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nLab > Latest Changes: D=5 Maxwell theory

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMay 4th 2020
- (edited May 4th 2020)
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Author: Urs
Format: MarkdownItexstarting something, for the moment mainly to give a home to _[[relation between 5d Maxwell theory and self-dual 3-forms in 6d -- section]]_ <a href="https://ncatlab.org/nlab/revision/D%3D5+Maxwell+theory/1">v1</a>, <a href="https://ncatlab.org/nlab/show/D%3D5+Maxwell+theory">current</a>

starting something, for the moment mainly to give a home to relation between 5d Maxwell theory and self-dual 3-forms in 6d – section

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeMay 4th 2020
- PermaLink
Author: Urs
Format: MarkdownItexadded a section "Relation between 5d Maxwell theory and massive vector mesons in 4d" ([here](https://ncatlab.org/nlab/show/D%3D5+Maxwell+theory#RelationBetween5dMaxwellTheoryAndMassiveVectorMesons)) spelling out a key step in * [[Paul Sutcliffe]], Section 3 of: _Skyrmions, instantons and holography_, JHEP 1008:019, 2010 ([arXiv:1003.0023](https://arxiv.org/abs/1003.0023)) (but not the full construction there, and also using some other assumptions, as I have highlighted in the text -- maybe more later) <a href="https://ncatlab.org/nlab/revision/diff/D%3D5+Maxwell+theory/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D5+Maxwell+theory/3">v3</a>, <a href="https://ncatlab.org/nlab/show/D%3D5+Maxwell+theory">current</a>
added a section “Relation between 5d Maxwell theory and massive vector mesons in 4d” (here) spelling out a key step in
- Paul Sutcliffe, Section 3 of: Skyrmions, instantons and holography, JHEP 1008:019, 2010 (arXiv:1003.0023)
(but not the full construction there, and also using some other assumptions, as I have highlighted in the text – maybe more later)

diff, v3, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeMay 6th 2020
- (edited May 6th 2020)
- PermaLink
Author: Urs
Format: MarkdownItexI am starting a subsection "Relation to massive vector mesosns in 4d -- The Sakai-Sugimoto model" ([here](https://ncatlab.org/nlab/show/D%3D5+Maxwell+theory#SakaiSugimotoModel)), following Sutcliffe. So far I have added a (somewhat lengthy) proof ([here](https://ncatlab.org/nlab/show/D%3D5+Maxwell+theory#SakaiSugimotoEquation)) that the 5d vacuum Maxwell equations on the Sakai-Sugimoto spacetime $$ g \;\coloneqq\; H^{+1} \, \eta_{\mu \nu} d x^\mu \otimes d x^\nu + H^{-1} d z \otimes d z $$ with $$ H(\{x^\mu\}, z) \;\coloneqq\; H(z) \;\coloneqq\; \left( 1 + \frac{z^2}{L^2} \right)^{2/3} $$ are equivalent to the following massive 4d wave equation: $$ \begin{aligned} & \star d \star d A \;=\; 0 \\ \Leftrightarrow \;\;\; & \eta^{\mu \nu}\partial_\mu \partial_\nu A \;+\; H^{1/2} \partial_z \left( H^{3/2} \partial_z A \right) \;=\; 0 \end{aligned} $$ This statement is sort of in the references cited, but in a somewhat roundabout manner. I wanted to write out an explicit statement and proof. <a href="https://ncatlab.org/nlab/revision/diff/D%3D5+Maxwell+theory/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/D%3D5+Maxwell+theory/7">v7</a>, <a href="https://ncatlab.org/nlab/show/D%3D5+Maxwell+theory">current</a>

I am starting a subsection “Relation to massive vector mesosns in 4d – The Sakai-Sugimoto model” (here), following Sutcliffe.

So far I have added a (somewhat lengthy) proof (here) that the 5d vacuum Maxwell equations on the Sakai-Sugimoto spacetime
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>g</mi><mspace width="0.27778em"/><mo>≔</mo><mspace width="0.27778em"/><msup><mi>H</mi> <mrow><mo lspace="0.11111em" rspace="0em">+</mo><mn>1</mn></mrow></msup><mspace width="0.16667em"/><msub><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mo>⊗</mo><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup><mo>+</mo><msup><mi>H</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>z</mi><mo>⊗</mo><mi>d</mi><mi>z</mi></mrow><annotation encoding="application/x-tex"> g \;\coloneqq\; H^{+1} \, \eta_{\mu \nu} d x^\mu \otimes d x^\nu + H^{-1} d z \otimes d z </annotation></semantics></math>$
with
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo stretchy="false">}</mo><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>≔</mo><mspace width="0.27778em"/><mi>H</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>≔</mo><mspace width="0.27778em"/><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><msup><mi>z</mi> <mn>2</mn></msup></mrow><mrow><msup><mi>L</mi> <mn>2</mn></msup></mrow></mfrac><mo>)</mo></mrow> <mrow><mn>2</mn><mo stretchy="false">/</mo><mn>3</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> H(\{x^\mu\}, z) \;\coloneqq\; H(z) \;\coloneqq\; \left( 1 + \frac{z^2}{L^2} \right)^{2/3} </annotation></semantics></math>$
are equivalent to the following massive 4d wave equation:
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd/> <mtd><mo>⋆</mo><mi>d</mi><mo>⋆</mo><mi>d</mi><mi>A</mi><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mn>0</mn></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="0.27778em"/><mspace width="0.27778em"/><mspace width="0.27778em"/></mtd> <mtd><msup><mi>η</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msup><msub><mo>∂</mo> <mi>μ</mi></msub><msub><mo>∂</mo> <mi>ν</mi></msub><mi>A</mi><mspace width="0.27778em"/><mo>+</mo><mspace width="0.27778em"/><msup><mi>H</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><msub><mo>∂</mo> <mi>z</mi></msub><mrow><mo>(</mo><msup><mi>H</mi> <mrow><mn>3</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><msub><mo>∂</mo> <mi>z</mi></msub><mi>A</mi><mo>)</mo></mrow><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} &amp; \star d \star d A \;=\; 0 \\ \Leftrightarrow \;\;\; &amp; \eta^{\mu \nu}\partial_\mu \partial_\nu A \;+\; H^{1/2} \partial_z \left( H^{3/2} \partial_z A \right) \;=\; 0 \end{aligned} </annotation></semantics></math>$
This statement is sort of in the references cited, but in a somewhat roundabout manner. I wanted to write out an explicit statement and proof.

diff, v7, current

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nLab > Latest Changes: D=5 Maxwell theory