U-duality transformation of membrane on Tn revisited

The problem with the U-duality transformation of membrane on Tn is recently addressed in [arXiv:1509.02915]. We will consider the U-duality transformation rule of membrane on Tn ×R. It turns out that winding modes on Tn should be taken into account, since the duality transformation may bring the membrane configuration without winding modes into the one with winding modes. With the winding modes added, the membrane worldvolume theory in lightcone gauge is equivalent to the n + 1 dimensional super-Yang-Mills (SYM) theory in T˜n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\tilde{T}}^n $$\end{document}, which has SL(2, Z) × SL(3, Z) and SL(5, Z) symmetries for n = 3 and n = 4, respectively. The SL(2, Z) × SL(3, Z) transformation can be realized classically, making the on-shell field configurations transformed into each other. However, the SL(5, Z) symmetry may only be realized at the quantum level, since the classical 5d SYM field configurations cannot form the representation of SL(5, Z).


Introduction
It is well-known that string theory compactified on T n has the O(n, n; Z) T-duality symmetry, which is manifested in the classical equations of motion for both background fields and the 2d sigma model. In [1], it has been shown that string T-duality can be realized on the string worldsheet as a rotation of field equations into Bianchi identities. M theory compactified on T n also has the U-duality symmetry, which, when n = 3 and n = 4, are SL(2, Z)×SL(3, Z) and SL(5, Z) respectively. Equations of motion for the 11d supergravity are U-duality invariant. It remains to find the U-duality transformation rule of membrane on the given supergravity background. A natural generalization of [1] in the membrane case is to realize the U-duality transformation as the rotation of field equations into Bianchi identities on the 3 dimensional worldvolume [2]. However, the approach has some difficulties as discussed recently in [3]. Concretely, when n = 3, the SL(2, R)×SL(3, R) transformation is well-defined, but for n = 4, only a subgroup GL(4, R) × R 4 can be realized.
Nevertheless, because M theory compactified on T n is U-duality symmetric, there must be a U-duality transformation rule for membrane. In this note, we will reconsider this problem. Different from the approaches in [2,3], where the worldvolume theory of membrane is covariant, we will impose the lightcone gauge, and then, after the discrete regularization, membrane worldvolume theory becomes U(∞) Matrix model. Another difference is that in [2,3], membrane worldvolume, including the time direction, is totally embedded in T n , but in our discussion, membrane lives in T n × R with the time direction identified as R. As a result, some identities in [3] are not valid here. For example, membrane in T 3 is a topological object with the winding current proportional to the momentum current due to some algebraic relations [3]. On the other hand, membrane in T 3 × R carries dynamical degrees of freedom, while the winding current and the momentum current are independent.

JHEP08(2016)138
Branes in compact space contain more degrees of freedom than branes in non-compact space, coming from the winding modes on non-trivial 1-cycles. For example, the worldvolume theory of a single D2 in 10d non-compact spacetime is the 3 dimensional U(1) super-Yang-Mills (SYM) theory. If n transverse dimensions are compactified to T n , open string winding modes along T n should also be added so that the theory is equivalent to the (3 + n) dimensional U(1) SYM theory. Duality transformation usually makes branes without winding modes transformed into branes with winding modes. As an instance, let us consider the type IIA theory compactified on T 4 ∼ x 1 × x 2 × x 3 × x 4 and a D2 wrapping x 1 ×x 2 with no winding modes. If D2 is not translation invariant along x 1 and x 2 , after two successive T-duality transformations along x 1 and x 2 , it will become a D0 with winding modes along x 1 × x 2 included. The further T-duality transformations along x 3 and x 4 give D2, which wraps x 3 × x 4 and is translation invariant along x 3 and x 4 , still including the winding modes along x 1 × x 2 . Similarly, we may expect that for membrane in T n , winding modes should also be added and the U-duality transformation may make a configuration without winding modes transform into the one with winding modes.
Membrane in T n × R in lightcone gauge with all winding modes included is equivalent to the (n + 1)-dimensional U(∞) SYM theory in the dualT n × R. Without the winding modes, membrane configuration is mapped to the 0-mode of the SYM field. We will consider the SL(2, Z) × SL(3, Z) and SL(5, Z) transformations for the 4d and 5d SYM theories, respectively. It turns out that the duality transformation does not always convert the 0-mode into the 0-mode, so starting from the membrane configuration without winding modes, the U-dual configuration may involve winding modes. For the SL(2, Z) transformation of the 4d YM fields, based on the loop space formulation, [13,14] gives a prescription converting the classical on-shell fields into each other. On the other hand, the SL(5, Z) transformation of the 5d SYM fields cannot always be realized classically. The 5d SYM theory in T 4 × R, with all instanton configurations taken into account, could be taken as the 6d (2, 0) theory in T 5 × R, which is supposed to be SL(5, Z) symmetric. Each 5d SYM field configuration carries the definite instanton number, or in other words, the definite P 5 momentum, and thus could not form the representation of SL(5, Z), except for the SL(4, Z) subgroup on T 4 . Nevertheless, the SL(5, Z) symmetry may be realized at the quantum level, for example, the partition function of the 5d SYM theory on T 4 × R may be SL(5, Z) invariant, although the explicit verification is still lacking.
The rest of the paper is organized as follows. In section 2, we review the SO(n, n) transformation for strings on T n , based on [1]. In section 3, we comment on the problems for the U-duality transformation of membrane on T n , following [3]. In section 4, we review the matrix theory description of the membrane on T n × R. In section 5, we discuss the U-duality transformation of membrane on T 3 × R. In section 6, we consider the U-duality transformation of membrane on T 4 × R. The conclusion is in section 7.
2 SO(n, n) transformation for string on T n In [1], it was pointed out that string T-duality originates from transforming field equations into Bianchi identities on the string worldsheet. Consider the worldsheet theory of the JHEP08(2016)138 string on T n with the constant background fields g µν and b µν , the equations of motion and the Bianchi identity can be interpreted as the conservation equations for the currents p i µ and j iµ : where γ ij = ∂ i x µ ∂ j x ν g µν , i = 0, 1, µ = 1, 2, · · · , n, and Integrations of p i µ and j iµ give P µ and J µ , the momentum and winding number of the string in µ direction. P µ and J µ are integers. Under the SO(n, n; Z) T-duality transformation Λ, (P, J) transforms as Although the exact symmetry is SO(n, n; Z), one can nevertheless first consider the continuous group SO(n, n; R), under which the infinitesimal transformation of (P, J) is A stronger requirement is that the current (p i µ , j iν ) transforms in the same way as the charge (P µ , J ν ): The background fields g µν and b µν can be compactly written as a 2n × 2n metric G M N The SO(n, n) transformation rule for G M N is G → G ′ = ΛGΛ T , which gives From (2.6) and (2.8), δ∂ i x µ is determined to be With the equations of motion imposed, ǫ ij ∂ i δ∂ j x µ = 0, so (2.9) can be integrated to give some δx µ . δx µ together with (2.8) composes the SO(n, n; R) transformation rule with the momentum and winding number current (p i µ , j iν ) transforming as (2.6).

JHEP08(2016)138
3 Problem for the U-duality transformation of membrane on T n Similar to the string situation, it is naturally expected that the M theory U-dualities originate from transforming field equations into Bianchi identities on the membrane worldvolume [2]. However, unless the target space has dimension D = p + 1, there is a problem that is identified as the non-integrability of the U-duality transformation assigned to the pull-back map [3].
For the worldvolume theory of membrane on T n with the constant background fields g µν and b µνρ , the equations of motion and the Bianchi identity are (3.1) can be interpreted as the conservation laws of the current p i µ and j iµν , and the integration of which gives the momentum P µ and membrane wrapping number J µν , where i = 0, 1, 2, µ = 1, 2, · · · , n. P µ and J µν are also integers.
For n = 3, there is the identity also note that p i µ and j i µ can be simplified as which can be satisfied when

JHEP08(2016)138
For n = 4, the U-duality symmetry of M theory is SL(5, Z). Let us define K µ5 ≡ P µ , K µν ≡ 1 2 ǫ µνρσ J ρσ , K ≡ (K µ5 , K µν ) forms a 5 × 5 antisymmetric matrix, which, under the action of Λ ∈ SL(5, Z), transforms as (3.12) If one requires the worldvolume Lorentz invariance, the current (p i µ , j iνρ ) will transform in the same way as the charge (P µ , J νρ ), and if the SL(5, Z) symmetry is relaxed to SL(5, R), the infinitesimal transformation of (p i µ , j iνρ ) will be g µν and b µνρ can also be assembled into a symmetric 5×5 matrix G M N with M, N = 1,· · ·, 5, From (3.13), (3.15) and the constraint γ ij = ∂ i x µ ∂ j x ν g µν , δ∂ i x µ is determined to be (3.17) ǫ ijk ∂ j δ∂ k x µ = 0 does not necessarily hold when C αβγ = 0, even if the equations of motion is imposed. This just indicates that we cannot find a δx µ with (p i µ , j iνρ ) transforming as (3.13).
A possible reason is that maybe (3.13) is a too strong requirement. We may only need the charge K M N transforming as a 5×5 antisymmetric matrix like that in (3.12). In fact, we do have examples for which, the U-duality transformation is implemented respecting (3.12) but violating (3.13). Consider the membrane configuration in T 4 corresponding to string in T 3 . Membrane should wrap one direction, for example, x 4 in T 4 . Decompose the membrane worldvolume coordinate as ξ i = (ξî, ξ 2 ),î = 0, 1 and the target-space coordinate as x µ = (xμ, x 4 ),μ = 1, 2, 3. One can let ξ 2 = x 4 so that ∂îx 4 = 0, ∂ 2 x 4 = 1, and also JHEP08(2016)138 suppose ∂ 2 xμ = 0. For simplicity, assume g 4μ = bμνρ = 0. One can prove that the transformation is integrable on-shell. j 3μν does not transform as in (3.13) but (3.12) is respected. So for such kind of the particular membrane configurations, the symmetry generated by C µν4 can indeed be realized at the price of the violation of (3.13). Compared with (2.9), C µν4 ∼ B µν generates part of the SO(3, 3) T-duality transformation for string in T 3 . Besides, there are several subtleties in the U-duality transformation of the membrane. Consider M theory on T 4 and then make a dimensional reduction along x 4 to get type IIA in T 3 . The SL(5, Z) U-duality also has its manifestation in type IIA. For example, M2 without wrapping T 4 becomes D2 without wrapping T 3 , which is dual to D4 wrapping 12 or 13 or 23, or equivalently, M5 wrapping 124 or 134 or 234. In this case, M2 is U-dual to M5. M2 wrapping 23 is also D2 wrapping 23, which, after the T-duality transformation along 2 becomes D1 winding 3 with all winding modes along 2 included, if the original M2 configuration is not translation invariant along the x 2 direction. A further T-duality transformation along 1 gives D2 wrapping 13, or equivalently, M2 wrapping 13, which is translation invariant along 1 but still with all winding modes along 2 included. Obviously, the appearance of M5 and the infinite number of the winding modes after the U-duality transformation cannot be realized in the worldvolume transformation of the membrane.
It is well-known that matrix theory [4], which is supposed to be the non-perturbative description of M theory, is U-duality invariant [5][6][7][8][9]. On the other hand, matrix theory can also arise as the discrete regularization of the membrane theory in lightcone gauge [10], so U-duality transformation rule of matrix theory can also be translated as the U-duality transformation rule of membrane. Moreover, matrix theory description is also complete enough to incorporate the M5 brane as well as the winding modes.
In the following, we will consider the membrane worldvolume theory in lightcone gauge, which is equivalent to the matrix theory. We will discuss the U-duality transformation rule for matrix theory in T n × R, and then reinterpret it as the U-duality transformation rule of the membrane theory in T n × R. This is a little different from the approaches in [2,3], where membrane is the instanton in T n . As a result, some conclusions in [2,3] do not hold here. For example, membrane in T 3 is topological with the identity (3.8) holds, while the membrane in T 3 × R is dynamical with the momentum density and the winding number density independent. 4 A review of the matrix theory description of the membrane in T n × R Let us consider (the bosonic part of) the supermembrane action with the background fields g µν and b µνλ , µ = 0, 1, · · · , 10 [11].
Suppose the membrane world-volume is of the form Σ × R, where Σ is a Riemann surface of the fixed topology and R is the time direction σ 0 . In lightcone gauge, with the gauge fixing γ 0a = 0, (4.1) is equivalent to where In temporal gauge, we have ω = 0 together with the constraint The equation of motion isẍ With the matrix regularization [10] x(t, σ 1 , where ν/N = κ. (4.10) is the matrix theory action on the background g ij and b ijk with X i the N × N matrix, N = ∞. The extension to the supersymmetric case is straightforward. (4.10) is also the low energy effective action for N D0-branes. When a particular transverse dimension xî is compactified to S 1 with the radius Rî, X i for i = 1, 2, · · · , 9 should be replaced by the infinite block matrix X i mn with constraints [12] Xî nn = 2πRî + Xî (n−1)(n−1) = 2nπRî + Xî 0 .
where i∂î = diag(· · · , −4πRî, −2πRî, 0, 2πRî, 4πRî, · · · ). After the successive n times T duality transformations, D0 branes with n transverse dimension compactified to Rî becomes Dn branes with n longitudinal dimensions compactified tõ Rî,î = 1, · · · , n. (4.10) becomes (the bosonic part of) the action of the (n + 1)-dimensional SYM theory on the dualT n in temporal gauge with A 0 = 0. The winding modes on T n are converted into the momentum modes of the gauge fields on the dualT n . In the following, we will consider two particular situations with n = 3 and n = 4.
The original membrane configuration xî(t, σ 1 , σ 2 ) corresponds to Xî(t), which is the zero mode Aî 0 (t) of the YM fields onT 3 . For the zero mode, while the momentum becomes (5.7) In (5.6) and (5.7), Aî is the 0-mode of the gauge field. If Aî is the N × N matrix with N finite, there will be Wîĵ = 0. In matrix theory, the non-trivial wrapping number can be produced due to N = ∞.

JHEP08(2016)138
which is also the string in type IIA theory compactified on T 2 with x 3 the M theory direction. U-duality transformation (5.30) can be realized via two successive O(2, 2; Z) T duality transformations together with a SL(2, Z) transformation on T 2 . For the generic on-shell non-abelian gauge field A I , the S-dual field strength cannot satisfyF IJ = −ǫ IJKL F KL /2, since the related gauge fieldÃ I may not exist. The key point for the S transformation is that the equations of motion for F IJ can be reinterpreted as the integrable condition for the dual field strengthF IJ , and vice versa. Based on the loop space formulation, [13,14] give a prescription to get the S-dual for the generic non-abelian gauge field. Consider the loops passing through a fixed reference point ξ 0 For each loop, one may define a path-ordered phase factors (Wilson loops) The derivatives in loop space can be defined as can be introduced as follows (5.40) or more concretely,

JHEP08(2016)138
As is required, the dual transformation is reversible apart from a sign, i.e.,Ẽ = −E. For N = 1, (5.41) gives the abelian S-dualF IJ = − 1 2 ǫ IJKL F KL . From (5.40), one can prove which is the non-abelian extension of Taking the trace on both sides of (5.41), we get Thus, although the relation between F IJ and its S-dualF IJ may be complicated, their traces respect the simple hodge dual relation. For the 0-mode, tr F 0î and 1 2 ǫîĵk tr Fĵk are the membrane momentum and wrapping number respectively. If the original field A and dual fieldÃ are both 0-modes, since H IJ5 = ǫ IJ5KL4 H KL4 /2, or equivalently,F IJ = −ǫ IJKL F KL /2. In the non-abelian case, the loop space formulation may give the dualF IJ , which could be taken as H IJ5 in 6d.
In this sense, [13,14] also give a generalization of the self-dual relation (5.48) to the nonabelian case when the 6d theory is reduced on T 2 . It is the 0-mode A I (t) that is mapped to the membrane configuration. It remains to see whether the SL(2, Z) × SL(3, Z) transformation for N = 4 SYM theory could map A I (t) to A ′ I (t) which is also constant in space. For the SL(3, Z) transformation, the answer is obviously yes, but for the SL(2, Z) transformation, there are counter-examples in U(1) case. Consider the on-shell gauge field

JHEP08(2016)138
The dual field strength is for which, the related Aî(x, t) cannot be constant in space. In fact, S-transformation in 6d theory just makes (5.52) B 4î and B 5î are not independent but are related via the self-duality relation. For B 4î /B 5î that is constant in space, there is no reason to expect that B 5î /B 4î is also constant. Especially, in the U(1) case, the self-duality relation requires Membrane on T n contains more degrees of freedom than that in non-compact space, coming from the winding modes on T n . Without the winding modes included, the membrane configuration corresponds to the constant gauge field of the SYM theory inT n . With the winding degrees of freedom added, membrane is then related to the generic gauge field inT n . When n = 3, there is a SL(2, Z) × SL(3, Z) transformation, mapping the given A I (x, t) to A ′ I (x, t). If the 0-mode subspace is invariant under the U-duality transformation, i.e. A I (t) → A ′ I (t), there will be a well-defined U-duality transformation for membrane without the need of involving the winding modes. However, this is not quite likely to be the case and the U-duality transformation can only be complete with the winding modes also taken into account.

JHEP08(2016)138
Moreover, D3 coupling with the type IIB supergravity is S duality invariant, so N = 4 SYM theory in type IIB background should also be SL(2, Z) invariant.
The situation becomes complicated since the background fields may have the dependence on the six transverse coordinates xī, or in SYM theory, the six scalar fields Xī. Multiple D branes in curved background was discussed in [18]. For the term like T r[gīj(Xk)(DîXīDîXj)] with gīj depending on xk, one can make a Taylor expansion T r[gīj(Xk)(DîXīDîXj)] = ∞ n=0 1 n! ∂k 1 · · · ∂k n gīj(0)ST r[DîXīDîXjXk 1 · · · Xk n ] , (5.55) giving rise to an infinite number of the higher dimensional operators in the Lagrangian. We will not consider this generic situation here, since according to the counterexample given in [3], even for the constant background, only a two-parameter subgroup of SL(2) can be realized in T 3 × R 8 . We will focus on the simplest situation: N = 4 SYM theory in 10d flat spacetime with the constant axion-dilaton field τ = b + ie −Φ . The remaining problem is to find a S transformation for SYM fields, extending the S transformation (5.41) for YM fields. In [19], with the 4d spacetime extended to the N = 1 superspace, the loop space formulation and the S transformation for YM theory is extended to N = 1 SYM theory. The further extension to the N = 4 supersymmetrty is straightforward. However, just as the YM case, nothing could guarantee that the zero mode of the N = 4 SYM field would necessarily be mapped into the zero mode under the S transformation. So the SL(2, Z) duality cannot always be realized at the membrane worldvolume level, in agreement with [3,16,17]. Nevertheless, with the winding modes on T 3 taken into account, membrane worldvolume theory on T 3 × R 8 is equivalent to the N = 4 SYM theory onT 3 × R 7 , for which, the SL(2, Z) × SL(3, Z) symmetry is definite.

JHEP08(2016)138
Let I = 0, 1, 2, 3, 4 and set Xī = 0 for simplicity, the equations of motion and the Bianchi identity for YM field onT 4 are The discussion is parallel to the T 3 situation. N = 2 SYM theory is the 6d (2, 0) theory reduced along x 5 . For simplicity, let bîĵk = 0, gkĵ = δkĵ. A I ≡ B I5 , F IJ ≡ H IJ5 . H 0ĩj ≡ (H 0î5 , H 0îĵ ) forms a 5 × 5 antisymmetric matrix. In the abelian case, according to the self-duality relation (5.48), H 0îĵ is determined from Hîĵ 5 via H 0îĵ = ǫ 0îĵkl5 Hkl 5 /2, or equivalently, In the non-abelian case, H 0îĵ and Hîĵ 5 may still be related by some self-duality relation, whose form is unknown at present. One possibility is that we can still use the prescription in [13,14] but with the indices restricted to the 4d space (6.9) (6.9) reduces to (6.8) in the abelian case as is required. Recall that (5.41) is supposed to determine H 5ij /H 4ij from H 4ij /H 5ij when the 6d theory is translation invariant along x 4 and x 5 , i = 0, 1, 2, 3. The similar relation (6.9) may be able to determine H 0îĵ from H 5îĵ when the 6d theory is translation invariant along x 0 and x 5 ,î = 1, 2, 3, 4. So, (6.9) is valid at least for the time-independent system. For the 0-mode, tr H 0î5 = Pî and tr Hîĵ 5 = Wîĵ correspond to the membrane momentum and the wrapping number respectively. H 0îĵ determined from (6.9) satisfies the relation tr H 0îĵ = ǫîĵkl tr Hkl 5 /2. So if (6.9) is valid, tr H 0ĩj ≡ (Pî, ǫîĵklWkl/2) forms a 5 × 5 antisymmetric matrix.

JHEP08(2016)138
However, for the 5d SYM theory, (6.10) is not well-defined due to a subtlety in x 5 direction: fields are translation invariant along x 5 but not necessarily so in 1234 space. In fact, gauge field configurations in N = 2 SYM theory are classified by the instanton number n = − 1 8 d 4 x tr(ǫîĵklFîĵFkl) . (6.11) The momentum in the 5 th direction is P 5 = n/R. All fields are translation invariant along x 5 carrying the definite P 5 momentum and then could not form the representation of SL(5, Z) in the sense of (6.10). Nevertheless, SL(5, Z) symmetry may have the manifestation at the quantum level. Partition function of the N = 2 SYM theory on T 4 , with the contribution from the instanton configurations included, may have the SL(5, Z) invariance, although the explicit calculation is still lacking.
We are especially interested with the 0-mode A I (t) that could be mapped to the membrane configuration. In this case, field strength F IJ ≡ H IJ5 is constant in space. It is reasonable to expect that H IJK , which is determined by H IJ5 , is constant as well. Also suppose the instanton number n = 0, then (6.10) is well-defined and becomes H 0ĩj (t) → Ul i Um j H 0lm (t) . (6.12) However, the SL(5, Z) transformation does not always convert the 0-mode A I (t) to another 0-mode A ′ I (t), which is already manifested in U(1) situation. Consider the on-shell gauge field A 0 (t) = 0 , Aî(t) = Kît (6.13) with H 0î5 (t) = Kî , H 0îĵ (t) = 1 2 ǫ 0îĵkl5 Hkl 5 = 0 . (6.14) The SL(5, Z) transformed field strength (H ′ 0î5 (t), H ′ 0îĵ (t)) may have H ′ 0îĵ (t) = 0, which, cannot be obtained from the space independent A ′ i (t). Intuitively, the SL(5, Z) transformation on Bĩj acts as Bĩj(xk, t) → Ul i Um j Blm(U ñ k xñ, t) . (6.15) The components in Bĩj are not independent but are related via the self-duality relation. For B I5 (t) ≡ A I (t) that is constant in space, B IJ is not guaranteed to be constant and so, the SL(5, Z) transformation does not always map the 0-mode to 0-mode. Moreover, the transformed A ′ I (t) may even be x 5 -dependent, making the SL(5, Z) transformation ill-defined. Nevertheless, for some special 0-mode configurations like (5.22) and (5.29), SL(5, Z) transformation is well defined, making the 0-mode converted into the 0-mode and M ∼ (Pî, ǫîĵklWkl/2) transformed into U M U −1 for U ∈ SL(5, Z).
When mapped to the membrane, we may conclude that for the generic membrane configuration, SL(5, Z) symmetry may only be realized at the quantum level with winding modes on T 4 also taken into account.

JHEP08(2016)138 7 Conclusion
We have studied the U-duality transformation of membrane in T n × R. In lightcone gauge, membrane worldvolume theory is equivalent to the matrix theory, which, with the winding modes on T n taken into account, becomes (n + 1)-dimensional SYM theory. The original membrane configuration is mapped to the 0-mode of the SYM field. When n = 3 and n = 4, 4d SYM theory and 5d SYM theory are SL(2, Z) × SL(3, Z) and SL(5, Z) U-duality symmetric. However, the U-duality transformation does not always bring the 0-mode into 0-mode, so the truncation to the 0-mode subspace is not valid. In membrane's respect, this means the duality transformation may convert the configuration without the winding mode into the one with the winding mode. The SL(2, Z) × SL(3, Z) transformation in 4d SYM theory can be realized classically, making the on-shell SYM fields transformed into each other. On the other hand, the SL(5, Z) symmetry in 5d SYM theory may only be realized at the quantum level.