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f21fe5887d557a228fd28824d2a54744d1ff296f | subsection | 20 | 130 | Body | According to the relation (REF ) between the tension and the vector, the tension associated with a fundamental weight \lambda ^1 is \mathcal {T}_1=R_1/l_p^3 .
This is the tension of a string in the external d-dimensional spacetime.
More concretely, this string can be interpreted as an M2-brane wrapped along the interna... | {
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f4b9e555ffa645746aaf78d685b35361d2d9abef | subsection | 21 | 130 | Body | The tension is given by T^{(\text{\tiny KKM})}_5=R_1^2R_2/l_p^9 and it corresponds to \lambda ^{(\text{\tiny KKM})}=\lambda _1 + \lambda _2 .
Namely, the second 5-brane multiplet has the Dynkin label [1,1,0,\cdots ,0].
Higher p-brane multiplets can also be constructed similarly.
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8948bc802bbe9575d9c6993f09e1a5bb338d67de | subsection | 22 | 130 | Body | Again by performing a formal T-duality along the x^8-direction, we obtain the background of the 5^3_2(12345,678)-brane,{\mathrm {d}}s^2 = {\mathrm {d}}x^2_{01\cdots 5} + \frac{\tau _2}{\vert {\tau } \vert ^2} \, {\mathrm {d}}x^2_{67} + \tau _2^{-1} \,{\mathrm {d}}x_{8}^2 + \tau _2 \,{\mathrm {d}}x_{9}^2\,,\quad \operat... | {
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5387ee206ba574fdf4899ff5f0cbb583af52819d | subsection | 23 | 130 | Body | More explicitly, we obtainLet us note that this map can be singular in certain backgrounds, for example, when E_{mn} is not invertible.\begin{split}
&\tilde{g}_{mn} = E_{mp}\,E_{nq}\,g^{pq}\,,\qquad \beta ^{mn} = E^{mp}\,E^{nq}\,B_{pq}\,,\qquad \operatorname{e}^{-2\tilde{\phi }} \equiv \frac{\det (g_{mn})}{\det (E_{mn}... | {
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"Jose J. Fernandez-Melgarejo",
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d7c0874078f690345d8fcaa483261805293661aa | subsection | 24 | 130 | Body | In fact, as we show in the next section, in all of the “elementary” domain-wall solutions, the winding-coordinate dependence appears only in a certain gauge field linearly.With the above parameterization, the linear map (REF ) (a T-duality along the y-direction) between the generalized coordinates can be summarized as ... | {
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c2add1306fc11a822022cf5b22728c9f2c46e672 | subsection | 25 | 130 | Body | However, even if a minus sign can appear, it does not affect the following computations for obtaining exotic-brane solutions in EFT since the minus sign can be absorbed into a free parameter m .&x^a\ \leftrightarrow \ x^a\,,\qquad \tilde{x}_a\ \leftrightarrow \ \tilde{x}_a\,,\qquad x^y\ \leftrightarrow \ \tilde{x}_y\,,... | {
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4779746b11e6a0b59ecd91b861b1c5cecdc95d39 | subsection | 26 | 130 | Body | The dual fields in M-theory and type IIB theory can be summarized as\text{M-theory}:\quad &\bigl \lbrace \tilde{G}_{ij},\,\Omega ^{i_1i_2i_3},\,\Omega ^{i_1\cdots i_6},\,\Omega ^{i_1\cdots i_8,\,j}\bigr \rbrace \,,
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6d4bf6bab5eec4fe85665f221988456890c0c83f | subsection | 27 | 130 | Body | In the following, for simplicity, we drop the subscript {\text{\tiny (A)}} for the type IIA fields.In summary, the T-duality transformation (REF ) and (REF ) maps a solution of EFT to another solution of EFT.
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1f080563d976041b445c5b09170d27b9f727a0bf | subsection | 28 | 130 | Body | \end{split}The S-duality rule for \beta ^{m_1\cdots m_7,\,n} will be \beta ^{\prime m_1\cdots m_7,\,n}=\beta ^{m_1\cdots m_7,\,n}+\text{(non-linear terms)} . | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
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fbc696928d4c48d9effcd34b3ec99b9aa614d136 | subsection | 29 | 130 | Body | The S-duality transformation also rotates the generalized coordinates as\begin{split}
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
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01b8b48d217e5c3c8f39b4d0cb9c68bfd1db755b | subsection | 30 | 130 | Body | By performing the T-duality along the x^8-direction, we obtain the p_3^{(1,7-p)}(1\cdots p,p+1\cdots 7,8)-brane solution\begin{split}
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6513a62ce9970ce8e9d7d2a8ca1f48c0e593e929 | subsection | 31 | 130 | Body | \end{split}By performing a formal T-duality along the x^8-direction, we obtain the 1_4^{(1,0,6)}(1,234567,,8) background,{\mathrm {d}}s^2= \frac{\vert {\tau } \vert ^2}{\tau _2} \, \bigl ({\mathrm {d}}x_{01}^2 + \tau _2\,{\mathrm {d}}x_9\bigr )+ {\mathrm {d}}x^2_{2\cdots 8} \,, \quad \operatorname{e}^{-2\Phi } = \frac{... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
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3ee886b373aa82e4de326e9b2fb912d07798504d | subsection | 32 | 130 | Body | In this case, the linear winding-coordinate dependence is contained in the \beta -field \beta ^{67}=m\,\tilde{x}_8 .
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2ea94a6a2ec5bd299e126fc51cbd5d9c2cabff1d | subsection | 33 | 130 | Body | In particular, when we choose c=0 , the solution is simplified as{\mathrm {d}}s^2 = {\mathrm {d}}x^2_{01\cdots 5} + \tau _2^{-1} \,{\mathrm {d}}x_{678}^2 + \tau _2 \,{\mathrm {d}}x_{9}^2\,, \qquad \operatorname{e}^{-2\Phi }= \tau _2^2 \,,where the asymmetry between \lbrace 6,7\rbrace and 8 disappears.According to the g... | {
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008ed5a292dae7706179e7bee5f5d888cc6e9820 | subsection | 34 | 130 | Body | In other words, we are essentially describing a seven-dimensional supergravity.In this manner, we have transformed the DFT solution (REF ) into a winding-coordinate-independent solution (REF ) of the deformed supergravity.
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52b01c6fc42b87c8807046526dd12fff43c5f4cb | subsection | 35 | 130 | Example: string multiplet in | As an example, let us consider a string multiplet in M-theory compactified on T^6 , where the U-duality group is E_{6(6)} .
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cb0747ab8a20615c5a99fdffafb6b70c1b17ecae | subsection | 36 | 130 | Example: string multiplet in | It is noted that, in this case, all of the states correspond to weight vectors with the same length, and the 27 states are connected via the Weyl reflections, or the U-duality transformations (REF ). | {
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7de64f7e10a748423b920305b24b74f5ee92da2c | subsection | 37 | 130 | Web of supersymmetric branes | Utilizing the duality transformation rule (REF ), we can generate a chain of exotic branes in M-theory.
Indeed, by brute force applications of duality (REF ) to the tensions of the standard branes, we obtain Tables REF –REF , which show the explicit brane charges and the degeneracies in each multiplet.
By summing up th... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
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"Yuho Sakatani"
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9d5e1c1ef3cfd73f4cfcc76a67fb7fdfd3551f60 | subsection | 38 | 130 | Web of supersymmetric branes | In M-theory compactified to d-dimensions with d\ge 3, all of the branes appearing in the Weyl orbit are summarized as follows (potentials that couple to the following branes are listed in ):\begin{split}
&\text{P}\,,\ 2_3\,,\ 5_6\,,\ 6_{9}^{1}\,,\ {purple}{5_{12}^{3}}\,,\ {blue}{8_{12}^{(1,0)}}\,,\ {purple}{2_{15}^{6}}... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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5bcb7b7d8aaaf5f0ab4b1f1308e5e1ab6c5db11a | subsection | 39 | 130 | Web of supersymmetric branes | \end{split}In the literature, 2_3, 5_6, 6_{9}^{1}, and 8_{12}^{(1,0)} are respectively called M2, M5, KKM, and M9-brane while the others do not have familiar common names.
We consider a b_{n}^{(c_s,\cdots ,c_2)}-brane as a kind of (b+c_2+\cdots +c_s)-brane, and the codimension is given by 10-(b+c_2+\cdots +c_s).
If the... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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e4767a664622d4637c4c6d3893996697252a8046 | subsection | 40 | 130 | Web of supersymmetric branes | The tension is \mathcal {T}_7=R_1^3R_2/l_p^{12} , corresponding to 2\,\lambda _1 + \lambda _2 and [2,1,0] in d=8.We can also consider the type II branes by using the 11D/10D relation (REF ), and the type IIA branes associated with all of the “elementary” M-branes are obtained in Table REF .
[Table: A map between branes... | {
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... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
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4e5844fd291196e8f7cac5fa8cf0aba66d54873c | subsection | 41 | 130 | Web of supersymmetric branes | In addition, 7_{3}^{(1,0)} is known as the KK8A-brane in .
As one can clearly see, in dimensions d\ge 3 , there exist the type IIA branes with tensions proportional to g_s^{\alpha } with -11\le \alpha \le 0 .In order to obtain all of the “elementary” type IIB branes, we act a T-duality to each of the type IIA branes.
S... | {
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a94aa7b1cb2f7ef33d64b3633b3dc7112d6735d4 | subsection | 42 | 130 | Web of supersymmetric branes | A list of all of the “elementary” type IIB branes is as follows:&1_0\,,\ \text{P}\,,\ 1_1\,,\ 3_1\,,\ 5_1\,,\ {purple}{7_1}\,,\ {darkcyan}{9_1}\,,\ 5_2\,,\ 5_2^1\,,\ {purple}{5_{2}^{2}}\,,\ {blue}{5_{2}^{3}}\,,\ {darkcyan}{5_{2}^{4}}\,,\ {purple}{7_{3}}\,,\ {purple}{5_{3}^{2}}\,,\ {purple}{3_{3}^{4}}\,,\ {purple}{1_{3}... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
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b9e7638154dc55e4963c382d7720526b6a966c67 | subsection | 43 | 130 | Web of supersymmetric branes | For example, 2^{(7,0,0,0)}_{11(8)} in Figure REF represents the 2^{(7,0,0,0)}_{11}-brane, and also denotes that its S-dual partner is the 2^{(7,0,0,0)}_{8}-brane.
The characters in the squared brackets are not important here, and will be explained in Section REF .
Each (solid or dashed) line corresponds to a T-duality ... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
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53e1beb304b6b6fcded607eef91c406572a597fb | subsection | 44 | 130 | Web of the missing states | In the previous subsection, we have only considered the branes that are connected to the standard branes via T- and S-duality transformations, i.e. the Weyl reflections (REF ).
However, as we can clearly see from Table REF , if we consider the non-standard branes (i.e. colored branes with codimension 2 or less), these ... | {
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326ac03cf8b91e2bb13d1066476044c3d1645b11 | subsection | 45 | 130 | Example: 4-brane multiplet in | Let us start with a simple example, a 4-brane multiplet in M-theory compactified on T^4.
In this case, Table REF shows that the number of supersymmetric branes is 20, although the dimension of the 4-brane multiplet is 24.
Thus, there are four missing states.
In order to identify the missing states, let us consider the ... | {
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"Jose J. Fernandez-Melgarejo",
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26cbeb92d77dfde24c75971cd5da6e2027123a18 | subsection | 46 | 130 | List of missing states | Generalizing the above procedure, we can compute the tensions of missing states in all of the multiplets.
In order to obtain a list of the weights for the exceptional groups E_{6(6)}, E_{7(7)}, and E_{8(8)}, it will be useful to use a computer program such as SimpLie .
By transforming the Dynkin labels into the tension... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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b77032f68c7efff99d55d2f5ae24b2f263b7e6d5 | subsection | 47 | 130 | List of missing states | The states contained in a single column have the weights with the same length, and we have checked that they are indeed in a single U-duality orbit of (REF ).In terms of M-theory, the following states are contained in Tables REF –REF :&8_9\,,
7_{12}^{2}\,,
9_{12}^{1}\,,
4_{15}^{5}\,,
6_{15}^{4}\,,
7_{15}^{(1,2)}\,,
1_{... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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adfeaffa89464d658b99b0cfadcdb8f58893c631 | subsection | 48 | 130 | List of missing states | For example, the 8_9-brane in the p-brane multiplet (1\le p\le 6) has degeneracy (8-p) , although for p=6 the degeneracy becomes 1.
The p-dependence is non-trivial, but the degeneracy is independent of d for all missing states.
The missing states in higher d can be obtained from the missing states in lower d just by tr... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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f2c21df04951720d0e86c2d4535009429335a5bd | subsection | 49 | 130 | Web of mixed-symmetry potentials | The standard branes in type II theory couple to certain potentials in type II supergravity.
For example, the F1 and the pp-wave (electrically) couple to the B-field B_2 and the graviphoton A_1^m , and Dp-branes couple to the R–R potentials C_{p+1} .
Following a series of works , , , , , , , , , , we call F1 and the pp-... | {
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"Jose J. Fernandez-Melgarejo",
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17ca05e24d5b8cb3d03621319a9c5ca990b500fb | subsection | 50 | 130 | Web of mixed-symmetry potentials | E^{(4)}=F, E^{(5)}=G, E^{(6)}=H, and so on) and denote the corresponding brane the E^{(n)}-brane.
[Table: Exotic branes with the tension proportional to g_s^{\alpha } (\alpha \le -3) and the corresponding mixed-symmetry potentials in type II string theories.Here, n and m are non-negative integers while p (q) runs over ... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
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1c36e0651efffb3257b15929140a2ca8bb490511 | subsection | 51 | 130 | Exotic-brane solutions in DFT | In this section, we explain how to construct the supergravity solutions for the variety of exotic branes discussed in the previous section.
If we consider only the standard branes or the defect branes, we can (at least locally) write down the solutions satisfying the standard supergravity equations of motion.
However, ... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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def29b089ca060a30fd8db3a974a85ea4cb1aed5 | subsection | 52 | 130 | D7-brane solution | Let us begin with the standard D7(1234567)-brane solution,{\mathrm {d}}s^2 = \tau _2^{-1/2} \,\bigl ({\mathrm {d}}x^2_{01\cdots 7} + \tau _2 \,{\mathrm {d}}x_{89}^2\bigr )\,,\qquad \operatorname{e}^{-2\Phi }= \tau _2^{2} \,, \qquad \vert {A} \rangle = \bigl (\tau _1 - \tau _2^{-1}\,\gamma ^{0 \cdots 7}\bigr )\vert {0} ... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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"hep-th"
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da823a7c2f47970a248c26115b59aaec494229de | subsection | 53 | 130 | A quick review of DFT | In order to perform a formal T-duality, we utilize the DFT on a 20-dimensional doubled spacetime with the generalized coordinates (x^M)=(x^m,\,\tilde{x}_m) .For our purpose, it is not necessary to double the time direction, but just for notational simplicity, we double all of the directions.
In DFT, all of the bosonic ... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
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"hep-th"
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668181e646a9b7851502c23d55bd7d732064a202 | subsection | 54 | 130 | A quick review of DFT | The generalized metric and the dilaton can be parameterized asIn our convention, the sign of the B-field is opposite to the conventional DFT.(\mathcal {H}_{MN}) = \begin{pmatrix} (g-B\,g^{-1}\,B)_{mn} & -B_{mp}\,g^{pn} \\ g^{np}\,B_{pn} & g^{mn} \end{pmatrix} , \qquad \operatorname{e}^{-2d} = \sqrt{-g}\,\operatorname{e... | {
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} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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"hep-th"
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7112f040416cb7079063ed27f487bb66706859b3 | subsection | 55 | 130 | A quick review of DFT | The field strength is defined as\vert {F} \rangle \equiv \partial\partial / A (
\partial / M M) .Unlike the standard supergravity fields, the DFT fields can depend on the generalized coordinates x^M but the consistency condition, namely the SC,\eta ^{MN}\,\partial _M \otimes \partial _N =0 \,,requires that the D... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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2f8634a6c15bbe6c4e821fd33659efa4331bdb4f | subsection | 56 | 130 | D8-brane solution | By using the above setup, let us construct the D8-brane solution in DFT.
We start from the smeared D7 solution (REF ), and perform the formal T-duality (REF ) along the x^8-direction.
We then obtain{\mathrm {d}}s^2 = \tau _2^{-1/2} \,\bigl ({\mathrm {d}}x^2_{01\cdots 8} + \tau _2 \,{\mathrm {d}}x_{9}^2\bigr )\,,\qquad ... | {
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"doi": "10.1007/jhep11(2011)086",
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"raw": "O. Hohm and S. K. Kwak, “Massive type II in double field theory,” JHEP 1111, 086 (2011) [arXiv:1108.4937 [hep-th]].",
"source_ref_id": "2... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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0c66aa45960ca4235439f850a9a7d634998d3f2a | subsection | 57 | 130 | Non-geometric fluxes and mixed-symmetry potentials | In the dual parameterization, we can define the so-called non-geometric Q-flux , , , asQ_1^{pq}\equiv Q_m{}^{pq}\,{\mathrm {d}}x^m \equiv {\mathrm {d}}\beta ^{pq} \,.The non-geometricity of the 5^2_2-brane (or the Q-brane) background was pointed out in , and as shown in , , the 5^2_2(12345,67) background has a constant... | {
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"raw": "F. Marchesano and W. Schulgin, “Non-geometric fluxes as supergravity backgrounds,” Phys. Rev. D 76, 041901 (2007) [arXiv:0704.3272 [hep-th]].",
"source_ref_id": "7c2f101314da499d3b97dacbc33620... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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99cb05324fa71a68e4b685dbf9c7082d86767518 | subsection | 58 | 130 | Non-geometric fluxes and mixed-symmetry potentials | The equation of motion for the \beta -field takes the form\partial _m \bigl (\operatorname{e}^{-2\tilde{\phi }}\sqrt{-\tilde{g}}\,\tilde{g}^{mn}\,\tilde{g}_{pq,\,rs}\, Q_n{}^{rs} \bigr ) = 0 \,,and this suggests to introduce the dual field strength as Q_{9,2} \equiv \operatorname{e}^{-2\tilde{\phi }} \tilde{g}_{pq,\,rs... | {
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"doi": "10.1007/jhep03(2015)135",
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"source_ref_id": "ae050b5b2c3a... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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] | 2,018 | en | Physics | [
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5830ca01126b51dfbee0f6c2e3666fad86931863 | subsection | 59 | 130 | Non-geometric fluxes and mixed-symmetry potentials | In , the effective Lagrangian for the R-flux was derived from the DFT Lagrangian as (see also , , , , )\mathcal {L}\sim \sqrt{-\tilde{g}}\operatorname{e}^{-2\tilde{\phi }}\Bigl (\tilde{R} + 4\,\vert {{\mathrm {d}}\tilde{\phi }} \vert ^2 - \frac{1}{2}\,\vert {R} \vert ^2 \Bigr ) \,,where \vert {R} \vert ^2\equiv \frac{1... | {
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"raw": "D. Andriot, O. Hohm, M. Larfors, D. Lüst and P. Patalong, “A geometric action for non-geometric fluxes,” Phys. Rev. Lett. 108, 261602 (2012) [arXiv:1202.3060 [hep-th]].",
"source_ref_id": "3ce... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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ef1f5102098582893ddcf2dcd4f8a2c6c132788f | subsection | 60 | 130 | Non-geometric fluxes and mixed-symmetry potentials | As it has been discussed there, D_{9,3} is the T-dual of D_{8,2} ,D_{a_1\cdots a_8,\,b_1b_2} \ \overset{T_z}{\longleftrightarrow } \ D_{a_1\cdots a_8y,\,b_1b_2y}\qquad \bigl (y\notin \lbrace a_1,\cdots ,a_8\rbrace \,,\quad \lbrace b_1,b_2\rbrace \in \lbrace a_1,\cdots ,a_8\rbrace \bigr )\,.By observing the explicit for... | {
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"raw": "B. Wecht, “Lectures on nongeometric flux compactifications,” Class. Quant. Grav. 24, S773 (2007) [arXiv:0708.3984 [hep-th]].",
"sourc... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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06fb63d1e7d668f312b8dbdd38dcbebff56cac71 | subsection | 61 | 130 | Non-geometric fluxes and mixed-symmetry potentials | In such cases, we provide heuristic definitions of the R-fluxes by considering the symmetry of exotic branes and providing an appropriate antisymmetrization, like R^{mnp}=3\,\tilde{\partial }^{[m}\beta ^{np]} .We can again consider a definition of a non-geometric flux , Q_1^{n_1\cdots n_6}\equiv Q_m{}^{n_1\cdots n_6}\,... | {
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"doi": "10.1007/jhep03(2015)135",
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"source_ref_id": "ae050b5b2c3a... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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3d1ae1346754ea4f21ecce65a685ecfbe19b4a28 | subsection | 62 | 130 | Non-geometric fluxes and mixed-symmetry potentials | The mixed-symmetry potential may be defined through Q_{9,6}\equiv {\mathrm {d}}E^{(4)}_{8,6} , and in the 1_4^6(1,234567) background, we obtainE^{(4)}_{\bar{0}\cdots \bar{7},\,\bar{2}\cdots \bar{7}} = -m\,\tau _2^{-1} \,.Similarly, in the 1_4^{(1,0,6)}(1,234567,,8) background, a new locally non-geometric flux may be de... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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18c4e9a2dd889e5355167c3b7871cd2fc6b2a33e | subsection | 63 | 130 | All exotic-brane solutions in EFT | In this section, we give a prescription to construct all of the elementary exotic-brane solutions in EFT.
After introducing duality transformations in Sections REF and REF , in Sections REF to REF we construct all the domain-wall solutions contained in (REF ), (REF ), and (REF ) as well as their associated R-fluxes and... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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c10e7425cb61f136bdad390fadd62d1a8512133b | subsection | 64 | 130 | Duality rotations in EFT | Type II string theory compactified on an (n-1)-torus has the E_{n(n)} U-duality symmetry, which contains the \mathrm {O}(n-1,n-1) T-duality symmetry as a subgroup.
The E_{n(n)} EFT is a generalization of DFT that manifests the U-duality symmetry in supergravity , , , , , , , , , , , , , , .
Similar to DFT, it is define... | {
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"source_ref_id": "213d910a15b48... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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46d6dac1c47e74bfe5505eaca0a2c8c525948d73 | subsection | 65 | 130 | Duality rotations in EFT | In the type IIB parameterization, all of the coordinates are associated with the type IIB branes, such as P, F1/D1, D3, NS5/D5 etc., and the index \alpha represents the \mathrm {SL}(2) S-duality doublet.
The winding-coordinates \mathsf {y}^{\alpha \beta }_{m_1\cdots m_7}=\mathsf {y}^{(\alpha \beta )}_{m_1\cdots m_7} co... | {
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{
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"doi": "10.1093/ptep/ptx038",
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d289d16c142c5bc2afd468ad4378a6607adcdc2a | subsection | 66 | 130 | Dual parameterization in the whole bosonic sector | In the case of DFT, the conventional fields and the dual fields are related through the expression (REF ).
Here, we briefly explain how to generalize the relation (REF ) to EFT.As we already explained, the generalized metric \mathcal {M}_{IJ} in EFT can be parameterized by the bosonic fields in type IIB supergravity, w... | {
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"doi": "10.1007/jhep03(2015)144",
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"raw": "C. D. A. Blair and E. Malek, “Geometry and fluxes of SL(5) exceptional field theory,” JHEP 1503, 144 (2015) [arXiv:1412.0635 [hep-th]].",
... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d4e515591b9f87ece5be941d36ca4571303c3f64 | subsection | 67 | 130 | Reorganization of the generalized coordinates | In order to simplify the T-duality rule, we here consider the following redefinitions of the generalized coordinates in type II theory.
The winding coordinates for P and F1 (that appear also in DFT) are defined as\big (x^m,\,\tilde{x}_m\bigr ) \equiv {\left\lbrace \begin{array}{ll}
\bigl (x^m,\, y_{m\text{\tiny M}}\big... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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44a4db16c4d57ed14d91d2c09f69ea1a4d8cc9ec | subsection | 68 | 130 | Reorganization of the generalized coordinates | The winding coordinates for the D-branes y^{\text{\tiny D}}_{m_1\cdots m_p} (p=0,\cdots ,7) and the solitonic branes 5_2^n (n=0,1,2) are denoted as&(y^{\text{\tiny D}}_{m_1\cdots m_p}) \equiv \bigl (\underbrace{-x^\text{\tiny M}}_{\text{D0}},\,\underbrace{-\mathsf {y}^2_m}_{\text{D1}},\,\underbrace{y_{m_1m_2}}_{\text{D... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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67521efee13d5348e7b42aa348c9cde5cbcc8ca9 | subsection | 69 | 130 | Reorganization of the generalized coordinates | \end{array}\right.}The winding coordinates for the exotic p_3^{7-p}-branes (p=0,\cdots ,7) and the (1^6_4,\,0^{(1,6)}_4)-branes are called\begin{split}
&(y^{\text{\tiny E}}_{m_1\cdots m_7,\,n_1\cdots n_{7-p}})
\equiv \bigl (\underbrace{y_{m_1\cdots m_7\text{\tiny M},\,n_1\cdots n_7\text{\tiny M},\,\text{\tiny M}}}_{0^7... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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1014a9e7e5ee455b1e502888e7de22e0767cecd5 | subsection | 70 | 130 | Reorganization of the generalized coordinates | \end{array}\right.}The remaining eight coordinates in the E_{8(8)} exceptional space,&y_{m_1\cdots m_6\text{\tiny M},\,n}\quad \bigl (n\notin \lbrace m_1\cdots m_6\rbrace \bigr )\,,\quad y_{m_1\cdots m_7\text{\tiny M}}\qquad (\text{IIA})\,,
\\
&\mathsf {y}_{m_1\cdots m_6,\,n}\quad \bigl (n\notin \lbrace m_1\cdots m_6\r... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
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03e4d2ad232bd3dcf767131c6e810921502053c8 | subsection | 71 | 130 | First two examples of domain-wall solutions in EFT | Before considering all of the “elementary” domain-wall solutions, let us begin with two simple examples. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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3d87e17d49621cd15e4e558c2c58e02766c4a6fc | subsection | 72 | 130 | Dual parameterization for the | For the 7_3 background, the non-vanishing fields are the (g_{\mu \nu },\,g_{mn},\,\Phi ,\,C_0) .
In this case, (REF ) and (REF ) are reduced to\begin{split}
&g^{\text{\tiny E}}_{mn} = \tilde{g}^{\text{\tiny E}}_{mn}\,,\qquad \bigl (m_{\alpha \beta }\bigr ) = \operatorname{e}^{\Phi }\,\begin{pmatrix}
\operatorname{e}^{-... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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a63f890b7c1776c7f930bcf70818ba4770cae4b2 | subsection | 73 | 130 | Dual parameterization for other | We can similarly obtain the dual parameterizations for other p-brane solutions in (REF ) and (REF ).
However, in general, a direct comparison of the generalized metrics is very complicated.For simplicity, we instead use the T-duality rules (REF ) and (REF ).
Then, we can easily obtain the dual parameterization of the p... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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f04daab466f19454436c616e646b04d64a341380 | subsection | 74 | 130 | Non-geometric flux and mixed-symmetry potentials | As discussed in , backgrounds of the exotic p_3^{7-p}-branes are the magnetic sources of the non-geometric P-fluxes.
The non-geometric P-fluxes were introduced in , , and in particular, a P-flux P_m{}^{pq} is S-dual of the Q-flux.
They are roughly defined as (see for more details)P_1^{n_1\cdots n_{7-p}} \equiv P_m{}^{n... | {
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{
"arxiv_id": "",
"doi": "10.1007/jhep03(2015)135",
"end": 116,
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"raw": "Y. Sakatani, “Exotic branes and non-geometric fluxes,” JHEP 1503, 135 (2015) [arXiv:1412.8769 [hep-th]].",
"source_ref_id": "ae050b5b2c3a... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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79b8de35378668974ac13faf1c77a82c1f39368a | subsection | 75 | 130 | Non-geometric flux and mixed-symmetry potentials | In the p_3^{7-p}(1\cdots p,p+1\cdots 7) solution, we obtainE_{\bar{0}\bar{1}\bar{2}\bar{3}\bar{4}\bar{5}\bar{6}\bar{7},\,\overline{p+1}\cdots \bar{7}} = -m\,\tau _2^{-1} \,.Similarly, in the p_3^{(1,7-p)}(1\cdots p,p+1\cdots 7,8) background, the derivative of the \gamma -field gives a locally non-geometric flux introdu... | {
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"arxiv_id": "",
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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f25a463e223d1b5b875efa29ab3af4b6df6e573e | subsection | 76 | 130 | Non-geometric flux and mixed-symmetry potentials | If we also introduce the potential as R_{10,8-p,1} \equiv {\mathrm {d}}E_{9,8-p,1} , we obtainE_{\bar{0}\bar{1}\bar{2}\bar{3}\bar{4}\bar{5}\bar{6}\bar{7}\bar{8},\,\overline{p}\cdots \bar{8},\bar{8}} = -m\,\tau _2^{-1} \,,in the p_3^{(1,7-p)}(1\cdots p,p+1\cdots 7,8) background, suggesting the T-duality ruleE_{a_1\cdots... | {
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"source_ref... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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07bf12a4234d3060caa6bf38a3ed2f4a167e0c0b | subsection | 77 | 130 | A short summary | Up to here, we have discussed the defect-brane solutions\text{D7}\ (\ref {eq:D7-soln}),\quad 5^2_2\ (\ref {eq:522-soln}),\quad p_3^{7-p}\ (\ref {eq:p(7-p)3-soln}),\quad 1_4^{(1,0,6)}\ (\ref {eq:164-soln})\,,and the domain-wall-brane solutions\text{D8}\ (\ref {eq:D8-soln}),\quad 5^3_2\ (\ref {eq:532-soln}),\quad p_3^{(1... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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65a077cdd209ce376e709f75988894ebd5b50c87 | subsection | 78 | 130 | A short summary | Further, the set of indices \lbrace A\rbrace in the R-fluxes can be found from the set of indices \lbrace A\rbrace in the mixed-symmetry potentials, which are consistent with the general rule (REF ).
In fact, this appears to be a general structure as we see below.In the following, we will firstly introduce a generaliza... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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a2037dba8308172aeb7b24e16e7bf29c750f368c | subsection | 79 | 130 | Locally non-geometric fluxes | As we have already discussed, a domain-wall brane, say the b^{(c_s,\cdots ,c_2)}_n-brane, is the magnetic source of the non-geometric flux with a set of antisymmetrized indices, R^{c_2+\cdots +c_s,\cdots ,c_{s-1}+c_{s},c_s}, which is a U-duality version of the familiar R-flux (see , , for definitions of locally non-geo... | {
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... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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2ec160a1d8d64ddfff52d804edc9f333cb214f50 | subsection | 80 | 130 | Locally non-geometric fluxes in type IIA theory/ | The obtained R-fluxes in type IIA theory and the corresponding domain-wall branes can be summarized as follows:\begin{split}
&\underline{R_{(2)}^{3}\ \leftrightarrow \ \text{$5^3_2$-brane:}}
\\
&R_{(2)}^{m_1m_2m_3} \equiv 3\,\tilde{\partial }^{[m_1} \beta ^{m_2m_3]} \,,
\end{split}
\\
\begin{split}
&\underline{R_{(3)}^... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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f21b32779d6b9d8e5f28952c30308cdb57bdccb5 | subsection | 81 | 130 | Locally non-geometric fluxes in type IIA theory/ | \end{split}Here, the vector field A^m_n\equiv \tilde{g}^{mn}/\tilde{g}^{nn} is the graviphoton, which is T-dual of the \beta -field; A^m_y\ \overset{T_y}{\leftrightarrow }\ \beta ^{my} .
We have attached the subscript (n) to the R-flux that is associated with the exotic brane b^{(c_s,\cdots ,c_2)}_{n} .In the type IIA ... | {
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... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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1f6ae555561be4ee41ae8b3b987a3d583619d563 | subsection | 82 | 130 | Locally non-geometric fluxes in type IIB theory/ | In type IIB theory, the R-fluxes and the corresponding domain-wall branes are as follows:\begin{split}
&\underline{R_{(2)}^{3}\ \leftrightarrow \ \text{$5^3_2$-brane:}}
\\
&R_{(2)}^{m_1m_2m_3} \equiv 3\,\tilde{\partial }^{[m_1} \beta ^{m_2m_3]} \,,
\end{split}
\\
\begin{split}
&\underline{R_{(3)}^{6,1}\ \leftrightarrow... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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6bcaf73861b85c760ba4029e8d761f5f3e712a05 | subsection | 83 | 130 | Locally non-geometric fluxes in type IIB theory/ | \end{split}For an exotic brane that is self-dual under the S-duality, we can check that the associated R-flux also behaves as a singlet.
Under the S-duality, the scalar \gamma is mapped to the R–R 0-form -C_0 , but since C_0 is non-linear in terms of the dual fields, we have just truncated C_0 in the above expressions. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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2f4ad66dd5780db886a77e9d06279ecb32bb0ed1 | subsection | 84 | 130 | Locally non-geometric fluxes in M-theory/ | We can easily uplift the R-fluxes obtained in type IIA theory to M-theory. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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137535004f3d526dfbe76aabe336b7dc1c85c1f8 | subsection | 85 | 130 | Locally non-geometric fluxes in M-theory/ | The results are as follows:\begin{split}
&\underline{R^{1,1}\ \leftrightarrow \ \text{$8^{(1,0)}_{12}$-brane:}}
\\
&R^{i,\,j} \equiv \partial ^{ki}A^{j}_k \,,
\end{split}
\\
\begin{split}
&\underline{R^{4,1}\ \leftrightarrow \ \text{$5^{(1,3)}_{15}$-brane:}}
\\
&R^{i_1\cdots i_4,\,j} \equiv \frac{4!}{2!\,2!}\,\partial ... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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ef722dc7bc90afd3aa20e0b940fa908e1f484584 | subsection | 86 | 130 | Locally non-geometric fluxes in M-theory/ | \end{split}Note that the A_{\text{\tiny M}}^m is equal to the -\gamma ^m in type IIA theory while A^{\text{\tiny M}}_m is a complicated non-linear expression that will be related to R–R 1-form C_m.By using the identities such as (REF ), the fluxes R^{4,1}, R^{7,4}, and R^{7,7} appear to be consistent with the locally n... | {
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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4afa8585c042ed92172882cf1935d7a29e0e949d | subsection | 87 | 130 | Mixed-symmetry potentials in EFT | In the previous subsection, we have introduced various R-fluxes on a heuristic basis.
Similar to the R-fluxes in DFT, we here consider the introduction of the dual field strength to the R-flux in type II theory, R_{(n)}^{m_1\cdots m_{a_1},\,\cdots ,\,p_1\cdots p_{a_s}} .
As we check in the next subsection, if we define... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d2a9585611f28d2e48a795a64907fbacc2aecac7 | subsection | 88 | 130 | Mixed-symmetry potentials in EFT | The transformation rule (at the linearized level) is perfectly consistent with the rule , ,E^{(n)}_{\underbrace{{\tiny \cdots y,\cdots y,\cdots y}}_{p}}\quad \overset{T_y}{\leftrightarrow }\quad E^{(n)}_{\underbrace{\scriptsize \cdots y,\cdots y,\cdots y}_{n-p}} \,.We thus expect that the dual potentials E^{(n)}_{9,a_1... | {
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"source_ref... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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ce5d47246678b75411ede2a2251ae9310ff7dd49 | subsection | 89 | 130 | Exotic-brane solutions in type II theory | Utilizing the technique of the duality rotations in EFT, we here provide a full list of the type II domain-wall solutions in EFT.
The structure of the solutions is quite similar to the domain-wall solutions discussed above, and only a certain gauge field contains a winding-coordinate dependence.
Similar to the domain-w... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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a624892079b5ff9c1d4b65e3d8d291ef76e10b9b | subsection | 90 | 130 | E | By considering S-dual of the 5^3_2-brane or the 4^{(1,3)}_3-brane, we obtain the backgrounds of a T-duality family, E^{(4;3)}-branes. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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7f66b62f4f8f4cdd2534f18011d0f12cf8b7ea7c | subsection | 91 | 130 | E | The explicit forms of the dual fields and the locally non-geometric fluxes are as follows:\begin{split}
&\underline{{5^3_4(12345,678):} \qquad \bigl \lbrace \,R_{(4)}^{3},\,E^{(4)}_{9,3}\,\bigr \rbrace }
\\
&{\mathrm {d}}\tilde{s}^2 = \tau _2\, {\mathrm {d}}x^2_{01\cdots 5} + {\mathrm {d}}x^2_{678} + \tau _2^2 \,{\math... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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e870a031ad462f21827b62f0917b205a7f1c624a | subsection | 92 | 130 | E | \end{split}Similarly, by performing the S-duality in the 2^{(1,5)}_3 or the 2^{(3,3)}_4 solution, we obtain a T-duality chain of the E^{(5;6)}-branes. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d7bcb72316bf7b560f65523a659fe87b31c238b9 | subsection | 93 | 130 | E | The dual fields and the locally non-geometric fluxes are as follows:\begin{split}
&\underline{{2^{6}_5(12,345678):} \qquad \bigl \lbrace \,R_{(5)}^{6},\,E^{(5)}_{9,6}\,\bigr \rbrace }
\\
&{\mathrm {d}}\tilde{s}^2 = \tau _2^{3/2}\, \bigl ({\mathrm {d}}x^2_{012} + \tau _2\,{\mathrm {d}}x_{9}^2\bigr ) + \tau _2^{1/2}\,{\m... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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2c693efe24e2cd2ce421e7f091e03356a67ba228 | subsection | 94 | 130 | E | \end{split}We can repeat the duality transformations and obtain the following solutions,\begin{split}
&\underline{{1^{(4,3)}_6(1,678,2345):} \qquad \bigl \lbrace \,R_{(6)}^{7,4},\,E^{(6)}_{9,7,4}\,\bigr \rbrace }
\\
&{\mathrm {d}}\tilde{s}^2 = \tau _2^2\, {\mathrm {d}}x^2_{01} + {\mathrm {d}}x_{2345}^2 + \tau _2\,{\mat... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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1b00f956703bc639fc1f71698db1a046830551f3 | subsection | 95 | 130 | E | \end{split}Note that the 1^{(2,4,1)}_6-brane is self-dual under the S-duality transformation.
Apparently, the above 1^{(2,4,1)}_6 is not invariant under the S-duality transformation, but since the R-flux is invariant under the S-duality, the apparent non-invariance is due to a particular gauge choice. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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f51c3e94432d78bb40f45032d61d95f726372820 | subsection | 96 | 130 | E | The R-flux, or the magnetic charge of the 1^{(2,4,1)}_6-brane is invariant under the S-duality.We can further obtain the following family of solutions:\begin{split}
&\underline{{1^{(7,0)}_7(1,,2345678):} \qquad \bigl \lbrace \,R_{(7)}^{7,7},\,E^{(7)}_{9,7,7}\,\bigr \rbrace }
\\
&{\mathrm {d}}\tilde{s}^2 = \tau _2^{5/2}... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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5a4dcc768097696694ffc3173ee99861b7ea225b | subsection | 97 | 130 | E | \end{split}Finally, by performing the S-duality in the 1^{(7,0,0)}_7 background, we obtain\begin{split}
&\underline{{1^{(7,0,0)}_8(1,,,2345678):} \qquad \bigl \lbrace \,R_{(8)}^{7,7,7},\,E^{(8)}_{9,7,7,7}\,\bigr \rbrace }
\\
&{\mathrm {d}}\tilde{s}^2 = \tau _2^{3}\, \bigl ({\mathrm {d}}x^2_{01} + \tau _2\,{\mathrm {d}}... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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8b899bc33f8f3cc4dc649502cb25392d5e4b6767 | subsection | 98 | 130 | Exotic-brane solutions in M-theory | By uplifting the defect-brane solutions in type II theories to M-theory, we obtain the following defect-brane solutions:\begin{split}
&\underline{{5_{12}^{3}(12345,67z):} \qquad \bigl \lbrace \,S_1^{3},\,E_{9,3}\,\bigr \rbrace }
\\
&{\mathrm {d}}\tilde{s}^2 = \tau _2^{1/3}\, \bigl ({\mathrm {d}}x^2_{012345} + \tau _2\,... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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adf68beffe299581ca69ce1be1f9f1bb242d3dd3 | subsection | 99 | 130 | Exotic-brane solutions in M-theory | The direction z represents one of the internal ones that is not necessary to be the M-theory direction, which we denote M. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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c9937a17cc29f984b8b1a6f2af86b4ad74e17b34 | subsection | 100 | 130 | Exotic-brane solutions in M-theory | If we define the dual field strengths,\begin{split}
S_{10,\bar{m}_1\bar{m}_2\bar{m}_3} &\equiv \tilde{G}_{\bar{m}_1\bar{m}_2\bar{m}_3,\bar{n}_1\bar{n}_2\bar{n}_3}\,\tilde{*}_{11} S_{1}^{\bar{n}_1\bar{n}_2\bar{n}_3} \equiv {\mathrm {d}}E_{9,\bar{m}_1\bar{m}_2\bar{m}_3}\,,
\\
S_{10,\bar{m}_1\cdots \bar{m}_6} &\equiv \til... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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48563d1c16768b43ca93d0619455efe15b541cd4 | subsection | 101 | 130 | Exotic-brane solutions in M-theory | \end{split}Again, by computing the dual mixed-symmetry potentials through (REF ), we obtainE_{\bar{0}\cdots \bar{9},\cdots ,\cdots ,\cdots } = -m\,\tau _2^{-1}\,. | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0f9caa1401a1733eeeb70d91173090e65063e3f8 | subsection | 102 | 130 | Solutions for space-filling branes | We have completed the full list of the “elementary” domain-wall solutions.
We can straightforwardly continue the duality rotations to obtain all of the space-filling branes given in Figures REF –REF or their M-theory extensions.
Since there are too many space-filling branes in EFT, we will just show several examples an... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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f390a30546b0be9bdb2b3caf09e6ad0c738b1068 | subsection | 103 | 130 | Solutions for space-filling branes | By performing a T-duality along the x^9-direction in the 2^{6}_5(12,345678) solution, we obtain the 2^{(1,0,0,6)}_5(12,345678,9) solution,{\mathrm {d}}\tilde{s}^2 = \tau _2^{3/2}\, {\mathrm {d}}x^2_{012} + \tau _2^{1/2}\,{\mathrm {d}}x^2_{345678} + \tau _2^{-5/2}\,{\mathrm {d}}x_{9}^2 \,,\qquad \operatorname{e}^{-2\til... | {
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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8c6edd4bc7ed821dd1c74277e5eca24a0ba17a28 | subsection | 104 | 130 | Projection condition for Killing spinors | As it is well known, actions of the standard type II branes, such as the D-branes, are invariant under the half of the spacetime supersymmetry, which is generated by the 32-component Majorana–Weyl Killing spinors \varepsilon _1 and \varepsilon _2 satisfying\Gamma ^{11}\,\varepsilon _1 = \varepsilon _1 \,,\qquad \Gamma ... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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52dcac2f794770c3a0fafc7fd965c611d8f7337a | subsection | 105 | 130 | Projection condition for Killing spinors | Then, the projection condition for each type II brane is expressed as follows (see for a textbook):\begin{split}
\text{P(1)}:&\quad \bigl ({1}\mp \gamma ^{01}\,\mbox{1}\hspace{-2.5pt}\mbox{l}\bigr )\,\epsilon =0\,, \qquad \text{F1(1)}:\quad \bigl ({1}\mp \gamma ^{01}\,\sigma _3\bigr )\, \epsilon =0 \,,
\\
\text{NS5(123... | {
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... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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2980bcd55c37fe532278bc0edcd6f8c700d595a7 | subsection | 106 | 130 | Projection condition for Killing spinors | The projection condition for an exotic brane that electrically couples to the mixed-symmetry potential E^{(n)}_{m_1\cdots m_{a_1},\,\cdots ,\,n_1\cdots n_{a_s}} is given by\bigl ({1}\mp \gamma ^{m_1\cdots m_{a_1}}\cdots \gamma ^{n_1\cdots n_{a_s}}\,\mathcal {O}\bigr )\,\epsilon = 0\,,where \mathcal {O} is a 2\times 2 m... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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e0a89e4d556dc34f9795b1afe32321d5a6467bd9 | subsection | 107 | 130 | Exotic brane solutions in deformed supergravities | In the previous sections, we have constructed various exotic-brane solutions in DFT/EFT.
Unlike the case of the standard branes or the defect branes, the obtained solutions explicitly depend on the dual winding coordinates.
In this section, we explain that the winding-coordinate dependence in the domain-wall solutions ... | {
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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"hep-th"
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a98fcdaa9d99b17df8d3d5e1d04c429988553e2e | subsection | 108 | 130 | Generalized type II supergravity | In order to get a feeling of the deformed supergravity, it is instructive to review the derivation of GSE , from DFT , (see also for a derivation from EFT). | {
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
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2aa74e39b407d09dde9439d1c2d689b7e14e6279 | subsection | 109 | 130 | Bosonic sector of type II DFT | The equations of motion of the type II DFT are given as\mathcal {R}_{MN} + \mathcal {E}_{MN} = 0 \,,\qquad \mathcal {R}= 0 \,, \qquad \partial\partial / K F = 0 ,
where \mathcal {R}_{MN} and \mathcal {R} are the generalized Ricci tensor/scalar, and \mathcal {K} contains the information of \mathcal {H}_{MN} , and t... | {
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"raw": "O. Hohm, S. K. Kwak and B. Zwiebach, “Unification of type II strings and T-duality,” Phys. Rev. Lett. 107, 171603 (2011) [arXiv:1106.5452 [hep-th]].",
"source_ref_id": "303d0b963988aa94e475328... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
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"hep-th"
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ee2b97282cd45aa2b475da477296fdfe0d0e94ea | subsection | 110 | 130 | Bosonic sector of type II DFT | Regarding the R–R field, since the field strength F takes the formF=\operatorname{e}^{-\Phi }\operatorname{e}^{-B_2\wedge }\hat{\mathcal {F}} \,,from (REF ), we suppose that the R–R fields have the following winding-coordinate dependence:F(x) = \operatorname{e}^{-I^m\,\tilde{x}_m} \mathsf {F}(x^i) \,.Since I^m triviall... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
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d0fcf3a850d4d889f90b6ada322c652834e7b99f | subsection | 111 | 130 | Bosonic sector of type II DFT | \end{split}These are precisely the generalized type II supergravity equations of motion , .
When the winding-coordinate dependence vanishes (i.e. I^m=0), they have the same form as the usual supergravity equations of motion.In this manner, we can consider a slight modification of the supergravity equations of motion by... | {
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"raw": "G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A. A. Tseytlin, “Scale invariance of the \\eta -deformed AdS_5\\times S^5 superstring, T-duality and modified type II equations,” Nucl. Phys. B 903, ... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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251841411b9be35bce275b58b102cb523b3c1aa7 | subsection | 112 | 130 | Another viewpoint in terms of the Scherk–Schwarz reduction | As discussed in the addendum of , the ansatz for the dilaton (REF ) can be understood as the Scherk–Schwarz ansatz in DFT , , , , (see also where the derivation of GSE from a Scherk–Schwarz compactification of EFT was originally discussed).
An ansatz\mathcal {H}_{MN}(x,y) = (U^{\mathrm {T}})_M{}^K(y)\, \hat{\mathcal {H... | {
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{
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"doi": "10.1007/jhep04(2017)123",
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"raw": "Y. Sakatani, S. Uehara and K. Yoshida, “Generalized gravity from modified DFT,” JHEP 1704, 123 (2017) [arXiv:1611.05856 [hep-th]].",
"sou... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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8aeff5f9217bf25e9e24b3669f713bbd631a0efb | subsection | 113 | 130 | D8 solutions in the Romans massive type IIA supergravity | Before considering further new examples, let us briefly go back to the well-studied D8-brane solution (REF ).
In this case, the R–R 1-form potential A_1 includes a linear winding-coordinate dependence,A_1(x) = \hat{A}_1(x^i) + m\,\tilde{x}_8\,{\mathrm {d}}x^8 \,,where \hat{A}_1(x^i) does not include the x^8 dependence ... | {
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"doi": "10.1007/jhep11(2011)086",
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"raw": "O. Hohm and S. K. Kwak, “Massive type II in double field theory,” JHEP 1111, 086 (2011) [arXiv:1108.4937 [hep-th]].",
"source_ref_id": "2... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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06c23ed2aac15c1badb2ceefed2e7af963f41d63 | subsection | 114 | 130 | KK8A and M9 solutions | Let us consider the solution of the 7_3^{(1,0)}(1\cdots 7,,8)-brane (), which is also known as the KK8A-brane.
At the same time, we consider its eleven-dimensional uplift, the solution of the 8_{12}^{(1,0)}(1234567z,,8)-brane (REF ).
In this case, the linear winding-coordinate dependence is included in \gamma ^{8} = m\... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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be600ecfb3a1ef07b7bfa3d9dcdfc2f17efa8807 | subsection | 115 | 130 | KK8A and M9 solutions | \end{split}By translating the dual fields into the conventional fields (recall (REF )), we obtain\begin{split}
&{\mathrm {d}}s^2 = \Bigl (\frac{c^2+\tau _2^2}{\tau _2}\Bigr )^{1/2}\, \bigl ({\mathrm {d}}x^2_{01\cdots 7}+\tau _2\,{\mathrm {d}}x_{9}^2\bigr )
+ \tau _2^{-1}\,\Bigl (\frac{c^2+\tau _2^2}{\tau _2}\Bigr )^{-1... | {
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"raw": "P. Meessen and T. Ortín, “An Sl(2,Z) multiplet of nine-dimensional type II supergravity theories,” Nucl. Phys. B 541, 195 (1999) [hep-th/9806120].",
"source_ref_id": "9df00eb9eecea1a7634d6298a... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0ae64f9ea74cf9fdfd9c3fb9d780b4a50511ead2 | subsection | 116 | 130 | KK8A and M9 solutions | This is a solution of a deformed type IIA supergravity.
The eleven-dimensional uplift corresponds to the bound state of the 8_{12}^{(1,0)}(1\cdots 8,,\text{M})-brane and the 8_{12}^{(1,0)}(1\cdots 7\text{M},,8)-brane, and the corresponding background will be a solution of the \mathrm {SL}(2)-covariant eleven-dimensiona... | {
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"source_ref_id": "9df00eb9eecea1a7634d6298a... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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919faeee643a1b535fe69e4f27028ea6fd15ebbe | subsection | 117 | 130 | Conventions | We define the totally antisymmetric delta functions as\delta ^{m_1\cdots m_p}_{n_1\cdots n_p} \equiv \delta ^{[m_1}_{[n_1}\cdots \delta ^{m_p]}_{n_p]} \,,where the antisymmetrization is defined asA_{[m_1\cdots m_n]} \equiv \frac{1}{n!}\,\bigl (A_{m_1\cdots m_n} \pm \text{permutations}\bigr ) \,.We also define the antis... | {
"cite_spans": []
} | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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da68cd6e4432b8f735157d7dd12f88c5f302955e | subsection | 118 | 130 | Parameterizations of the generalized metric in EFT | In this appendix, we review the parameterization of the generalized metric in E_{n(n)} EFT (n\le 7).
We follow the convention used in . | {
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{
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"doi": "10.1093/ptep/ptx038",
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"raw": "Y. Sakatani and S. Uehara, “Connecting M-theory and type IIB parameterizations in exceptional field theory,” PTEP 2017, no. 4, 043B05 (2017) [arXiv... | 10.1007/JHEP09(2018)072 | 1805.12117 | Weaving the Exotic Web | [
"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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fc17188070e5d23d3a0fb53f9f936d8cecf0bdae | subsection | 119 | 130 | M-theory parameterization | When we consider M-theory, we can parameterize the generalized metric \mathcal {M}_{IJ} in terms of the conventional supergravity fields G_{ij}, A_{i_1i_2i_3}, and A_{i_1\cdots i_6} as follows (see , , for earlier works):\begin{split}
&\mathcal {M}_{IJ} = (L_6^\mathrm {T}\,L_3^\mathrm {T}\,\hat{\mathcal {M}}\,L_3\,L_6)... | {
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"Jose J. Fernandez-Melgarejo",
"Tetsuji Kimura",
"Yuho Sakatani"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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