Strings and membranes from ${\cal A}$-theory five brane

Superstring theory is a strong candidate for a unified theory that describes the four fundamental forces in a unified framework. However, as it stands, there is not a single unified theory but six different theories: five superstring theories and M-theory. These six theories are interconnected through S-duality and T-duality, forming a single chain that creates a hexagon. Each of these six theories is described on distinct branes, complementarily expressing the unique characteristics of superstring theory. The question is what kind of theory could provide a unified description of these six theories.

S-duality and T-duality are unified under U-duality, which is determined by the group-theoretic inclusiveness of duality symmetries. Specifically, the S-duality symmetry GL(D+1) (an extension of spacetime diffeomorphism by GL(2) ) and the T-duality symmetry O(D,D) are encompassed within the U-duality symmetry represented by the exceptional group E_D+1. The U-duality symmetry should comprehensively relate the six superstring theories. Therefore, the theory that unifies these six theories is expected to exhibit covariance under U-duality. We term a theory explicitly possessing such U-duality as ”${\cal A}$-theory,” and we aim to establish a perturbative formulation of this theory to discuss its quantum aspects, see for a review [1].

In 1995, Witten proposed M-theory as a theory related to IIA superstring theory through S-duality (Witten). The low-energy limit of this theory is 11-dimensional supergravity. The diffeomorphism symmetry of this spacetime is GL(11), combining the 10-dimensional spacetime of the original string theory and the S-duality symmetry SL(2). We refer to a worldvolume theory explicitly exhibiting GL(D+1) symmetry as ${\cal M}$-theory. In 1996, Vafa proposed F-theory as a theory related to IIB superstring theory through S-duality [2]. For a review, see, for example [3].

In 1993, Siegel proposed a string theory guided by the T-duality symmetry O(D,D) [4, 5, 6]. This framework employs the idea of doubling spacetime coordinates, where the 2D-dimensional spacetime coordinates are treated as vectors in the O(D,D) representation[7]. The geometry generated by the O(D,D) covariant current algebra is the stringy gravity theory with $B$-field, where gauge fields are parameters of the coset O(D,D) over the doubled Lorentz group. We refer to this as ${\cal T}$-theory and the T-duality covariant string as ${\cal T}$-string. Later, Hull, Zwiebach, and Hohm proposed a theory of O(D,D) covariant background fields, known as Double Field Theory (DFT) [8, 9]. This is the low-energy gravitational theory of ${\cal T}$-string theory. To consistently reduce the doubled 2D-dimensional spacetime coordinates to the physical D-dimensional spacetime, the section condition was employed, which is the Virasoro constraint ${\cal S}$=0. For reviews, see [10, 11]. Since IIA and IIB superstring theories are related by T-duality, ${\cal T}$-string theory was constructed [12, 13, 14, 15, 16]. All bosonic component fields represent supersymmetrized O(D,D) (the doubled non-degenerated super-Poincare group) while fermions represent only O(D-1,1)². To incorporate S-duality the exceptional group U-duality symmetry is necessary. The exceptional group E_D+1 relates to the D-dimensional subgroup of the doubled non-degenerated super-Poincare group.

Siegel, Linch and Polacek subsequently proposed brane theories guided by the U-duality symmetry ( E_D+1 ) [12, 17, 18]. Since the representation of exceptional groups varies with dimension, the theory is labeled by the spacetime dimension $D$ when reduced to string theory. This is called D$=D$ ${\cal A}$-theory. According to the classification of Lie algebras, removing a specific node in the Dynkin diagram of E_D+1 reduces it to the GL(D+1) Dynkin diagram. This corresponds to reducing the spacetime dimensions of ${\cal A}$-theory using the Virasoro constraint, recovering the aforementioned ${\cal M}$-theory. Conversely, removing another node in the E_D+1 Dynkin diagram reduces it to the O(D,D) Dynkin diagram. This corresponds to reducing the spacetime dimensions of ${\cal A}$-theory using the Gauss law constraint ${\cal U}$=0, recovering the aforementioned ${\cal T}$-theory. ${\cal A}$-theory is described by branes covariant under the exceptional group. Both the spacetime coordinates and world-volume coordinates of the branes are representations of the exceptional group, ensuring that the brane current algebra is covariant under the exceptional group. Moreover, a perturbative Lagrangian describing these branes on their worldvolume has been constructed. In the previous papers [17, 18, 24, 25, 26, 27, 28, 29, 30, 19, 20, 21, 22, 28, 23] we refer these models as “F-theory”, but we renamed our “F-theory” to ${\cal A}$-theory since our latest paper [1] as a generalization of F-theory for all dimensions with all duality symmetries.

Exceptional Field Theory (EFT) applies DFT concepts to exceptional groups [31, 35, 36, 37, 38, 39, 40, 41, 42, 32, 33, 34]. The symmetry of exceptional groups was initially discovered as a partial dimensional symmetry of background fields in 11-dimensional supergravity [43, 44]. Active research continues on expressing exceptional groups through brane current algebra [45, 46, 47, 48, 49, 50, 51, 52].

This paper focuses on the 5-brane that describes D=3 ${\cal A}$-theory with SL(5) U-duality symmetry. We clarify how the usual string, and ${\cal T}$-string and membrane of 11-dimensional supergravity (M2) emerge. Specifically, we derive the Lagrangians for the conventional non-perturbative M2-brane, the O(D,D)-covariant ${\cal T}$-string, and the conventional string from the ${\cal A}$-theory 5-brane (${\cal A}$5) Lagrangian.

In this paper we focus on the SL(5) U-duality symmetry and clarify the relation between the ${\cal A}$-theory five brane (${\cal A}$5-brane) and conventional branes. The ${\cal A}$5-brane is a 5-brane with manifest SL(5) U-duality symmetry and it is described by the perturbative Lagrangian with the SL(5) two rank anti-symmetric tensor coordinate [1]. The ${\cal A}$5 theory forms the apex, allowing a diamond-shaped contour to be drawn, with the ${\cal T}$-string with O(3,3) T-duality symmetry, ${\cal M}$5-brane with GL(4) duality symmetry, and $S$tring with GL(3) symmetry as the vertices of the diamond diagram. In this paper we present the sectioning procedure along the diamond contour. In addition to this we present reducing procedures from the ${\cal T}$-string Lagrangian to the conventional string Lagrangian and from the ${\cal M}$5-brane Lagrangian to the conventional non-perturbative M2-brane Lagrangian [53]. We generalize the dimensional reduction of S-duality for the 11-dimensional supergravity to the type IIA supergravity [54] to T-duality in subsection 1.3. For S-duality, $\lambda_{\rm string}\leftrightarrow 1/\lambda_{\rm string}$, the dimensional reduction is achieved by taking the $\lambda_{\rm string}\ll 1$ limit in the metric. For T-duality, $R/\sqrt{\alpha^{\prime}}\leftrightarrow\sqrt{\alpha^{\prime}}/{R}$ with the string length $l_{\rm string}=\sqrt{\alpha^{\prime}}$, the dimensional reduction is achieved by taking the $R\gg\sqrt{\alpha^{\prime}}$ in the metric, so that the O(D,D) spacetime for the ${\cal T}$-string reduces to the D-dimensional spacetime for the conventional string. We also propose the perturbative ${\cal M}$5-brane Lagrangian in the supergravity background obtained from ${\cal A}$5-brane (6.95).

In section 2 the relation among ${\cal A}$5-brane, ${\cal M}$5-brane, ${\cal T}$-string and $S$tring theories is explained by diamond diagrams based on duality symmetries. The diamond diagram is a contour in the U-duality plane which is spanned by two parameters, the string coupling and the scale based on the string length. The branes are described by the field strengths where the spacetime coordinate has the gauge symmetry generated by the Gauß law constraint. These field strengths and gauge parameters as well as the world-volume coordinate and the spacetime coordinate are representations of duality symmetries. The world-volume diffeomorphism is generated by the Virasoro constraints ${\cal S}=0$. Dimensions to reduce are determined by solving the Virasoro constraints ${\cal S}=0$ for spacetime and by solving the Gauß law constraints ${\cal U}=0$ for world-volume.

In section 3 both the SL(5) and the SL(6) covariant Lagrangians of the ${\cal A}$5-brane are given, where the SL(6) manifests the 5-brane world-volume Lorentz symmetry. The SL(6) vielbein includes both the SL(5) spacetime vielbein and the 6-dimensional world-volume vielbein, so the spacetime and world-volume are mixed under the new duality symmetry SL(6). The SL(6) formulation is useful to reduce to other branes: Since the string world-sheet directions and the spacetime directions are direct sum, the SL(6) vielbein is in a block diagonal form as shown in subsection 5.1. On the other hand, the brane world-volume directions share the spacetime directions unlike the string as shown in subsection 6.1.

In section 4 we begin by the O(D,D) string Hamiltonian, and apply the double zweibein method [55, 56] to obtain the ${\cal T}$-string Lagrangian. Then reducing procedure from the O(D,D) ${\cal T}$-string Lagrangian to the conventional string Lagrangian in D dimensions is presented analogously to subsection 1.3. We show that the Wess-Zumino term can be obtained by adding a total derivative term.

In section 5 we begin by the ${\cal A}$5-brane Lagrangian and present the reduction procedure to ${\cal T}$-string Lagrangian. We rewrite the O(D,D) background gauge field in the SL(4) tensor index such a way that the SL(4) tensor coordinate of the ${\cal T}$-string couples. Then we apply the procedure given by section 4 to obtain the conventional $S$tring Lagrangian.

In section 6 we begin by the SL(6) covariant ${\cal A}$5-brane Lagrangian to lead the new perturbative ${\cal M}$5-brane Lagrangian. We further reduce to the conventional M2-brane Lagrangian. The Nambu-Goto Lagrangian is obtained by the gauge choice of the world-volume vielbein, while the Wess-Zumino term is obtained by adding the total derivative term.

In [54] it has been pointed out that under an S-duality transformation between the 10-dimensional type IIA theory and the 11-dimensional supergravity theory, the structure of the supersymmetry algebra remains invariant, though the meaning of the central charge changes. The global superalgebra with supercharges $Q$, $Q^{\prime}$ of different chiralities, 10 dimensional momenta $P$ and the central charge $W$ is given as $\left\{Q,Q\right\}\sim P\sim\left\{Q^{\prime},Q^{\prime}\right\}$, $\left\{Q_{\alpha},Q^{\prime}_{\dot{\beta}}\right\}\sim\delta_{\alpha\dot{\alpha}}W$. The central charge $W$ is recognized as the D0-brane Ramond-Ramond (RR) charge in 10 dimensions and as the 11-th dimensional momentum in 11 dimensions. The 11-dimensional spacetime reduces into the 10-dimensional spacetime in the weak coupling limit $e^{2\phi}\ll 1$,

with the 11-th dimensional coordinate $y$ and the string coupling $\lambda_{\rm string}=e^{3\phi/2}$. The 11-dimensional momentum $W$ is maintained as the D0-brane charge in the 10-dimeniosnal IIA theory after the dimensional reduction.

This is generalized to T-duality. We compare the superalgebra for the 2D-dimensional ${\cal T}$-string with manifest T-duality and the type II superalgebra for the conventional string in D dimensions. The global type II superalgebra with two supercharges $Q$, $Q^{\prime}$ and the D-dimensional momentum $P$ is given as $\left\{Q,Q\right\}\sim(P+\tilde{P})$ and $\left\{Q^{\prime},Q^{\prime}\right\}\sim(P-\tilde{P})$ with the central charge $\tilde{P}$. It is recognized as the NS-NS charge in D dimensions, while the extra D dimensional momenta in 2D dimensions. If we restore $\alpha^{\prime}$ in the momentum and the winding mode constituting O(D,D) vector $(p_{m},~{}\frac{1}{\alpha^{\prime}}\partial_{\sigma}x^{m})$ $\to$ $(p_{m},~{}\frac{1}{\alpha^{\prime}}\tilde{p}^{m})$, then their canonical conjugates are written as $(x^{m},~{}\alpha^{\prime}y_{m})$. In the small scale $R\ll\sqrt{\alpha^{\prime}}$ the winding modes become massless and easily excited, while in the large scale $R\gg\sqrt{\alpha^{\prime}}$ the winding mode gets heavy so only the momentum modes are excited. Manifest T-duality is broken by considering the particular background in such a way that the 2D-dimensional spacetime reduces into the D-dimensional spacetime in the $R\gg\sqrt{\alpha^{\prime}}$ limit

Here $y_{m}$ is the extra D-dimensional coordinate and $B_{mn}$ is the NS-NS gauge field. The extra D-dimensional momentum is maintained as the NS-NS charge after the dimensional reduction.

This dimensional reduction procedure corresponds to the gauge fixing of the dimensional reduction constraint which is the first class constraint in Hamiltonian formulation. The dimensional reduction constraint and the gauge fixing to reduce the conventional string are discussed in [57] for a flat space case. The dimensional reduction constraint is the $y$ component of the symmetry generator, $\tilde{\hbox{\,\Large$\triangleright$}}_{y}=0$. The gauge fixing condition $\partial_{\sigma}y=0$ reduce the set of conventional string operators, the physical momentum ${P}_{x}\neq 0$ and left/right covariant derivatives $\hbox{\,\Large$\triangleright$}=P_{x}\pm\partial_{\sigma}x$. In Lagrangian formulation the momentum is replaced by $P_{X}=\partial L/\partial\dot{X}$. It is generalized to brane case, then the dimensional reduction constraint turns out to be the Virasoro constraint in which one of the momenta is replaced by the 0-mode [18], i.e. for zero-mode momenta $p_{x},~{}p_{y}$ and momenta including both the 0-mode and the non-0-modes $P_{x},~{}P_{y}$ , the dimensional reduction constraint $\tilde{\hbox{\,\Large$\triangleright$}}_{y}=0$ is also written as $p_{y}\cdot P_{x}+p_{x}\cdot P_{y}=0$ $\to$ $P_{y}=0$.

Although the equation of motion derived from the doubled Lagrangian with using the selfduality condition coincides with the equation of motion derived from the original Lagrangian, the selfduality condition makes the doubled Lagrangian vanish [58]. The selfduality condition $\partial_{\mu}{x}=\epsilon_{\mu\nu}\partial^{\nu}y$ reduces the Lagrangian of the O(D,D) ${\cal T}$-string in flat space to 0 as

It is also mentioned that the naive section $y=0$ the ${\cal T}$-string Lagrangian in curved background does not reduce to the expected string Lagrangian in curved background as

The followings are also noted. Integrating out $(dy+\cdots)^{2}$ is possible for the constant background, but for general non-constant background cases the $dy$-path integral of $exp[-\int(dy+\cdots)g(dy+\cdots)]$ produces the Jacobian factor $\sqrt{g}$ in the path integral measure, which effectively produces the additional term in the action. Using the equation of motion is possible for the constant background again, but it does not reduce to the conventional string Lagrangian for the non-constant background $g(x,y),~{}B(x,y)$. Imposing two conditions $\partial_{+}y-\partial_{+}xB=0$ and $\partial_{-}y-\partial_{-}xB=0$ is not consistent, since the integrability condition is not satisfied for the curved background $[\partial_{+},\partial_{-}]y\neq 0$ for non-constant background. Studies to refine the reduction of conventional string Lagrangians have yielded several interesting approaches [58, 59, 60, 61, 48, 62].

Instead we propose the reducing procedure from ${\cal T}$-string Lagrangian to the conventional $S$tring Lagrangian: (1) adding the total derivative term ${-}\partial_{\mu}(\epsilon^{\mu\nu}x^{m}\partial_{\nu}y_{m})$ to derive the Wess-Zumino term, then (2) the dimensional reduction (1.2) as

This is the expected string Lagrangian up to the normalization factor two which can be absorbed by the Lagrange multiplier. The section conditions of spacetime fields $\Phi(x,y)$ are consistent with the Lagrangian where the section $y=0$ can be chosen as $\Phi(x)$.

This procedure is similar to the usual dimensional reduction where the reduction is done in the local flat Lorentz coordinate. i.e. Suppose that we have a line element $dx^{A}\equiv dx^{M}E_{M}{}^{A}$. We decompose the doubled coordinate $dx^{A}$ into $dx^{a}$ and $dy_{a}$ in the local Lorenz frame, and then discard $(dy_{a})^{2}$. Since the metric ($\hat{\eta}$-tensor) in local flat spacetime is already diagonal, in practice we can just apply this reduction by deleting certain blocks of $\hat{\eta}$-tensor similar to (1.2).

The main purpose of this paper is to perform this reduction for various specific cases. But in general, the reduction procedure could be schematically summarized as follows.

1. 1.

   We start with the current algebra with extended coordinates (both momentum and winding modes have their conjugate coordinates). The Hamiltonian is just the sum of selfdual and anti-selfdual constraints $H=g{\cal H}+\tilde{g}\tilde{\mathcal{H}}+s_{m}{\cal S}^{m}+\tilde{s}_{m}\tilde{{\cal S}}^{m}+Y^{m}{\cal U}_{m}$, where ${\cal H},\tilde{{\cal H}}$ are the Hamiltonian constraints (and its dual counterpart), ${\cal S}^{m},\tilde{{\cal S}}^{m}$ are the Virasoro constraints. ${\cal U}_{m}$ is the Gauß law constraint which only exits for brane Hamiltonians.

2. 2.

   We find the Lagrangian by the Legendre transformation of the above Hamiltonian $H$. This has been done in previous papers for various theories [1]. Schematically, it can be written as $L=\Phi J_{\textrm{SD}}\cdot\hat{\eta}\cdot J_{\overline{\textrm{SD}}}+\Lambda\cdot J_{\overline{\textrm{SD}}}\cdot\hat{\eta}\cdot J_{\overline{\textrm{SD}}}+\cdots$, where $J^{A}_{\textrm{SD}/\overline{\textrm{SD}}}$ are the selfdual and anti-selfdual currents. They are coupled with vielbein, and thus they have flat indices. $\Phi$ and $\Lambda$ are Lagrange multipliers which are functions of $g,s^{m},\tilde{g},\tilde{s}^{m}$. One can gauge fix $\Lambda=0$ by the suitable choice of original parameters.

3. 3.

   Separate coordinates and currents into the physical part and the auxiliary part as $X^{M}\rightarrow x^{m},y^{\mu}$, and $J^{A}\rightarrow J^{a},J^{\alpha}$ where $x^{m}$ is the physical coordinate whatever string/brane we want to maintain and $y^{\mu}$ is the auxiliary coordinate. Then perform the dimensional reduction (1.2).

4. 4.

   The reduced Lagrangian $L{}^{\prime}=J^{a}{}_{\textrm{SD}}\hat{\eta}_{ab}J^{b}{}_{\overline{\textrm{SD}}}$ could be explicitly shown to be to the string/brane action we want, but without the Wess-Zumino term. We found that adding a total derivative term to the Lagrangian gives the Wess-Zumino term

   $$L{}+\textrm{Total derivative}=J^{a}{}_{\textrm{SD}}\hat{\eta}_{ab}J^{b}{}_{\overline{\textrm{SD}}}+\tilde{J}^{\alpha}_{\textrm{SD}}\hat{\eta}_{\alpha\beta}\tilde{J}^{\beta}{}_{\overline{\textrm{SD}}}+L_{\textrm{WZ}}.$$

   The current $\tilde{J}^{\alpha}{}$ is modified after adding the total derivative, then it eventually would be reduced by the dimensional reduction (1.2). This is how we get the Wess-Zumino term.

The duality web of $G$-symmetry of ${\cal A}$-theory is represented by a diamond diagram as studied in [18] in (2.19). The $G$-symmetry of the coset group $G/H$ is duality symmetry. The coset parameter is the gauge field of the duality covariant geometry including the spacetime veilbein field, the NS-NS and R-R gauge fields of the superstring theory.

The relation of these duality groups is explained by the Dynkin diagrams [1]. Removing one node from the Dynkin diagram of E_D+1(D+1) reduces to the one of GL(D+1) or O(D,D) depending of the position of the removed node. Further removing one node from the Dynkin diagram of GL(D+1) or O(D,D) reduces to the one of GL(D).

In this paper we focus on D=3 case where the $G$-symmetry is SL(5) and the diamond diagram becomes Fig.2 in (2.40). This SL(5) duality symmetry is enlarged to SL(6) for the (5+1)-dimensional world-volume covariance in Lagrangian [1]. We named this enlarged symmetry “$A$-symmetry”. This ${\cal A}$-theory unifies the spacetime and the world-volume, in a sense that the coset parameter of $A/L$=SL(6)/GL(4) includes not only the spacetime vielbein field but also the world-volume vielbein field.

It is denoted that we use $H=$SO($D$) instead of SO($D-1$) for simplicity, so the Wick rotation is necessary for realizing the time component in this section and other places.

The duality covariant theories are described by the spacetime coordinate and the world-volume coordinate which are representations of the duality symmetries, ${A}$-symmetry or $G$-symmetry. Then the world-volume dimension is determined by the duality group. The Gauß law constraint generates the gauge symmetry of the duality covariant spacetime, therefore the brane current becomes the field strength.

The D=3 ${\cal M,~{}T},~{}S$-theories are obtained from the D=3 ${\cal A}$-theory [17, 22]. We list representations of duality groups as below; the world-volume derivative $\partial^{m}$, the gauge parameter $\lambda^{m}$, the spacetime coordinate $X^{M}$, and the field strength (the current) $F_{M}=\eta_{MNm}\partial^{m}X^{N}$ ( $J_{\mu}{}^{M}=\partial_{\mu}X^{M}$, $\mu=(\tau,\sigma)$). $\eta_{MNm}$ is the $G$-symmetry invariant tensor which enters the current algebra.

The world-volume dimension of ${\cal M}$-theory is still $1\oplus 5$ where four dimensions are embedded in the 4 spacetime $x^{\underline{m}}$ and one dimension is embedded in the internal space, so we denote as $1\oplus 4(5)$.

The field strengths and currents together with the gauge transformations are given concretely as follows.

1. 1.

   ${\cal A}$5-brane field strengths

   1. (a)

      World-volume covariant ${\cal A}$5-brane field strength

      The SL(6) $A$-symmetry covariant ${\cal A}$-theory is described by a 5-brane with the manifest SL(6) new duality symmetry which manifests 6-dimensional world-volume Lorentz symmetry, namely world-volume covariant ${\cal A}$5-brane.

      $\displaystyle F^{\hat{m}\hat{n}\hat{p}}=\frac{1}{2}\partial^{[\hat{m}}X^{\hat{n}\hat{p}]}~{}~{},~{}~{}\delta_{\lambda}X^{\hat{m}\hat{n}}=\partial^{[\hat{m}}\lambda^{\hat{n}]}~{}~{},~{}~{}\hat{m}=0,1,\cdots,5$

      (2.64)

   2. (b)

      ${\cal A}$5-brane field strength

      The SL(5) $G$-symmetry covariant ${\cal A}$-theory is described by a 5-brane with manifest SL(5) U-duality symmetry, namely ${\cal A}$5-brane.

      $\displaystyle{\left\{\begin{array}[]{l}F_{\tau}{}^{mn}=\dot{X}^{mn}-\partial^{[m}Y^{n]}\\ F_{\sigma}{}_{;m_{1}m_{2}}=\frac{1}{2}\epsilon_{m_{1}\cdots m_{5}}\partial^{m_{3}}X^{m_{4}m_{5}}\\ m=1,\cdots,5\end{array}\right.,~{}\left\{\begin{array}[]{l}\delta_{\lambda}X^{mn}=\partial^{[m}\lambda^{n]}\\ \delta_{\lambda}Y^{m}=\dot{\lambda}^{m}-\partial^{m}\lambda^{0}\end{array}\right.}$

      (2.70)

2. 2.

   ${\cal M}$5-brane field strength

   The GL(4) ${\cal M}$-theory is described by a 5-brane with the manifest GL(4) duality symmetry, namely ${\cal M}$5-brane. We focus only on 4-dimensional subspace of the 5-dimensional world-volume which is embedded in the 4-dimensional spacetime. Physical currents are as follows.

   $\displaystyle{\left\{\begin{array}[]{l}F_{\tau}{}^{\underline{m}}=\dot{x}^{\underline{m}}+\partial^{\underline{m}}Y\\ F_{\sigma}{}_{;\underline{m}_{1}\underline{m}_{2}}=-\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial^{\underline{m}_{3}}x^{\underline{m}_{4}}\\ \underline{m}=1,\cdots,4\end{array}\right.,~{}\left\{\begin{array}[]{l}\delta_{\lambda}x^{\underline{m}}=\partial^{\underline{m}}\lambda\\ \delta_{\lambda}Y=-\dot{\lambda}\end{array}\right.}$

   (2.76)

   The following currents are auxiliary written by auxiliary coordinates $y^{\underline{mn}}$, $Y^{\underline{m}}$.

   $\displaystyle{\left\{\begin{array}[]{l}F_{\tau}{}^{\underline{m}\underline{n}}=\dot{y}^{\underline{m}\underline{n}}-\partial^{[\underline{m}}Y^{\underline{n}]}\\ F_{\sigma}{}_{;\underline{m}_{1}}=\frac{1}{2}\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial^{\underline{m}_{2}}y^{\underline{m}_{3}\underline{m}_{4}}\end{array}\right.,~{}\left\{\begin{array}[]{l}\delta_{\lambda}y^{\underline{m}\underline{n}}=\partial^{[\underline{m}}\lambda^{\underline{n}]}\\ \delta_{\lambda}Y^{\underline{m}}=\dot{\lambda}^{\underline{m}}\\ \delta_{\lambda}\lambda^{\underline{m}}=\partial^{\underline{m}}{\lambda}\end{array}\right.}$

   (2.82)

   These currents constitute the SL(5) $A$-symmetry together with (2.76), and they are used to lead the non-perturbative M2-brane Lagrangian.

3. 3.

   ${\cal T}$-string currents

   The O(3,3) ${\cal T}$-theory is described by a string with the manifest O(3,3) T-duality symmetry, namely ${\cal T}$-string.

   $\displaystyle{\left\{\begin{array}[]{l}J_{\tau}{}^{\underline{m}_{1}\underline{m}_{2}}=\dot{X}^{\underline{m}_{1}\underline{m}_{2}}\\ J_{\sigma}{}_{\underline{m}_{1}\underline{m}_{2}}=\frac{1}{2}\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial_{\sigma}X^{\underline{m}_{3}\underline{m}_{4}}\end{array}\right.}$

   (2.85)

   It is convenient to represent in terms of $x^{\bar{m}}$ and $y_{\bar{m}}=\frac{1}{2}\epsilon_{\bar{m}\bar{n}\bar{l}}y^{\bar{n}\bar{l}}$.

   $\displaystyle{\left\{\begin{array}[]{l}J_{\tau}{}^{\bar{m}}=\dot{x}^{\bar{m}}\\ J_{\sigma}{}_{\bar{m}_{1}\bar{m}_{2}}=-\epsilon_{\bar{m}_{1}\bar{m}_{2}\bar{m}_{3}}\partial_{\sigma}x^{\bar{m}_{3}}\\ J_{\tau}{}^{\bar{m}_{1}\bar{m}_{2}}=\dot{y}^{\bar{m}_{1}\bar{m}_{2}}\\ J_{\sigma}{}_{\bar{m}_{1}}=\frac{1}{2}\epsilon_{\bar{m}_{1}\bar{m}_{2}\bar{m}_{3}}\partial_{\sigma}y^{\bar{m}_{2}\bar{m}_{3}}\\ \bar{m}=1,2,3\end{array}\right.}$

   (2.91)

4. 4.

   $S$-tring currents

   The GL(3) $S$-theory is described by a string with the manifest GL(3) spacetime diffeomorphism symmetry, namely a 3-dimensional string.

   $\displaystyle{\left\{\begin{array}[]{l}J_{\tau}{}^{\bar{m}}=\dot{x}^{\bar{m}}\\ J_{\sigma}{}_{\bar{m}_{1}\bar{m}_{2}}=-\epsilon_{\bar{m}_{1}\bar{m}_{2}\bar{m}_{3}}\partial_{\sigma}x^{\bar{m}_{3}}\end{array}\right.}$

   (2.94)

Some minus signs come from the mere notation $\epsilon_{1234}=1=-\epsilon_{4123}$. It is denoted that these currents are flat currents, and in later sections flat current symbols ${\buildrel\mkern 2.5mu\raise-1.00006pt\hbox{$\scriptstyle\circ$}\mkern-2.5mu\over{F}}$ or ${\buildrel\mkern 2.5mu\raise-1.00006pt\hbox{$\scriptstyle\circ$}\mkern-2.5mu\over{J}}$ will be used to distinguish from curved background currents.

The theories in Hamiltonian formulation are constructed by the current algebra with manifest duality symmetries [17]. The Spacetime translation is generated by the covariant derivative $\hbox{\,\Large$\triangleright$}_{M}(\sigma)$. The $p$-brane current algebra with $G$-symmetry covariance is given by

Branes are governed by the brane Virasoro constraints ${\cal S}^{m}=\frac{1}{2}\hbox{\,\Large$\triangleright$}_{M}\eta^{MNm}\hbox{\,\Large$\triangleright$}_{N}=0$ and ${\cal H}=\frac{1}{2}\hbox{\,\Large$\triangleright$}_{M}\hat{\eta}^{MN}\hbox{\,\Large$\triangleright$}_{N}=0$ together with the Gauß law constraints ${\cal U}_{m}=0$ which is required by the closure of the Virasoro algebra. $\hat{\eta}^{MN}$ is the $H$-invariant metric. Theories are related by sectionings; The Virasoro constraint ${\cal S}^{m}=0$ gives the section conditions to reduce the spacetime dimensions, and the Gauß law constraint ${\cal U}_{m}=0$ is used to reduce the world-volume dimension as Fig.3 in (2.113).

The spacetime covariant derivatives, constraints and section conditions are given [17, 22] in Fig.3 (2.113) concretely as follows.

1. 1.

   ${\cal A}$5-brane in 10-dimensional spacetime

   The 10-dimensional spacetime is described by the two rank anti-symmetric tensor covariant derivative $\hbox{\,\Large$\triangleright$}_{m_{1}m_{2}}$ as

   $\displaystyle\hbox{\,\Large$\triangleright$}_{m_{1}m_{2}}$

   $\displaystyle=$

   $\displaystyle P_{m_{1}m_{2}}+\frac{1}{2}\epsilon_{m_{1}\cdots m_{5}}\partial^{m_{3}}X^{m_{4}m_{5}}$

   (2.115)

   where $P_{mn}$ is canonical conjugate of $X^{mn}$ with $[P_{mn}(\sigma),X^{lk}(\sigma^{\prime})]=\frac{1}{i}\delta_{m}^{[l}\delta_{n}^{k]}\delta^{(5)}(\sigma-\sigma^{\prime})$ with $m=1,\cdots,5$.

   The SL(5) covariant current algebra of ${\cal A}$5-brane is given by

   $\displaystyle\left[\hbox{\,\Large$\triangleright$}_{{m}_{1}{m}_{2}}(\sigma),\hbox{\,\Large$\triangleright$}_{{m}_{3}{m}_{4}}(\sigma^{\prime})\right]$

   $\displaystyle=$

   $\displaystyle 2i\epsilon_{{m}_{1}\cdots{m}_{5}}\partial^{m_{5}}\delta^{(5)}(\sigma-\sigma^{\prime})~{}~{}~{}.$

   (2.116)

   The 5-dimensional world-volume diffeomorphism is generated by the Virasoro constraints ${\cal S}^{m}=0$ while the world-volume time diffeomorphism is generated by ${\cal H}=0$. The Gauß law constraint ${\cal U}_{m}=0$ generates the gauge symmetry of the spacetime coordinate. These constraints are given by [1] as:

   $\displaystyle{\left\{\begin{array}[]{ccl}{\cal S}^{m}&=&\frac{1}{16}\hbox{\,\Large$\triangleright$}_{m_{1}m_{2}}\epsilon^{mm_{1}\cdots m_{4}}\hbox{\,\Large$\triangleright$}_{m_{3}m_{4}}=0\\ {\cal H}&=&\frac{1}{16}\hbox{\,\Large$\triangleright$}_{m_{1}m_{2}}\delta^{m_{1}[n_{1}}\delta^{n_{2}]m_{2}}\hbox{\,\Large$\triangleright$}_{n_{1}n_{2}}=0\\ {\cal U}_{m}&=&\partial^{n}\hbox{\,\Large$\triangleright$}_{mn}=0\end{array}\right.}$

   (2.120)

   The SL(5) covariant constraints ${\cal S}^{m}=0$ and ${\cal U}_{m}=0$ are background independent. These constraints are used as the dimensional reduction and section condition by replacing the spacetime momentum $P_{M}(\sigma)$ with the derivative of the 0-mode of the spacetime coordinate $X_{0}^{M}$.

   $\displaystyle{\begin{array}[]{|l|c|c|}\hline\cr&{\rm Virasoro}:~{}{\cal S}^{m}&{\rm Gau\ss{}~{}law}:~{}{\cal U}_{m}\\ \hline\cr\begin{array}[]{l}{\rm dimensional}~{}{\rm reduction}\end{array}&\displaystyle\epsilon^{mm_{1}\cdots m_{4}}P_{m_{1}m_{2}}(\sigma)\frac{\partial}{\partial X^{m_{3}m_{4}}_{0}}&P_{mn}(\sigma)\displaystyle\frac{\partial}{\partial\sigma_{n}}\\ \hline\cr\begin{array}[]{l}{\rm section}~{}{\rm condition}\end{array}&\displaystyle\epsilon^{mm_{1}\cdots m_{4}}\frac{\partial}{\partial X^{m_{1}m_{2}}_{0}}\frac{\partial}{\partial X^{m_{3}m_{4}}_{0}}&\displaystyle\frac{\partial}{\partial\sigma_{n}}\frac{\partial}{\partial X^{mn}_{0}}\\ \hline\cr\end{array}}$

   (2.126)

   These operators act on fields $\Phi(X)$ and $\Psi(X)$ as

   	$\displaystyle\displaystyle\frac{\partial}{\partial X^{M}_{0}}\frac{\partial}{\partial X^{N}_{0}}\Phi(X_{0})~{}=~{}0~{}=~{}\frac{\partial}{\partial X^{M}_{0}}\Phi(X_{0})\frac{\partial}{\partial X^{N}_{0}}\Psi(X_{0})$
   	$\displaystyle\displaystyle\frac{\partial}{\partial\sigma_{n}}\frac{\partial}{\partial X^{M}}\Phi(\sigma,X(\sigma))~{}=~{}0~{}=~{}\frac{\partial}{\partial\sigma_{n}}\Phi(\sigma,X(\sigma))\frac{\partial}{\partial X^{M}}\Psi(\sigma,X(\sigma))~{}~{}~{}$		(2.127)

   where fields may be functions on $\sigma$ as $\Phi(\sigma,X(\sigma))$ and $\Psi(\sigma,X(\sigma))$.

   
   
2. 2.

   ${\cal M}$5-brane in 4-dimensional spacetime

   The dimensional reduction of the spacetime is obtained by solving the Virasoro constraint in (2.126) as

   $\displaystyle P_{\underline{m}\underline{n}}(\sigma)=0~{}~{},~{}~{}{\underline{m}=1,\cdots,4}~{}~{}\Rightarrow~{}\displaystyle\epsilon^{mm_{1}\cdots m_{4}}\frac{\partial}{\partial X^{m_{1}m_{2}}_{0}}P_{m_{3}m_{4}}(\sigma)=0~{}~{}~{}.$

   (2.128)

   This condition makes $X^{\underline{m}_{1}\underline{m}_{2}}=y^{\underline{m}_{1}\underline{m}_{2}}$ to be non-dynamical and reduced dimensionally. The remaining spacetime is 4 dimensions $P_{5\underline{m}}=p_{\underline{m}}\neq 0$.

   The 4-dimensional spacetime is described by the covariant derivative $\hbox{\,\Large$\triangleright$}_{\underline{m}}$. The 6-dimensional covariant derivative $\hbox{\,\Large$\triangleright$}_{\underline{m}\underline{n}}$ is maintained to construct SL(5) current algebra

   $\displaystyle{\left\{\begin{array}[]{ccl}\hbox{\,\Large$\triangleright$}_{\underline{m}}&=&p_{\underline{m}}\\ \hbox{\,\Large$\triangleright$}_{\underline{m}_{1}\underline{m}_{2}}&=&-\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial^{\underline{m}_{3}}x^{\underline{m}_{4}}\end{array}\right.}$

   (2.131)

   with $X^{5\underline{m}}=x^{\underline{m}}$ and $P_{5\underline{m}}=p_{\underline{m}}$ which is not confused with the 0-mode momentum.

   The SL(5) current algebra of ${\cal M}$5-brane is

   $\displaystyle{\left\{\begin{array}[]{ccl}\left[\hbox{\,\Large$\triangleright$}_{\underline{m}}(\sigma),\hbox{\,\Large$\triangleright$}_{\underline{n}}(\sigma^{\prime})\right]&=&0\\ \left[\hbox{\,\Large$\triangleright$}_{\underline{m}_{1}}(\sigma),\hbox{\,\Large$\triangleright$}_{\underline{m}_{2}\underline{m}_{3}}(\sigma^{\prime})\right]&=&2i\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial^{\underline{m_{4}}}\delta^{(5)}(\sigma-\sigma^{\prime})\\ \left[\hbox{\,\Large$\triangleright$}_{\underline{m}_{1}\underline{m}_{2}}(\sigma),\hbox{\,\Large$\triangleright$}_{\underline{m}_{3}\underline{m}_{4}}(\sigma^{\prime})\right]&=&0\end{array}\right.~{}~{}~{},}$

   (2.135)

   where the last algebra forces to $\partial^{5}=0$.

   The Virasoro operators of ${\cal M}$5-brane are

   $\displaystyle{\left\{\begin{array}[]{ccl}{\cal S}^{\underline{m}}&=&\frac{1}{2}\partial^{[\underline{m}}x^{\underline{n}]}p_{\underline{n}}\\ {\cal S}^{5}&=&\frac{1}{4}\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}(\partial^{\underline{m}_{1}}x^{\underline{m}_{2}})(\partial^{\underline{m}_{3}}x^{\underline{m}_{4}})\\ {\cal H}&=&\frac{1}{4}p_{\underline{m}}\hat{\eta}^{\underline{m}\underline{n}}p_{\underline{n}}+\frac{1}{16}(\partial^{[\underline{m}_{1}}x^{\underline{m}_{2}]})\hat{\eta}_{\underline{m}_{1}[\underline{n}_{1}}\hat{\eta}_{\underline{n}_{2}]\underline{m}_{2}}(\partial^{[\underline{n}_{1}}x^{\underline{n}_{2}]})\\ {\cal U}_{5}&=&\partial^{\underline{n}}p_{\underline{n}}\end{array}\right.}~{}~{}~{}.$

   (2.140)

   These constraints lead to the following dimensional reductions and section conditions.

   $\displaystyle{\begin{array}[]{|l|c|c|}\hline\cr&{\rm Virasoro}:~{}{\cal S}^{\underline{m}}&{\rm Gau\ss{}~{}law}:~{}{\cal U}_{m}\\ \hline\cr{\rm dimensional}~{}{\rm reduction}&\frac{1}{2}p_{\underline{m}}(\sigma)(\partial^{[\underline{m}}x^{\underline{n}]})&p_{\underline{m}}(\sigma)\partial^{\underline{m}}\\ \hline\cr{\rm section}~{}{\rm condition}&{\rm none}&\displaystyle\partial^{\underline{m}}\frac{\partial}{\partial x^{\underline{m}}_{0}}\\ \hline\cr\end{array}}$

   (2.144)

   
   
3. 3.

   ${\cal T}$-string in 6-dimensional spacetime

   The dimensional reduction condition of the world-volume is obtained by solving the Gauß law constraint in (2.126) as

   $\displaystyle\partial^{\underline{n}}=0~{},~{}P_{\underline{m}}(\sigma)=0~{}~{}~{}\Rightarrow~{}\partial^{n}P_{mn}(\sigma)=0~{}~{}~{}.$

   (2.145)

   These conditions make $\partial^{5}=\frac{\partial}{\partial\sigma}\neq 0$ and $X^{\underline{m}}$ to be non-dynamical (constant). The remaining spacetime is 6 dimensional $P_{\underline{m}\underline{n}}=p_{\underline{m}\underline{n}}\neq 0$.

   The 6-dimensional spacetime is described by the covariant derivative $\hbox{\,\Large$\triangleright$}_{\underline{m}\underline{n}}$. The 4-dimensional covariant derivative vanishes $\hbox{\,\Large$\triangleright$}_{\underline{m}}=0$

   $\displaystyle\hbox{\,\Large$\triangleright$}_{\underline{m}_{1}\underline{m}_{2}}$

   $\displaystyle=$

   $\displaystyle p_{\underline{m}_{1}\underline{m}_{2}}+\frac{1}{2}\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial^{5}x^{\underline{m}_{3}\underline{m}_{4}}$

   (2.146)

   with $X^{\underline{m}\underline{n}}=x^{\underline{m}\underline{n}}$.

   The O(3,3) current algebra of ${\cal T}$-string is

   $\displaystyle\left[\hbox{\,\Large$\triangleright$}_{\underline{m}_{1}\underline{m}_{2}}(\sigma),\hbox{\,\Large$\triangleright$}_{\underline{m}_{3}\underline{m}_{4}}(\sigma^{\prime})\right]$

   $\displaystyle=$

   $\displaystyle 2i\epsilon_{\underline{m}_{1}\cdots\underline{m}_{4}}\partial^{5}\delta(\sigma-\sigma^{\prime})~{}~{}~{}.$

   The Virasoro operators of ${\cal T}$-string are

   $\displaystyle{\left\{\begin{array}[]{ccl}{\cal S}^{5}&=&\frac{1}{16}\hbox{\,\Large$\triangleright$}_{\underline{m}_{1}\underline{m}_{2}}\epsilon^{\underline{m}_{1}\cdots\underline{m}_{4}}\hbox{\,\Large$\triangleright$}_{\underline{m}_{3}\underline{m}_{4}}\\ {\cal H}&=&\frac{1}{16}\hbox{\,\Large$\triangleright$}_{\underline{m}_{1}\underline{m}_{2}}\hat{\eta}^{\underline{m}_{1}[\underline{n}_{1}}\hat{\eta}^{\underline{n}_{2}]\underline{m}_{2}}\hbox{\,\Large$\triangleright$}_{\underline{n}_{1}\underline{n}_{2}}\end{array}\right.}$

   (2.149)

   with ${\cal S}^{\underline{m}}=0={\cal U}_{m}$.

   The Virasoro constraint ${\cal S}^{5}=0$ lead to the following dimensional reduction and the section condition.

   $\displaystyle{\begin{array}[]{|l|c|c|}\hline\cr&{\rm Virasoro}:~{}{\cal S}^{\underline{m}}&{\rm Gau\ss{}~{}law}:~{}{\cal U}_{m}\\ \hline\cr{\rm dimensional}~{}{\rm reduction}&p_{\underline{m}_{1}\underline{m}_{2}}(\sigma)\epsilon^{\underline{m}_{1}\cdots\underline{m}_{4}}\displaystyle\frac{\partial}{\partial x^{\underline{m}_{3}\underline{m}_{4}}_{0}}&{\rm none}\\ \hline\cr{\rm section}~{}{\rm condition}&\displaystyle\frac{\partial}{\partial x^{\underline{m}_{1}\underline{m}_{2}}_{0}}\epsilon^{\underline{m}_{1}\cdots\underline{m}_{4}}\frac{\partial}{\partial x^{\underline{m}_{3}\underline{m}_{4}}_{0}}&{\rm none}\\ \hline\cr\end{array}}$

   (2.153)

   
   
4. 4.

   $S$tring in 3-dimensional spacetime

   1. (a)

      From ${\cal T}$-string to $S$tring

      The dimensional reduction of the spacetime is obtained by solving the Virasoro constraint in (2.153) as

      $\displaystyle P_{\bar{m}\bar{n}}(\sigma)=0~{}~{},~{}~{}{\bar{m}=1,2,3}~{}~{}\Rightarrow~{}P_{\underline{m}_{1}\underline{m}_{2}}(\sigma)\epsilon^{\underline{m}_{1}\cdots\underline{m}_{4}}\displaystyle\frac{\partial}{\partial x^{\underline{m}_{3}\underline{m}_{4}}_{0}}=0~{}~{}~{}~{}.$

      (2.154)

      This condition makes $X^{\bar{m}_{1}\bar{m}_{2}}=y^{\bar{m}_{1}\bar{m}_{2}}$ to be non-dynamical (constant). The remaining spacetime is 3 dimensions $P_{4\bar{m}}=p_{\bar{m}}\neq 0$.

   2. (b)

      From ${\cal M}5$-brane to $S$tring

      The dimensional reduction condition of the world-volume is obtained by solving the Gauß law constraint in (2.144) as

      $\displaystyle\partial^{\bar{n}}=0~{},~{}P_{4}(\sigma)=0~{}~{}~{}\Rightarrow~{}P_{\underline{m}}\partial^{\underline{m}}=0~{}~{}~{}.$

      (2.155)

      In the 4-dimensional spacetime $\partial^{5}$ is considered to be 0. These conditions make $\partial^{4}=\frac{\partial}{\partial\sigma}=\partial_{\sigma}\neq 0$ and $X^{54}$ to be non-dynamical (constant). The remaining spacetime is 3 dimensions $P_{4\bar{m}}=p_{\bar{m}}\neq 0$.

   The 3-dimensional spacetime is described by the covariant derivative $\hbox{\,\Large$\triangleright$}_{4\bar{m}}$. The 3-dimensional covariant derivative vanishes $\hbox{\,\Large$\triangleright$}_{\bar{m}_{1}\bar{m}_{2}}$

   $\displaystyle{\left\{\begin{array}[]{ccl}\hbox{\,\Large$\triangleright$}_{\bar{m}}&=&p_{\bar{m}}\\ \hbox{\,\Large$\triangleright$}_{\bar{m}_{1}\bar{m}_{2}}&=&\epsilon_{\bar{m}_{1}\bar{m}_{2}\bar{m}}\partial_{\sigma}x^{\bar{m}}\end{array}\right.}$

   (2.158)

   with $X^{4\bar{m}}=x^{\bar{m}}$ and $\sigma^{5}=\sigma$ via ${\cal T}$-string and $X^{5\bar{m}}=x^{\bar{m}}$ and $\sigma^{4}=\sigma$ via ${\cal M}5$-brane.

   The GL(3) current algebra becomes

   $\displaystyle\left[\hbox{\,\Large$\triangleright$}_{\bar{m}_{1}}(\sigma),\hbox{\,\Large$\triangleright$}_{\bar{m}_{2}\bar{m}_{3}}(\sigma^{\prime})\right]$

   $\displaystyle=$

   $\displaystyle i\epsilon_{\bar{m}_{1}\bar{m}_{2}\bar{m}_{3}}\partial_{\sigma}\delta(\sigma-\sigma^{\prime})~{}~{}~{}.$

   (2.159)

   This is equivalent to the O(D,D) current algebra which is given by $\hbox{\,\Large$\triangleright$}_{M}=(\hbox{\,\Large$\triangleright$}_{\bar{m}},~{}\hbox{\,\Large$\triangleright$}^{\bar{m}})$ with $\hbox{\,\Large$\triangleright$}^{\bar{m}}=\frac{1}{2}\epsilon^{\bar{m}\bar{m}_{1}\bar{m}_{2}}\hbox{\,\Large$\triangleright$}_{\bar{m}_{1}\bar{m}_{2}}$ and the O(D,D) invariant metric $\eta_{MN}=\epsilon_{\bar{m}_{1}\bar{m}_{2}\bar{m}_{3}}$ as

   $\displaystyle\left[\hbox{\,\Large$\triangleright$}_{M}(\sigma),\hbox{\,\Large$\triangleright$}_{N}(\sigma^{\prime})\right]$

   $\displaystyle=$

   $\displaystyle i\eta_{MN}\partial_{\sigma}\delta(\sigma-\sigma^{\prime})~{}~{}~{}.$

   (2.160)

   The Virasoro operators become

   $\displaystyle{\left\{\begin{array}[]{ccl}{\cal S}&=&p_{\bar{m}}\partial_{\sigma}x^{\bar{m}}~{}=~{}\frac{1}{2}\hbox{\,\Large$\triangleright$}_{M}\eta^{MN}\hbox{\,\Large$\triangleright$}_{N}\\ {\cal H}&=&\frac{1}{2}p_{\bar{m}}\hat{\eta}^{\bar{m}\bar{n}}p_{\bar{n}}+\frac{1}{2}\partial_{\sigma}x^{\bar{m}}\hat{\eta}_{\bar{m}\bar{n}}\partial_{\sigma}x^{\bar{n}}~{}=~{}\frac{1}{2}\hbox{\,\Large$\triangleright$}_{M}\hat{\eta}^{MN}\hbox{\,\Large$\triangleright$}_{N}\end{array}\right.}$

   (2.163)

   with the double Lorentz invariant metric $\hat{\eta}^{MN}$. There is no further conditions of the Virasoro and the Gauß law constraints; ${\cal S}^{\underline{m}}=0={\cal U}_{m}$.