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Perturbative F-theory 10-brane and M-theory 5-brane

The exceptional symmetry is realized perturbatively in F-theory which is the manifest U-duality theory. The SO(5,5) U-duality symmetry acts on both the 16 spacetime coordinates and the 10 worldvolume coordinates. Closure of the Virasoro algebra requires the Gauss law constraints on the worldvolume. This set of current algebras describes a F-theory 10-brane. The SO(5,5) duality symmetry is enlarged to the SO(6,6) in the Lagrangian formulation. We propose actions of the F-theory 10-brane with SO(5,5) and SO(6,6) symmetries. The gauge fields of the latter action is coset elements of SO(6,6)/SO(6;C) which includes both the SO(5,5)/SO(5;C) spacetime backgrounds and the worldvolume backgrounds. The SO(5,5) current algebra obtained from the Pasti-Sorokin-Tonin M5-brane Lagrangian leads to the theory behind M-theory, namely F-theory. We also propose an action of the perturbative M-theory 5-brane obtained by sectioning the worldvolume of the F-theory 10-brane.


Introduction
Duality is a cornucopia of string theory creating its unique properties. Superstring theory is considered to be a candidate for a unified theory of all forces, and five types of superstring theories have been shown to exist. Five superstring theories together with M-theory form pairs related by T duality or S duality. Then why are superstring theories related in a way of chain of dualities? T duality is the equivalence under the interchange of R ↔ α ′ /R for the radius of the compactified space R, while S duality is the equivalence under the interchange of g ↔ 1/g for the string coupling g. So they relate paired theories. These T duality and S duality are encompassed by U duality. Therefore we examine whether a theory with manifest U-duality exists which describes different superstring theories by its different sections. We call such a theory with manifest U-duality "F-theory".
The theory with manifest T-duality was presented in [1][2][3] which was named "T-theory" later. "T-theory" was defined by the O(D, D) current algebra which generates gauge symmetries of background gauge fields. It contains winding modes even in an uncompactified space, and the D-dimensional space includes time direction to describe dynamical gravity. All string modes including massive modes are described in the T-theory which has been developed [1][2][3][4][5][6][7][8][9][10][11]. Section conditions eliminate winding modes while the string field theory condition L 0 =L 0 mixes massive winding modes with massive oscillator states [12].
T-duality and S-duality are unified into U-duality by the exceptional symmetry group which involves non-perturbative branes [22]. M-theory was conjectured as a theory to unify superstring theories through dualities whose low energy effective theory is the 11-dimensional supergravity theory [23]. F-theory was firstly proposed by Vafa [24] to understand the IIB theory in the string duality web where similar ideas are also referred [25,26]. The O(D, D) T-duality symmetry is extended to the exceptional symmetry groups U-duality symmetry in generalized geometry for M-theory [27,28] and Exceptional Field Theory (EFT) [29][30][31][32][33][34][35][36][37][38]. Current algebras for branes were calculated to present generalized brackets and to derive background gauge symmetries [39][40][41] corresponding to the U-duality covariant formulation of the 11-dimensional supergravity [29,30]. a generalization of T-theory. F-theory is defined by the exceptional group current algebras on branes. The exceptional group acts both the spacetime coordinates and the worldvolume coordinates. Since the exceptional group includes both the O(D, D) T-duality symmetry and the SL(2;R) S-duality symmetry, F-theory reduces both the IIB theory and M-theory directly by sectioning or dimensional reduction. This solves the puzzle of the IIB theory in the duality web as shown in the duality diamond (2.1) including F-theory.
In this paper we focus on the E 5 =SO(5,5) F-theory. The SO (5,5) current algebra is realized by a 10-brane. The spacetime coordinate is the 16-dimensional spinor representation of SO (5,5) while the worldvolume coordinate is 10-dimensional vector representation of it. 16 is decomposed into 5 + 10 + 1 under GL(5) symmetry where 5, 10, 1 correspond to the 5-dimensional momentum, the M2-brane winding mode, the M5-brane winding mode respectively. This is a generalization of doubling the spacetime coordinate for linear realization of the O(D, D) T-duality symmetry as D momenta plus D winding modes. We propose two different ways of writing the F-theory 10-brane actions: 1. the Hamiltonian form action and 2. Lagrangian form action.
1. The Hamiltonian form action is based on the SO(5,5) "G-symmetry" current algebra.
The Lagrangian is written in terms of the selfdual and anti-selfdual field strengths, • F SD µ and • F SD µ with µ = 1, · · · , 16. The background gauge fields G µν are coset elements of G/H where H is a subgroup of G. The worldvolume index is m = 1, · · · , 10 and γ mµν is the 10-dimensional gamma matrix. We propose the SO(5,5) symmetric F-theory 10-brane action in curved backgrounds in (4.14) as where g,λ and λ m are Lagrange multipliers.
The organization of the paper is as follows. In the next section we present complete sets of the SO(5,5) current algebras in both the SO(5,5) spinor representation in subsection 2.1 and the GL(5) tensor representation in subsection 2.2. The former reveals the structure of the current algebra such as the bosonic κ-symmetry, while the later gives direct coupling to the 11-dimensional supergravity background. One of the author presented the SO(5,5) current algebra of the M5-brane [41] obtained from the Pasti-Sorokin-Tonin(PST) M5-brane Lagrangian [52]. This SO(5,5) current algebra is recognized as the F-theory 10-brane current algebra by doubling the worldvolume coordinate as shown in (3.21), (3.24) and (3.25). In subsection 3.1 we begin by reviewing the double zweibein method to obtain the worldsheet covariant action [10,11] as a method to overcome the problem of the chiral action [53]. By applying this method to F-theory we obtain the SO(5,5) F-theory 10-brane action in the Hamiltonian formalism in subsection 3.2. We extend it to the SO(6,6) F-theory 10-brane action in subsection 3.3. In the SO(6,6) Lagrangian the worldvolume vielbein merges with the spacetime vielbein. In subsection 3.4 we present F-theory 10-brane actions in terms of GL(6) and GL(5) tensors to couple the supergravity background. In section 4 we present an action for a 5-brane obtained from the F-theory 10-brane action. 5 worldvolume dimensions are reduced by solving the worldvolume section constraint V = ∂ m∂ m = 0. The obtained action for a M-theory 5-brane is sum of the free kinetic term and bilinears of the selfduality constraint.

Introduction to F-theory
We begin by an introduction to "theories" with manifest dualities such as T-theory and F-theory together with "S-theory" and "M-theory". S-theory is a string theory compactified to D-dimensions and M-theory is a brane theory compactified to (D + 1)-dimensions. "Theories" are defined by current algebras with G-symmetry in the Hamiltonian formalism. The background gauge fields are parameters of cosets G/H which are generalization of the GL(D)/SO(D − 1,1) for the vielbein gauge field of the Einstein gravity. All bosonic component fields are representation of G, while fermionic fields are representation of H. A new duality web given in the diamond diagram in (2.1) [45].
Theories are also defined by worldvolume actions. The Hamiltonian form action is obtained from the G-symmetry current algebra. The spacetime and the brane worldvolume are representations of G-symmetry. The action is written as bilinear of the field strength F . There is a gauge symmetry generated by the Gauss law constraint with the gauge parameter κ. The spacetime coordinate X σ plays the gauge field while the auxiliary coordinate X τ corresponds to A 0 in the usual gauge theory. The worldvolume coordinate is denoted by ∂ = ∂ ∂σ . We focus on the D=4 E 5 =SO(5,5) G-symmetry case in this paper. Representations of the G-symmetries of theories for the D=4 case are summarized in table 1. The D=4 F-theory is described by the 10-brane in the 16-dimensional spacetime. The 10-dimensional chiral spinor has a bosonic κ-symmetry-like structure similar to the Green-Schwarz super-string. The D=4 F-theory in the lightcone-like gauge fixing reduces to the T-theory in the 4-dimensional spacetime.
The G-symmetry is enlarged to F-symmetry in the Lagrangian formulation where the worldvolume Lorentz covariance is manifest. In the usual gauge theory the G-symmetric field strengths correspond to the nonrelativistic electric field and magnetic field while the Fsymmetric field strength corresponds to the Lorentz covariant field strength. The constraint V = 0 is a worldvolume section condition. The F-symmetries and representations of theories are summarized as in the table 2.    F-theory reduces to T, M, S-theories by reducing the spacetime dimensions or the worldvolume dimensions by the dimensional reduction or the section condition. In this paper we reduce from the F-theory 10-brane to the M-theory 5-brane with preserving the SO(5,5) G-symmetry, then the obtained M-theory 5-brane couple to SO(5,5) background gauge fields.
At first, we present current algebras in the SO(5,5) spinor representation. Next, we present it in the GL(5) tensor representation, where reduction to M-theory is straightforward and coupling to the 5-dimensional subspace of the 11-dimensional supergravity background is manifest.
Under the global SO(5,5) transformation the canonical coordinates are transformed with use of (3.5) in such a way that currents are transformed as a SO (5,5) spinor The 10-dimensional worldvolume σ m -diffeomorphism is generated by the Virasoro constraint S m = 0, and the τ -diffeomorphism is generated by T . Closure of the Virasoro algebra requires secondary constraints, U µ = 0 and V = 0 They satisfy the following algebra where ∂ m and U µ act on δ(σ − σ ′ ). The SO (5,5) indices are raised and lowered by η mn and η mn . It is also noted thatη µρ γ mρλη λν = η mn γ nµν . The set of Virasoro algebras is given by The spacetime coordinate derivative of a function Φ(X) is given by The worldvolume coordinate derivative of a function Φ (X(σ)) is given by (3.14) The SO(5,5) current algebra in curved backgrounds with torsion T αβ γ is given as follows.
The curved space current ⊲ α , the flat space current ⊲ µ and the curved space worldvolume derivative D a , the flat worldvolume derivative ∂ m are related by the spacetime vielbein E α µ and the worldvolume vielbein E m a as (3.16) The spacetime vielbein E α µ ∈ SO(5,5)/SO(5;C) and the worldvolume vielbein relate the curved background indices µ, m and the flat space indices α, a as The gauge transformation of E α µ is given by In curved backgrounds the τ component of Virasoro constraint T = 0 (3.10) is generalized as The spacetime gauge field G µν is parametrized by elements of the coset SO(5,5)/SO(5;C). Spacial components of Virasoro constraints S m in (3.10) is inert in curved backgrounds as (3.20) Background independence of the spacial components of Virasoro constraints makes possible to impose as the section conditions on fields. This is the same property with the T-theory.

GL(5) tensor representation
The SO(5,5) current algebra in the GL(5) tensor representation was obtained in the M5 brane Hamiltonian [41] from the PST action [52]. The M5-brane is a 11-dimensional supergravity solution which is described by the spacetime coordinate x m (σ), m = 0, 1, · · · , 10, the second rank selfdual gauge field A ij (σ), i = 1, · · · , 5 and their canonical conjugates p m (σ), E ij (σ). The currents are the vector, the 2-rank tensor corresponding to the M2 brane charge and the 5-rank tensor corresponding to the M5 brane charge. The τ -diffeomorphism constraint T = 1 2 p m 2 + · · · = 0 is written in bilinear of currents. On the other hand the 5-dimensional worldvolume diffeomorphism constraints are given as Multiplying the pullback matrices ∂ i x m and E ij ∂ j x m on H i = 0 gives bilinears of currents S m = S m 1 ···m 4 = 0 as Virasoro constraints.
In the 5-dimensional subspace where the 11-dimensional space is compactified on a 5dimensional torus the currents combine into the 16 dimensional SO(5,5) spinor representation.
The reducible set of Virasoro constraints S m = S m 1 ···m 4 = 0 become S m =S m = 0 with m = 1, · · · , 5 which is the 5 +5 vector representation of the SO(5,5). All these constraints T , S m andS m satisfy a closed algebra with secondary constraints.
The SO(5,5) current algebra is written in terms of the 16-dimensional spacetime currents ⊲ M = (⊲ m , ⊲ m 1 m 2 ,⊲) as the GL(5) decomposition of 16 component SO(5,5) spinor current 16 → 5 ⊕ 10 ⊕ 1. The commutator of the spinor currents gives the vector, so the worldvolume is 10-dimensional space with ∂ m = (∂ m ,∂ m ) with m = 1, · · · , 10, and m = 1, · · · , 5. The SO(5,5) current algebra in the GL(5) tensor representation is given with the 10-dimensional gamma matrices ρ lM N and ρ lM N based on [41] by The O(5,5) invariant metric η mn is given by Concrete expression of ρ lM N with arbitrary parameters s l = (s l ,s l ) and s l = η lm s m = (s l , s l ) is given by where infinitesimal SO(5,5) matircies are given by The GL (5) tensor coordinate X M = (X m , X m 1 m 2 ,X) and its canonical conjugate P M = (P m , P mam 2 ,P ) are introduced by The selfdual and anti-selfdual currents,⊲ M and⊲ M , together with their current algebras are given by The selfdual currents and their algebra in components are given as The set of Virasoro constraints in (3.10) is rewritten as The Virasoro constraints in components are given by The Virasoro constraint S m generates the shift of the worldvolume coordinate on the current The Virasoro algebra in the GL(5) tensor representation is the same as (3.12) by replacing the γ mµν -matrices with ρ mM N in (3.23) The GL(5) tensor expression of the above relation is as follows.
The Virasoro algebra in the GL(5) tensor expression is given by In order to couple to the 11-dimensional supergravity background the 5-dimensional indices are converted into the 11-dimensional tensor indices as Currents in the GL(5) tensor representation coupled to the 5-dimensional subspace of the 11-dimensional supergravity background are given as e a m e a n C [3] nm 1 m 2 −e a n 1 4! C [3] n[m 1 m 2 C whereη AB is the same matrix asη M N in (3.30).

F-theory 10-brane actions
The F-theory SO(5,5) current algebras (3.8) or (3.27) are realized on the 10-brane worldvolume which we call F10-brane for short from now on. The Hamiltonian is given by linear combinations of a set of Virasoro constraints (3.10) or (3.29) which are written in terms of the selfdual currents. In order to construct the worldvolume covariant Lagrangian we include the ones for the anti-selfdual currents. At first, we review how to construct the worldsheet covariant action by using the double zweibein method [10,11]. Then we propose actions for the F10-brane with both the SO(5,5) symmetric Hamiltonian formulation and the SO(6,6) symmetric Lagrangian formulation.

Double zweibein formulation in T-theory
The physical current in the T-theory is the selfdual current which is chiral in the doubled space. For the doubled space coordinates X = (x, y) the auxiliary coordinate y is introduced with the selfduality condition; the anti-selfdual current is zero ∂ m x−ǫ mn ∂ n y = 0. We impose the selfduality condition as the first class constraint by squaring of it. The action contains both the selfdual current and the anti-selfdual current leading to the worldsheet covariant action. The Virasoro constraints in terms of the selfdual current and the antiselfdual currents are given by 1g −s withē = detē a m and e = det e a m .
The Lagrangian with the usual single worldvolume zweibein e a m is given by It is useful to give the T-theory Lagrangian in terms of the selfdual and the anti-selfdual currents where the anti-selfdual current is the selfduality constraint.
Bilinears of the anti-selfdual currents relax the selfduality constraint • J SD M = 0 as shown in (D.5). It is the selfduality constraint in curved backgrounds g τ m ∂ m X N G N M = ǫ τ m ∂ m X N η N M with g τ τ = 1 g , g τ σ = − s g and G N M =η N M . In this formulation the worldsheet zweibein is not factored out in this Lagrangian.

SO(5,5) Hamiltonian form action
We apply the double zweibein formulation to construct F10-brane actions. The Hamiltonian is sum of the set of Virasoro constraints T = S m = U µ = 0 in (3.10) and Virasoro constraints for the anti-selfdual currentsT =S m = 0 in (4.2) with Lagrange multipliers g, s m , Y µ , g,s m respectively. Y µ plays the role of A 0 in the usual gauge theory. We begin by the following Hamiltonian form action in the SO(5,5) spinor representation The Lagrangian is written in terms of field strengths by using V = 0 constraint. This gauge transformation is generated by the Gauss law constraint as δ κ X µ = [ dσ κ ν U ν , X µ ]. There are gauge symmetries of the gauge symmetry as same as the κ-symmetry δκ = / ∂κ [1] , δκ [1] = / ∂κ [2] , · · · and δκ = / ∂κ [1] , δκ [1] = / ∂κ [2] , · · · . The infinite series of gauge symmetries reduce a half of the coordinates. The 10-dimensional covariant action requires the auxiliary coordinate Y µ , but it is removed by the bosonic κ symmetry in the temporal gauge. In the lightcone-like gauge a half of X µ is removed, giving the (4+4)-dimensional T-theory.
In F-theory currents F ± µ the worldsheet vielbein cannot be extracted from the spacetime vielbein because the SO(5,5) covariant γ-matrix γ mµν mixes the worldvolume index and the spacetime index unlikely to the O(D, D) invariant metric η M N in the T-theory in (4.4). The first term contains both the selfdual and the anti-selfdual field strengths so it is a free kinetic term. Other terms contain only the anti-selfdual field strength, then they are constraints. The F10-brane Lagrangian is further rewritten in terms of the selfdual and the anti-selfdual currents analogously to (4.7) as The selfdual and the anti-selfdual currents in F-theory mixes the worldsheet vielbein and the spacetime vielbein as in (4.7).
In curved background the SO(5,5) gauge fields G µν and the SO(5,5) currents are given with the SO(5,5)/SO(5;C) vielbein E A M as (4.12) The F10-brane Lagrangian in curved background is written as with SO(5,5) vector parameter (s +s) a = (s +s) m E m a , as (3.5).
Then now we propose a Lagrangian for a F10-brane in curved background in terms of the selfdual and the anti-selfdual currents is given as Theλ and λ a are Lagrange multipliers for selfduality constraints given in (4.11).
We propose a F10-brane Lagrangian in a flat space with the worldvolume vielbein by rewriting the SO(5,5) covariant action (4.14) with the SO(6,6) field strength (4.21) with the 32×32 matrix metric The field strength with the worldvolume vielbein in a flat space F α is given by with f +− = lnφ and f −a = −ψ a . The Gauss law constraint is derived from the Lagrangian analogously to the usual gauge theory action. Lagrange multipliers, e, λâb, and the worldvolume vielbein fields φ, ψ a , in the SO(6,6) action (4.24) correspond to Lagrange multipliers of Virasoro constraints for selfdual and anti-selfdual parts, g, s a andg,s a , in the SO(5,5) action (4.13) as with ϕ −1 = (g +g) 2 − (s +s) 2 . They are solved inversely as We finally obtain the SO(6,6) covariant action for the F10-brane in curved backgrounds where λmn and λâb are related by the worldvolume vielbein as (4.26). The SO(6,6) covariant field strength in a curved background F α and the one in a flat space • F µ are given as The SO(6,6) vielbein satisfies the following condition where the SO(5,5) vielbein field is embedded in the SO(6,6) vielbein field as Both the SO(5,5) spacetime vielbein and the worldvolume vielbein combine into the vielbein field of the SO(6,6) F-theory. The SO(6,6) F-theory background is described by the coset SO(6,6)/SO(6;C) with its dimension 36 = 25 + 11. The number of spacetime vielbein, SO(5,5)/SO(5;C) fields is 25, while the number of Virasoro constraints of a F10-brane is 11. The SO(6,6) vielbein is transformed under the SO(6,6) transformation as Λâb ∈ SO(6, 6) , Eμα → Eμβ exp ΛâbΓâb βα , Eâm → ΛâbEbm .

GL(6) field strengths
Gauge transformations Gauge transformations Using with the 16×16 matrix ρ mM N for X M = (X m , X mn ,X) and Y M = (Y m , Y mn , Y ) in (3.5) the field strengths are written as below.
The GL(5) tensors directly couple to the 11-dimensional supergravity background. The GL(5) covariant field strength in curved background F A is related to the one in a flat background • F M with the vielbein field E A M which is a coset element of SO(5,5)/SO(5;C) as well as the one in (4.12) as The Hamiltonian form action given in (4.9) gives the same form with replacing γ mµν with ρ mM N . The GL(5) covariant F10-brane in curved background is given as The GL(5) covariant F10-brane Lagrangian in curved background in terms of the selfdual and the anti-selfdual currents is given by (4.11) as In order to couple to the 11-dimensional supergravity background Y m andF are rewritten in the 5-dimensional dual as The background of SO(5,5) vielbein given in (3.38) is given by the metric e m a and the three form gauge field C [3] m 1 m 2 m 3 in the 5-dimensional space m = 1, · · · , 5. The field strength in the backgrounds are as follows.

Perturbative M-theory 5-brane action
In order to obtain an action for the perturbative M-theory 5-brane coupled to the 11dimensional supergravity background, we preserve the number of the SO(5,5) currents. The worldvolume dimensions of F-theory is reduced by solving V = ∂ m η mn ∂ n = 0 as ∂ m = 1 4! ǫ m 1 ···m 5 ∂ m 2 ···m 5 = 0 consistently. The 5-dimensional worldvolume theory is obtained by the following sectioning [51] The selfdual and the anti-selfdual currents for the M5-brane are given as The SO(5,5) current algebras in (3.28) is reduced to the subalgebra which is the same one for the SL(5) case [40] A set of the Virasoro constraints and the Gauss law constraints in (3.31) and (3.32) are reduced to An action for the M-theory 5-brane in curved background is given from the F-theory 10-brane (4.13) by sectioning the worldvolume into 5 dimensions as Now let us construct the action for the M-theory 5-brane in terms of the selfdual and the anti-selfdual currents where the anti-selfdual currents are auxiliary introduced to make a free kinetic term. The selfdual (SD) and the anti-selfdual (SD) field strengths are given by where the selfdual or the anti-selfdual currents picks up the + or − among ± sign respectively.
The GL(5) covariant field strengths in flat space are given by where ∂ + is replaced with the τ derivative denoted byẊ M . The SO (5,5) background n 3 n 4 n 5 ] e a 1 n 1 · · · e a 5 We propose a perturbative action for a M-theory 5-brane in the curved background in terms of the selfdual and the anti-selfdual currents in (4.42) as follow.
The first term is a free kinetic term for GL(5) tensor fields on a (5+1)-dimensional worldvolume, while the rest is constraints of the anti-selfdual currents.

Conclusions
We have presented the F-theory 10-brane actions with the SO(5,5) U-duality symmety and the SO(6,6) enlarged U-duality symmetry. At first the SO(5,5) current algebras on the 10-dimensional worldvolume are presented in both the SO(5,5) spinor representation and the GL(5) tensor representation. The former reveals the gauge symmetry structure generated by the Gauss law constraint where the 10-dimensional worldvolume translation and the 16-dimensional spacetime current satisfy the bosonic κ-symmetry structure. The latter gives direct coupling to the 11-dimensional supergravity background fields. Next the action of the F-theory 10-brane is obtained by the Legendre transformation of the Hamiltonian constructed by the set of Virasoro constraints. Applying the double zweibein method to F-theory allows to give the 10-dimensional worldvolume covariant actions. Then we have also constructed the F-theory 10-brane action with the SO(6,6) symmetry in the Lagrangian formalism. The worldvolume is enlarged to 12 dimensional brane spacetime, while the target spacetime is enlarged to the 32 dimensional spacetime. The background vielbein represents the coset SO(6,6)/SO(6;C) including both the spacetime background SO(5,5)/SO(5;C) and the worldvolume vielbein.
We have also presented the action for the perturbative M-theory 5-brane in curved spacetime by sectioning the worldvolume of the F-theory 10-brane action. The spacetime is 16 dimensional manifesting the SO(5,5) background coupling. The action is sum of the free kinetic term and the bilinears of the selfduality constraint. So now quantization of the F10brane and the M5-brane is challenging problem. It is also interesting to note that 5-brane is the only object that appears in common in all theories; type I, IIA, IIB superstrings, SO(32), E 8 ×E 8 heterotic superstrings, M-theory, F-theory. In F-theory the 5-brane represents the SL(5) U-duality symmetry [46]. Recently it was shown that current algebras of 5-branes are preserved under the S and T-duality transformations with renaming the spacetime coordinates, where 5-branes include the NS5-brane, the D5-brane, KK5-branes and exotic 5-branes in 32-supersymmetric string theories [54]. 5-brane may give a clue of duality web including 16 supersymmetric string theories.
Many interesting topics are unsolved such as supersymmetric actions of F-theory and M-theory, first quantization of branes and spectrum, amplitudes, and duality web including 16-supersymmetric theories.

A Indices
Indices are summarized as follows.

B Brackets
In the F-theory spacetime the Lie derivative is modified in such a way that it is the SO(5,5) Uduality symmetry covariant. We compute commutators in the SO(5,5) spinor representation which is easier than the GL(5) representation. For vector functions V i µ (X) with i = 1, 2 in the 16-dimensional spacetime a commutator brackets of these vectors is given by The exceptional Courant bracket is given by K = 0 as [12] µ ⊲ µ δ(σ − σ ′ ) while the exceptional Dorfman bracket is given by K = 1 as
We present the current algebras preserving the full tensor indices such as ⊲ m 1 ···m 5 with m = 1, · · · , 5 in order to manifest the 11-dimensional supergravity background. The SO (

D Double vielbein formulation
The double vielbein formulation in the simplest example is explain in this appendix. The selfduality constraint in the T-theory Hamiltonian with a flat worldsheet is the anti-selfduality current is 0: The 2D-dimensional selfdual current (the covariant derivative) and the 2Ddimensional selfduality current (the symmetry generator current) are given by When the 2D-dimensional coordinate X M is written in terms of the D-dimensional coordinates as (x, y) and the canonical conjugates as (p The selfduality constraint is the anti-selfdual current is 0⊲ M = 0 in the usual formulation. By using the selfduality constraint p y = ∂ σ x, the selfdual current reduces into the D-dimensional momenta and the winding modes ⊲ M → (p x , ∂ σ x).
When the Hamiltonian is made from only the selfdual currents, the Hamiltonian form Lagrangian gives a chiral scalar Lagrangian where the term (∂ σ x) 2 is absent.
But adding the squared anti-selfdual current as a constraint with the Lagrange multiplierg leads to the worldsheet covariant action as This is rewritten in terms of the selfdual and the anti-selfdual currents as J SD M = ẋ + g∂ σ ẏ y + g∂ σ x , J SD = ẋ − g∂ σ ẏ y − g∂ σ x . (D.5) The first term in the Lagrangian is the free kinetic term while the second term is the selfduality constraint in a bilinear form. The bilinear form constraint reduces into the anti-selfdual current to be 0, which relates the doubled coordinates x and y as the usual selfduality constraint.