# U-Dualities in Type II and M-Theory: A Covariant Approach

## Abstract

**:**

## 1. Introduction

#### 1.1. Dualities in String Theory

#### 1.2. U-duality in Maximal Supergravity

## 2. Duality-Covariant Field Theories

#### 2.1. Local Symmetries of Extended Space

#### 2.2. Winding Modes and Exotic Branes

#### 2.3. Exceptional Field Theories

#### 2.4. The Section Constraint

_{7}under the action of the $GL\left(7\right)$ subgroup as

_{7}theory decomposed under the GL(7) subgroup apparently fits the field content of the 11 dimensional supergravity in the 7 + 4 split form, as the former was constructed from the latter initially. Type IIA supergravity is obtain from this theory by further ${\mathbb{S}}^{1}$ reduction in the usual way.

#### 2.5. Double Field Theory

^{*}into lower indices

_{*}, and the coordinates ${\tilde{x}}_{*}$ becomes normal geometric coordinate in the new T-duality frame. Note however a different possibility, where one simply replaces dependence of background fields on ${x}^{*}$ by ${\tilde{x}}_{*}$, still counting the latter as a non-geometric dual coordinate. This will turn a solution of supergravity equations of motion into a proper string background, which however (i) does not solve e.o.m.s, (ii) is non-geometric. This way DFT and ExFT allow addressing exotic branes and the corresponding non-geometric backgrounds. See further Section 3.2 for more details.

## 3. Applications

#### 3.1. Non-Geometric Compactifications

#### 3.2. Exotic Brane Backgrounds

#### 3.3. Deformations of Supergravity Backgrounds

## 4. Conclusions and Discussion

## Funding

## Conflicts of Interest

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**Figure 1.**Reduction of 11D fields into four dimensions, dualization into the lowest possible rank tensor and recollection into the ${E}_{7}$ multiplet in the $\mathbf{56}$ and the coset space ${E}_{7\left(7\right)}/SU\left(8\right)$. Both magnetic and electric vectors potentials are included in the counting (see the text).

**Figure 2.**Reduction of 11D fields into seven dimensions, dualization into the lowest possible rank tensor and recollection into SL(5) multiplets. Dashed lines correspond to an alternative combination of the 2- and 3-forms into the 3-form ${C}_{\mu \nu \rho}{}^{m}$ dual to ${B}_{\mu \nu m}$ in 7 dimensions (see the text). Here the small Latin indices $m,n,k,\cdots $ label the fundamental irrep $\mathbf{5}$ of SL(5) and small Latin indices from the beginning of the alphabet label the fundamental irrep $\mathbf{4}$ of GL(4).

**Figure 3.**Relations between toroidal reductions of $\mathcal{N}=1$ $D=11$ supergravity, gaugings and more complicated dimensional reductions involving geometric and non-geometric fluxes.

**Figure 4.**Systematics of backgrounds with H, geometric f and Q fluxes and their relations. Note that in [77] Q-monopole localised in both ${X}^{8}$ and ${X}^{9}$ has been constructed with both these directions being compact.

**Figure 5.**Relationships between the relevant theories and their solutions. b-frame refers to the standard supergravity (possibly generalised), while $\beta $-frame is the theory of [98]. Yang-Baxter deformation acts within usual supergravity, but we interpret it as a composition of the open/closed string map with a deformation by ${\beta}^{mn}$. This leads to the constraints for ${\beta}^{mn}$ (essentially the CYBE) arising from supergravity field equations.

**Table 1.**Y-tensor for different T(U)-duality groups for string and M-theory on a d-torus. Here the Greek indices $\alpha ,\beta ,\gamma =1,\dots ,5$ label the representation $\mathbf{5}$ of $SL\left(5\right)$ and the index i labels the $\mathbf{10}$ of $SO(5,5)$, n denotes dimension of the representation generalized vectors of the theory transform in.

Duality Group | The Y-Tensor | Dimension of Extended Space |
---|---|---|

O(d,d) | $Y{}_{KL}^{MN}=\eta {}^{MN}{\eta}_{KL}$ | $n=2d$ |

SL(5) | $Y{}_{KL}^{MN}=\u03f5{}^{\alpha MN}{\u03f5}_{\alpha KL}$ | $n=10$ |

SO(5,5) | $Y{}_{KL}^{MN}=\frac{1}{2}\left(\gamma {}^{i}\right){}^{MN}{\left({\gamma}_{i}\right)}_{KL}$ | $n=16$ |

${E}_{6\left(6\right)}$ | $Y{}_{KL}^{MN}=10\phantom{\rule{0.166667em}{0ex}}d{}^{MNR}{d}_{KLR}$ | $n=27$ |

${E}_{7\left(7\right)}$ | $Y{}_{KL}^{MN}=12\phantom{\rule{0.166667em}{0ex}}c{}^{MN}{}_{KL}+\delta {}_{K}^{(M}\delta {}_{L}^{N)}+\frac{1}{2}\u03f5{}^{MN}{\u03f5}_{KL}$ | $n=56$ |

**Table 2.**Counting of winding modes of branes of M-theory on a background of the form ${M}_{4}\times {\mathbb{T}}^{d}$ with ${M}_{4}$ being a four-dimensional manifold. The first column contains dimensions of the compact torus, the second column lists the corresponding U-duality group, and the last column lists representations of G under which coordinates of the extended space transform.

d | G | P | M2 | M5 | KK6 | ${5}^{3}$ | ${2}^{6}$ | ${0}^{(1,7)}$ | ${\mathcal{R}}_{\mathbb{X}}$ |
---|---|---|---|---|---|---|---|---|---|

2 | SL(2) | 2 | 1 | - | - | - | - | - | 3 |

3 | SL(3)×SL(2) | 3 | 3 | - | - | - | - | - | (3,2) |

4 | SL(5) | 4 | 6 | - | - | - | - | - | 10 |

5 | SO(5,5) | 5 | 10 | 1 | - | - | - | - | 16_{s} |

6 | ${E}_{6\left(6\right)}$ | 6 | 15 | 6 | - | - | - | - | 27 |

7 | ${E}_{7\left(7\right)}$ | 7 | 21 | 21 | 7 | - | - | - | 56 |

8 | ${E}_{8\left(8\right)}$ | 8 | 28 | 56 | 56 | 56 | 28 | 8 | 248 |

Coordinates | Brane | State |
---|---|---|

${x}^{\mu}=(z,{y}^{1},{y}^{2},{y}^{3},{x}^{r})\phantom{|}$ | NS5-brane (H-monopole) | ${5}_{2}^{0}$ |

${x}^{\mu}=(\tilde{z},{y}_{1},{y}^{2},{y}^{3},{x}^{r})\phantom{|}$ | KK-monopole | ${5}_{2}^{1}$ |

${x}^{\mu}=(\tilde{z},{\tilde{y}}_{1},{y}^{2},{y}^{3},{x}^{r})\phantom{|}$ | Q-monopole | ${5}_{2}^{2}$ |

${x}^{\mu}=(\tilde{z},{\tilde{y}}_{1},{\tilde{y}}_{2},{y}^{3},{x}^{r})\phantom{|}$ | R-monopole | ${5}_{2}^{3}$ |

${x}^{\mu}=(\tilde{z},{\tilde{y}}_{1},{\tilde{y}}_{2},{\tilde{y}}_{3},{x}^{r})\phantom{|}$ | R’-monopole | ${5}_{2}^{4}$ |

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Musaev, E.T.
U-Dualities in Type II and M-Theory: A Covariant Approach. *Symmetry* **2019**, *11*, 993.
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U-Dualities in Type II and M-Theory: A Covariant Approach. *Symmetry*. 2019; 11(8):993.
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2019. "U-Dualities in Type II and M-Theory: A Covariant Approach" *Symmetry* 11, no. 8: 993.
https://doi.org/10.3390/sym11080993