivial infinitesimal expression \(\beta ^{0}_\mu \), which exists only in the coordinate dependent backgrounds, is the source of the closed string non-commutativity.

- 1.\(\{y^\prime _\mu ,y^\prime _\nu \}\)$$\begin{aligned} K_{\mu \nu }\left[ x\right] =\frac{3}{\kappa }h_{\mu \nu }\left[ x\right] = \frac{1}{\kappa }B_{\mu \nu \rho }x^\rho , \quad L_{\mu \nu }=0, \end{aligned}$$(3.5)
- 2.\(\{y^\prime _\mu , {\tilde{y}}^\prime _\nu \}\)with$$\begin{aligned}&K_{\mu \nu }\left[ x,\tilde{x}\right] \!=\! \frac{3}{\kappa }h_{\mu \nu }\left[ \tilde{x}\right] \!-\!\frac{6}{\kappa }\left[ h\left[ x\right] G^{-1}b\!+\!bG^{-1}h\left[ x\right] \right] _{\mu \nu }, \nonumber \\&L_{\mu \nu }\left[ x\right] = \frac{1}{\kappa } g_{\mu \nu } -\frac{6}{\kappa }\left[ h\left[ x\right] G^{-1}b+bG^{-1}h\left[ x\right] \right] _{\mu \nu },\nonumber \\ \end{aligned}$$(3.6)Using Eqs. (2.6) and (7.2), expressions (3.6) can be rewritten in terms of the fluxes:$$\begin{aligned} \tilde{x}^{\prime \mu }=\frac{1}{\kappa }(G^{-1})^{\mu \nu }\pi _\nu +2(G^{-1}B)^\mu _{\ \nu }x^{\prime \nu }. \end{aligned}$$(3.7)$$\begin{aligned}&K_{\mu \nu }\left[ x,\tilde{x}\right] = \frac{1}{\kappa }B_{\mu \nu \rho }\tilde{x}^\rho -\frac{3}{2\kappa }\Gamma ^{E}_{\rho ,\mu \nu }x^\rho , \nonumber \\&L_{\mu \nu }\left[ x\right] =\frac{1}{\kappa }g_{\mu \nu } -\frac{3}{2\kappa }\Gamma ^{E}_{\rho ,\mu \nu }x^\rho , \end{aligned}$$(3.8)
- 3.\(\{\tilde{y}^\prime _\mu , {\tilde{y}}^\prime _\nu \}\)In terms of fluxes it becomes$$\begin{aligned}&\!\!\!K_{\mu \nu }\left[ x\right] = \frac{3}{\kappa }h_{\mu \nu }\left[ x\right] +\frac{24}{\kappa }\left[ bh\left[ x\right] b\right] _{\mu \nu }\nonumber \\&+\frac{6}{\kappa }\left[ h\left[ \tilde{x}\right] b-bh\left[ \tilde{x}\right] \right] _{\mu \nu }, \quad L_{\mu \nu }=0. \end{aligned}$$(3.9)where \(\Gamma ^{E}_{\nu ,\mu \rho }\) and \(Q_{\mu \nu \rho }\) are defined in Eqs. (7.1) and (7.5).$$\begin{aligned}&\!\!\!K_{\mu \nu }= -\frac{1}{\kappa } \left[ B_{\mu \nu \rho } -6g_{\mu \alpha }Q^{\alpha \beta }_{\ \ \rho }g_{\beta \nu } \right] x^\rho \nonumber \\&+\left[ -\frac{3}{2\kappa }\left( \Gamma ^{E}_{\mu ,\nu \rho }-\Gamma ^{E}_{\nu ,\mu \rho } \right) +\frac{4}{\kappa } B_{\mu \nu \sigma } (G^{-1}b)^\sigma _{\ \rho } \right] \tilde{x}^\rho ,\nonumber \\ \end{aligned}$$(3.10)

*closed string non-commutativity relation*

## 4 Comparison with the previous results

Let us mention that the case considered in the present paper is different from that of Ref. [16]. In Ref. [16], the non-commutativity relations in the nongeometric background with \(Q\)-flux where established, which are given in terms of winding numbers on the twisted torus \(N^{3}=\frac{1}{2\pi }\left( Y^{3}(\sigma +2\pi )-Y^{3}(\sigma ) \right) \). In the present article, the non-commutativity of the nongeometric background, which is not standard even locally and for \(D=3\) turns to R-flux background, was obtained in terms of the winding numbers on the torus with \(H\)-flux \(N^\mu =\frac{1}{2\pi }\left( X^\mu (\sigma +2\pi )-X^\mu (\sigma ) \right) \).

### 4.1 The brief overview of the results of Ref.

### 4.2 Similarities and differences

Although we analyzed the different cases, let us compare some general features of the results considered. In both approaches the commutators are infinitesimally small and they close on some winding numbers. Note that, in general, we can connect any geometric background with every nongeometric background from the chain of T-duality (1.2). Using the T-duality transformations we can calculate the non-commutativity of the coordinates of the nongeometric background in terms of the winding numbers of the geometrical background.

The main difference between the two approaches is the origin of non-commutativity. The nontrivial boundary conditions given in Eq. (2.25) of Ref. [16] are the source of the non-commutativity in that article. Because Ref. [16] does not consider \(T_3\)-dualization, the \(\beta ^0_\mu \)-functions (introduced in Eq. (2.15)) are zero and there is no non-commutativity of this kind. On the other hand, in the case considered in this paper, just these \(\beta ^0_\mu \) functions are the sources of the non-commutativity, even in the absence of the nontrivial boundary conditions of Ref. [16]. For complete non-commutativity relations one should take into account both kinds of non-commutativity.

## 5 Concluding remarks

In the present article we derived the closed string non-commutativity relations. We considered the theory describing a string moving in a weakly curved background. Its T-dual theory is obtained performing the T-dualization procedure along all the coordinates [23]. The T-dual transformation laws play a central role in our approach. These laws connect the world-sheet derivatives of the coordinates and momenta in the original and the T-dual theory. The zero orders are transformation laws of the constant background and they do not lead to the non-commutativity. The term \(\beta ^0_\mu \), which is infinitesimally small and bilinear in the \(x^\mu \) coordinates, plays a key role in obtaining the non-commutativity relations.

In the original space we choose the standard Poisoon brackets. The T-dual coordinates \(y_\mu \) have two terms: one linear in the original momenta and the other bilinear in the original coordinates. This explains the nontrivial PB \(\{y_\mu ,y_\nu \}\) of Eq. (3.11), which is linear in the coordinates. Note that in the case of an open string moving in the flat background coordinate is linear function in both effective momenta and coordinates. Therefore, the corresponding PB is constant.

The T-dual momenta \({}^\star \pi ^\mu \) are bilinear expressions in the original coordinates. Thus, the PB of the T-dual momenta vanishes, see Eq. (3.21), but the PB between the T-dual coordinates and the momenta (3.19) obtained an additional term linear in the coordinates.

In the doubled space there exists the additional coordinate \(\tilde{y}_\mu \). It consists of a term linear in the original momenta, but with the coefficient linear in the original coordinate and the other terms bilinear in the original coordinates. Thus, it produces a nontrivial PB with all variables \((y_\mu ,\tilde{y}_\mu ,{}^\star \pi ^\mu )\), see Eqs. (3.12), (3.13), and (3.20).

In terms of Ref. [16] for the three-dimensional torus \(x^\mu \rightarrow X^{a},\,(a=1,2,3)\) our case corresponds to the non-commutativity of the nongeometric background with \(W^{a}\) coordinates and \(R\)-fluxes obtained after the successive performation of all three T-dualizations along all three coordinates. It relates the \(W^{a}\) with the \(X^{a}\) coordinates of the torus with \(H\)-flux, and so the PB closes on the winding number of the \(X^{a}\)-coordinates. We hope that these results will contribute to a better understanding of the strangest, uncommon R-flux configurations where the non-commutativity appears as a consequence of the nontrivial \(\beta ^{0}_\mu \)-functions. Note that Ref. [16] uses \(T_{2}\)-duality (performed along \(Y^{2}\)) and the relation \(Z^{a}=Z^{a}(Y^{a})\) to obtain the non-commutativity of the nongeometric background with \(Q\)-flux in terms of the winding of the \(Y^{a}\)-coordinates. There the non-commutativity originates from the nontrivial boundary conditions. To obtain the general structure of the closed string non-commutativity for arbitrary background of the chain (1.2) one should find its T-duality transformations with all other backgrounds of the chain and calculate both kinds of non-commutativity originating from nontrivial boundary conditions as well as from nontrivial \(\beta _\mu ^{0}\) functions.

The term of the action with the constant part of the Kalb–Ramond field \(b_{\mu \nu }\) is topological. Thus, it does not contribute to the equations of motion. In the open string case it contributes to the boundary conditions and it is a source of the open string non-commutativity. In the closed string case it is absent from boundary conditions as well. Classically, we can gauge it away and the Kalb–Ramond field becomes infinitesimally small. But if \(b_{\mu \nu }=0\) one loses topological contributions. In order to investigate the global structure of the theory with holonomies of the world-sheet gauge fields in quantum theory we should preserve such a term.

## Notes

### Acknowledgments

Work supported in part by the Serbian Ministry of Education, Science and Technological Development, under contract No. 171031.

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