Strings and membranes from A-theory five brane

We thank the referees and the editorial team for their constructive feedback. We have revised the manuscript accordingly and addressed all comments and suggestions. A detailed point-by-point reply to the referees is included as a separate document. We hope that the revised version meets the expectations for publication in SciPost.

======================================== Reply to the referees

We would like to sincerely thank the reviewer for carefully reading our manuscript and providing thoughtful and constructive comments. Please find below the points we have revised for your review.

================================================================================================= Report #1

Weaknesses 1) → changes 1) 2) → changes 2) 3) → typos and grammar errors were corrected.

Requested changes 1) Clarify dimensional dependence and generalization potential:

① Implications of working in low dimensions (D=3) ⇒Page 4, line 10 from the bottom: In the last paragraph of subsection 1.1, the following explanation is added.

In the case of D=3, the theory provides a nontrivial yet tractable example that includes various types of branes and permits explicit computations. The Virasoro algebra, when extended to incorporate brane degrees of freedom, involves Gauss law–type constraints that are intrinsic to the brane. These constraints facilitate the dimensional reduction of the brane world-volume. Such an extended Virasoro algebra serves as a useful prototype for generalization to higher dimensions. Furthermore, the construction of the Lagrangian in this framework constitutes a significant milestone toward extending the formulation to higher-dimensional cases.

② An M5-brane in 4-dimensional space ⇒Page 13, line 8: The following explanation is added.

This M5-brane extends over both the main space (i.e., the duality-covariant space) and the internal space. Four of its world-volume directions lie in the main space while the remaining directions lie in the internal space, specifically one world-volume direction in the Hamiltonian formalism, or two in the Lagrangian formalism. Considering the critical string action in the full spacetime structure is an interesting subject, although it lies beyond the scope of the present discussion. The relationship between the main space and the internal space is schematically illustrated in Figure 2 the ``slug diagram" (see page 27 of arXiv:1610.0162 or page 14 of \cite{Hatsuda:2023dwx}). In the case of D = 3 the main space coordinate is represented by a bispinor $X^{\alpha\beta}$, and the world-volume coordinate by an antisymmetric bispinor $\sigma^{[\alpha\beta]}$, with $\alpha = 1, \dots, 4$. The internal space coordinate is given by a bispinor $Y^{[\alpha'\beta']}$, where $\alpha' = 1, \dots, 8$. The total number of supersymmetries is 32, which corresponds to the product of the dimensions of the spinor indices $32 = 4 \times 8$.

It is noted that the assignment of the duality symmetric space in ${\cal A}$-theory differs from that in conventional formulations. In ${\cal A}$-theory, the duality-symmetric space is assigned to the main "spacetime" rather than the internal space, such that all tensor gauge fields are automatically incorporated into the coset parameter of $\mathrm{E}_{\mathrm{D+1}} / H$.

③ Enhancement from SL(5) to SL(6) for higher dimenisons ⇒Page 5, line 10 from the bottom: In the third paragraph of subsection 1.2, the following explanation is added.

In general, the symmetry of Lagrangian formulation is larger than that of the corresponding Hamiltonian formulation. In ${\cal A}$-theory, the U-duality symmetry in the Hamiltonian formulation, G-symmetry, is enhanced to a novel duality symmetry in the Lagrangian formulation, A-symmetry. This symmetry enhancement in higher-dimensional cases (D < 6) is summarized on page 6 of arXiv:1806.02423 and page 14 of arXiv:2307.04934. It was shown that the brane world-volume metric is also transformed conformally under the SL(5) duality transformation as well as the spacetime background fields in \cite{Duff and Lu (1990) and Duff et al. in 1509.02915}. This mixing between spacetime and world-volume is a manifestation of the extended SL(6) duality symmetry transformation.

④ Relevant literature discussing enhanced dualities in low-dimensional brane ⇒Page 5, line 11 from the bottom: In the third paragraph of subsection 1.2, these references are cited.

Duff and Lu \cite{Duff Lu 1990, Duff et al 1509.02915} showed that the membrane theory exhibits the SL(5) duality symmetry by the Gaillard-Zumino approach.

2) Discuss relation to existing DFT/EFT approaches: ⇒Page 4, line 18: At the second-to-last paragraph of subsection 1.1, the following explanation is added.

The generalized diffeomorphism in EFT is characterized by the "Y-tensor", which reflects the structure of the exceptional group. This Y-tensor, $Y^{MN}{}_{PQ}$, is related to the group-invariant metric in ${\cal A}$-theory, denoted by $\eta^{MNm}$, through the relation $Y^{MN}{}{PQ} = \eta^{MNm} \eta$ where $M$ is the spacetime index and $m$ is the world-volume index. These indices correspond to different representations of the exceptional group. The origin of the Y-tensor lies in the Schwinger term of the current algebra, ${\dd_N(\sigma),\dd_L(\sigma')}\sim \eta_{NLm}\partial^m \delta(\sigma-\sigma')$, where the world-volume derivative $\partial^m$ is defined through the commutator with the Virasoro constraint, ${\cal S}^m\sim \dd_N\eta^{NLm}\dd_L$.
The section condition given by the Y-tensor, $Y^{MN}{}_{PQ} \partial_M \partial_N = 0$, is related to the zero-mode condition of the Virasoro constraint in ${\cal A}$-theory. Specifically, the zero-mode component of the constraint ${\cal S}^m$ takes the form ${\cal S}^m|_{\rm 0\text{-}modes} = \eta^{MNm} \partial_M \partial_N = 0,$ establishing a direct connection between the section condition in EFT and the Virasoro structure of ${\cal A}$-theory.

================================================================================================= Report #2 Weakness 1) Derivation of "non-perturbative" M2-brane from "perturbative" M5-brane: ⇒Page 6, line 14: In the last paragraph of subsection 1.2, the following explanation is added.

The perturbative" M5-brane Lagrangian is formulated as a bilinear expression in terms of currents, while thenon-perturbative" M2-brane Lagrangian comprises the sum of the Nambu–Goto and Wess–Zumino terms, each exhibiting multilinear dependence on the spacetime coordinates. The dimensional reduction from the M5-brane to the M2-brane is implemented via the ``non-perturbative projection" $ \partial^m = \epsilon^{ij} \partial_j x^m \partial_i, $ in \bref{stwvMix} and the gauge fixing of the world-volume metric in \bref{gaugechoicewv}.

2) English language ⇒ We have revised the English to the best of our ability.

3) Notation ⇒We acknowledge that denoting the stringy covariant derivative with a tilted symbol $\tilde{\nabla}$ may be somewhat unconventional. We kindly ask for the reader’s understanding due to the limitation of available notation.

List of changes

1. Page 4, line 10 from the bottom: In the last paragraph of subsection 1.1, the following explanation is added.

In the case of D=3, the theory provides a nontrivial yet tractable example that includes various types of branes and permits explicit computations. The Virasoro algebra, when extended to incorporate brane degrees of freedom, involves Gauss law–type constraints that are intrinsic to the brane. These constraints facilitate the dimensional reduction of the brane world-volume. Such an extended Virasoro algebra serves as a useful prototype for generalization to higher dimensions. Furthermore, the construction of the Lagrangian in this framework constitutes a significant milestone toward extending the formulation to higher-dimensional cases.

1. Page 4, line 18: At the second-to-last paragraph of subsection 1.1, the following explanation is added.

The generalized diffeomorphism in EFT is characterized by the "Y-tensor", which reflects the structure of the exceptional group. This Y-tensor, $Y^{MN}{}_{PQ}$, is related to the group-invariant metric in ${\cal A}$-theory, denoted by $\eta^{MNm}$, through the relation $Y^{MN}{}{PQ} = \eta^{MNm} \eta$ where $M$ is the spacetime index and $m$ is the world-volume index. These indices correspond to different representations of the exceptional group. The origin of the Y-tensor lies in the Schwinger term of the current algebra, ${\dd_N(\sigma),\dd_L(\sigma')}\sim \eta_{NLm}\partial^m \delta(\sigma-\sigma')$, where the world-volume derivative $\partial^m$ is defined through the commutator with the Virasoro constraint, ${\cal S}^m\sim \dd_N\eta^{NLm}\dd_L$.
The section condition given by the Y-tensor, $Y^{MN}{}_{PQ} \partial_M \partial_N = 0$, is related to the zero-mode condition of the Virasoro constraint in ${\cal A}$-theory. Specifically, the zero-mode component of the constraint ${\cal S}^m$ takes the form ${\cal S}^m|_{\rm 0\text{-}modes} = \eta^{MNm} \partial_M \partial_N = 0,$ establishing a direct connection between the section condition in EFT and the Virasoro structure of ${\cal A}$-theory.

1. Page 5, line 10 from the bottom: In the third paragraph of subsection 1.2, the following explanation is added.

In general, the symmetry of Lagrangian formulation is larger than that of the corresponding Hamiltonian formulation. In ${\cal A}$-theory, the U-duality symmetry in the Hamiltonian formulation, G-symmetry, is enhanced to a novel duality symmetry in the Lagrangian formulation, A-symmetry. This symmetry enhancement in higher-dimensional cases (D < 6) is summarized on page 6 of arXiv:1806.02423 and page 14 of arXiv:2307.04934. It was shown that the brane world-volume metric is also transformed conformally under the SL(5) duality transformation as well as the spacetime background fields in \cite{Duff and Lu (1990) and Duff et al. in 1509.02915}. This mixing between spacetime and world-volume is a manifestation of the extended SL(6) duality symmetry transformation.

1. Page 5, line 11 from the bottom: In the third paragraph of subsection 1.2, these references are cited.

Duff and Lu \cite{Duff Lu 1990, Duff et al 1509.02915} showed that the membrane theory exhibits the SL(5) duality symmetry by the Gaillard-Zumino approach.

1. Page 6, line 14: In the last paragraph of subsection 1.2, the following explanation is added.

The perturbative" M5-brane Lagrangian is formulated as a bilinear expression in terms of currents, while thenon-perturbative" M2-brane Lagrangian comprises the sum of the Nambu–Goto and Wess–Zumino terms, each exhibiting multilinear dependence on the spacetime coordinates. The dimensional reduction from the M5-brane to the M2-brane is implemented via the ``non-perturbative projection" $ \partial^m = \epsilon^{ij} \partial_j x^m \partial_i, $ in \bref{stwvMix} and the gauge fixing of the world-volume metric in \bref{gaugechoicewv}.

1. Page 13, line 8: The following explanation is added.

This M5-brane extends over both the main space (i.e., the duality-covariant space) and the internal space. Four of its world-volume directions lie in the main space while the remaining directions lie in the internal space, specifically one world-volume direction in the Hamiltonian formalism, or two in the Lagrangian formalism. Considering the critical string action in the full spacetime structure is an interesting subject, although it lies beyond the scope of the present discussion. The relationship between the main space and the internal space is schematically illustrated in Figure 2 the ``slug diagram" (see page 27 of arXiv:1610.0162 or page 14 of \cite{Hatsuda:2023dwx}). In the case of D = 3 the main space coordinate is represented by a bispinor $X^{\alpha\beta}$, and the world-volume coordinate by an antisymmetric bispinor $\sigma^{[\alpha\beta]}$, with $\alpha = 1, \dots, 4$. The internal space coordinate is given by a bispinor $Y^{[\alpha'\beta']}$, where $\alpha' = 1, \dots, 8$. The total number of supersymmetries is 32, which corresponds to the product of the dimensions of the spinor indices $32 = 4 \times 8$.

It is noted that the assignment of the duality symmetric space in ${\cal A}$-theory differs from that in conventional formulations. In ${\cal A}$-theory, the duality-symmetric space is assigned to the main "spacetime" rather than the internal space, such that all tensor gauge fields are automatically incorporated into the coset parameter of $\mathrm{E}_{\mathrm{D+1}} / H$.

1. We have corrected typos and grammatical errors as much as possible.