# Romans Massive QP Manifolds

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. QP Manifolds for IIA Branes

#### 2.1. D2 ($P=3$ Algebroid)

**Remark 1.**

- 1
- For ${J}_{1}=0$, the identities arising from ${Q}^{2}=0$ correspond precisely to Bianchi identities and field equations for the fluxes of massive type IIA with the conventions of [6].
- 2
- The field ${J}_{1}$ can naturally be viewed as a gauge connection for a ${\mathbb{R}}^{+}$ gauge transformation generated by rescalings $\xi \to {e}^{\lambda}\xi $, $\varphi \to {e}^{-\lambda}\varphi $. For these transformations to still satisfy (3) then we also need to shift ${p}_{\mu}\to {p}_{\mu}-{\partial}_{\mu}\lambda \xi \varphi $. Putting this into (5), we see that this has the effect of transforming ${J}_{1}\to {J}_{1}+\mathrm{d}\lambda $. The constraint (11) then just says that this is a flat connection. Flat connections on ${\mathbb{R}}^{+}$ gauge bundles are always gauge trivial. That is, we can always pick a gauge where ${J}_{1}=0$ globally. Hence, this is classically equivalent to the massive IIA Bianchi identities.

#### 2.2. D4 ($P=5$ Algebroid)

**Remark 2.**

- 1
- Setting ${J}_{1}=0$, we recover the Bianchi identities for massive IIA supergravity.
- 2
- ${J}_{1}$ is a flat ${\mathbb{R}}^{+}$ connection coming from the same rescaling gauge symmetry. The charges of all the fluxes under this ${\mathbb{R}}^{+}$ are different, however, to the D2 algebroid.

#### 2.3. NS5 ($P=6$ Algebroid)

**Remark 3.**

- 1
- Setting ${J}_{1}=0$, we recover the Bianchi identities for massless IIA supergravity. This is always possible as ${J}_{1}$ is a flat ${\mathbb{R}}^{+}$ connection.
- 2
- Note that, unlike the D2 and D4 branes, we do not have the Romans mass appearing here. Viewing the Romans mass as the flux of the D8 brane, we note that the NS5 brane only couples to the D8 brane via a D6 brane [17]. Our construction here, however, does not include any D6 branes since we have no ${F}_{8}$ flux. We therefore, do not expect that the QP manifold for NS5 branes would allow for massive deformations and this is indeed what we find.

## 3. IIA/M-Theory Duality

#### 3.1. M5 to NS5

#### 3.2. M2 to D2

## 4. Applications

#### 4.1. AKSZ Models

#### 4.1.1. D2 Brane

#### 4.1.2. NS5 Brane

#### 4.2. The (Topologically Massive) D2 Brane Phase Space

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | A supermanifold with an extra $\mathbb{Z}$-grading on its structure sheaf, so all functions f have a well-defined degree $degf$; for us, $degf\ge 0$ and $deg\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}2$ equal the Grassmann parity, and this grading is equivalent to the existence of a degree-counting vector field (“Euler”), which we employ later. |

2 | The anchor map $a:E\to T$ is intrinsic to the definition of these higher algebroids; see, e.g., [1]. |

3 | A priori, one could have terms such as ${\alpha}^{\mu}{}_{\nu}{p}_{\mu}{\psi}^{\nu}$ and ${\beta}^{\mu}{p}_{\mu}\xi $. However, we will assume that $\alpha $ is invertible, which is equivalent to saying that the anchor map associated to the algebroid is surjective. Hence, one can always perform a canonical transformation to set ${\alpha}^{\mu}{}_{\nu}={\delta}^{\mu}{}_{\nu}$ and $\beta =0$. |

4 | The necessity of performing asymplectomorphism before the symplectic reduction was also observed in [18]. |

5 | In double field theory, Romans IIA theory arises by giving the Ramond–Ramond fields a linear dependence on the dual coordinates [22]. |

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Arvanitakis, A.S.; Malek, E.; Tennyson, D.
Romans Massive QP Manifolds. *Universe* **2022**, *8*, 147.
https://doi.org/10.3390/universe8030147

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Arvanitakis AS, Malek E, Tennyson D.
Romans Massive QP Manifolds. *Universe*. 2022; 8(3):147.
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Arvanitakis, Alex S., Emanuel Malek, and David Tennyson.
2022. "Romans Massive QP Manifolds" *Universe* 8, no. 3: 147.
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