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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMar 23rd 2019
- (edited Mar 23rd 2019)
- PermaLink
Author: Urs
Format: MarkdownItexan entry to glorify the combination $$ I_8 \;\coloneqq\; \tfrac{1}{48} \Big( p_2 \;-\; \big( \tfrac{1}{2} p_1\big)^2 \Big) $$ seriously, I'll need to refer to this from various other entries, so it's useful to have a link <a href="https://ncatlab.org/nlab/revision/I8/1">v1</a>, <a href="https://ncatlab.org/nlab/show/I8">current</a>

an entry to glorify the combination
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>I</mi> <mn>8</mn></msub><mspace width="0.27778em"/><mo>≔</mo><mspace width="0.27778em"/><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>48</mn></mfrac></mstyle><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>p</mi> <mn>2</mn></msub><mspace width="0.27778em"/><mo>−</mo><mspace width="0.27778em"/><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mn>2</mn></msup><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> I_8 \;\coloneqq\; \tfrac{1}{48} \Big( p_2 \;-\; \big( \tfrac{1}{2} p_1\big)^2 \Big) </annotation></semantics></math>$
seriously, I’ll need to refer to this from various other entries, so it’s useful to have a link

v1, current
- CommentRowNumber2.
- CommentAuthorUrs
- CommentTimeMar 24th 2019
- PermaLink
Author: Urs
Format: MarkdownItexmade a note on "Inflow to M5-brane anomaly" ([here](https://ncatlab.org/nlab/show/I8#InflowToM5BraneAnomaly)). This is a bit of a story. Should copy this to _[[M5-brane]]_, eventually. <a href="https://ncatlab.org/nlab/revision/diff/I8/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/I8/4">v4</a>, <a href="https://ncatlab.org/nlab/show/I8">current</a>

made a note on “Inflow to M5-brane anomaly” (here). This is a bit of a story. Should copy this to M5-brane, eventually.

diff, v4, current
- CommentRowNumber3.
- CommentAuthorUrs
- CommentTimeNov 10th 2019
- (edited Nov 10th 2019)
- PermaLink
Author: Urs
Format: MarkdownItexadded the statement of $I_8$ appearing as the $O(\ell^6)$ higher curvature correction to 11d supergravity ([here](https://ncatlab.org/nlab/show/I8#AsHigherCurvatureCorrectionTo11dSugra)): *** A systematic analysis of the possible [[supersymmetry|supersymmetric]] [[higher curvature corrections]] of [[D=11 supergravity]] makes the $I_8$ appear as the higher curvature correction at order $\ell^6$, where $\ell$ is the [[Planck length]] in 11d ([Souères-Tsimpis 17, Section 4](https://ncatlab.org/nlab/show/I8#SoueresTsimpis17)). At this order, the [[equation of motion]] for the [[supergravity C-field]] [[flux]] $G_4$ and its dual $G_7$ is ([Souères-Tsimpis 17, (4.3)](https://ncatlab.org/nlab/show/I8#SoueresTsimpis17)) $$ d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge G_4(\ell) + \ell^6 I_8 \,, $$ where the flux forms themselves appear in their higher order corrected form as [[power series]] in the [[Planck length]] $$ G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots $$ $$ G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots $$ ([Souères-Tsimpis 17, (4.4)](https://ncatlab.org/nlab/show/I8#SoueresTsimpis17)) <a href="https://ncatlab.org/nlab/revision/diff/I8/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/I8/11">v11</a>, <a href="https://ncatlab.org/nlab/show/I8">current</a>

added the statement of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">I_8</annotation></semantics></math>$ appearing as the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><msup><mi>ℓ</mi> <mn>6</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(\ell^6)</annotation></semantics></math>$ higher curvature correction to 11d supergravity (here):

A systematic analysis of the possible supersymmetric higher curvature corrections of D=11 supergravity makes the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">I_8</annotation></semantics></math>$ appear as the higher curvature correction at order $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">\ell^6</annotation></semantics></math>$ , where $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math>$ is the Planck length in 11d (Souères-Tsimpis 17, Section 4).

At this order, the equation of motion for the supergravity C-field flux $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">G_4</annotation></semantics></math>$ and its dual $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">G_7</annotation></semantics></math>$ is (Souères-Tsimpis 17, (4.3))
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>d</mi><msub><mi>G</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>∧</mo><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>ℓ</mi> <mn>6</mn></msup><msub><mi>I</mi> <mn>8</mn></msub><mspace width="0.16667em"/><mo>,</mo></mrow><annotation encoding="application/x-tex"> d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge G_4(\ell) + \ell^6 I_8 \,, </annotation></semantics></math>$
where the flux forms themselves appear in their higher order corrected form as power series in the Planck length
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><msub><mi>G</mi> <mn>4</mn></msub><mo>+</mo><msup><mi>ℓ</mi> <mn>6</mn></msup><msubsup><mi>G</mi> <mn>4</mn> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots </annotation></semantics></math>$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>G</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><msub><mi>G</mi> <mn>7</mn></msub><mo>+</mo><msup><mi>ℓ</mi> <mn>6</mn></msup><msubsup><mi>G</mi> <mn>7</mn> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots </annotation></semantics></math>$
(Souères-Tsimpis 17, (4.4))

diff, v11, current
- CommentRowNumber4.
- CommentAuthorDavidRoberts
- CommentTimeNov 10th 2019
- (edited Nov 10th 2019)
- PermaLink
Author: DavidRoberts
Format: MarkdownItexI think what was $$ d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R) $$ should be $$ d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - 2\ell^3 \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R) $$ as the $\ell^3$ correction term looks like it should be, based on the text, $\ell^3 G_4(\ell) \wedge \tfrac{1}{2} p_1(R)$, not $\tfrac{1}{2}G_4(\ell)\wedge \tfrac{1}{2}p_1(R)$. Is this right? <a href="https://ncatlab.org/nlab/revision/diff/I8/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/I8/12">v12</a>, <a href="https://ncatlab.org/nlab/show/I8">current</a>

I think what was
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>d</mi><msub><mi>G</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>+</mo><msup><mi>ℓ</mi> <mn>6</mn></msup><msub><mi>I</mi> <mn>8</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R) </annotation></semantics></math>$
should be
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>d</mi><msub><mi>G</mi> <mn>7</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mspace width="0.27778em"/><mo>=</mo><mspace width="0.27778em"/><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>∧</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><msup><mi>ℓ</mi> <mn>3</mn></msup><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>+</mo><msup><mi>ℓ</mi> <mn>6</mn></msup><msub><mi>I</mi> <mn>8</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - 2\ell^3 \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R) </annotation></semantics></math>$
as the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\ell^3</annotation></semantics></math>$ correction term looks like it should be, based on the text, $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>3</mn></msup><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>∧</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ell^3 G_4(\ell) \wedge \tfrac{1}{2} p_1(R)</annotation></semantics></math>$ , not $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>G</mi> <mn>4</mn></msub><mo stretchy="false">(</mo><mi>ℓ</mi><mo stretchy="false">)</mo><mo>∧</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}G_4(\ell)\wedge \tfrac{1}{2}p_1(R)</annotation></semantics></math>$ . Is this right?

diff, v12, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeNov 10th 2019
- (edited Nov 10th 2019)
- PermaLink
Author: Urs
Format: MarkdownItexYeah, but it's more complicated: They write $tr(R^2)$ instead of $p_1$. The conversion factor for thee two should be $1/4\pi^2$, but of course it depends on conventions. Worse, when deriving the $\ell^6$-correction, they ignore the $\ell^3$-correction but then "absorb a numerical factor" in $\ell^6$ (below (4.3)). So at this point the two $\ell$-s in the two prefactors are not even the same anymore. So what I did in the entry was that I took for the $\ell^3$-term the prefactor that I know from other reasons to be there. Then the $\ell^6$ term is correct for _some_ rescaling of $\ell$. Of course there should really be a comment on that step in the entry. And it would be good to sort out the actual relative factor between these two correction terms. If you are interested in digging into this, please let me know.

Yeah, but it’s more complicated:

They write $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>tr</mi><mo stretchy="false">(</mo><msup><mi>R</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">tr(R^2)</annotation></semantics></math>$ instead of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math>$ . The conversion factor for thee two should be $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>4</mn><msup><mi>π</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1/4\pi^2</annotation></semantics></math>$ , but of course it depends on conventions. Worse, when deriving the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">\ell^6</annotation></semantics></math>$ -correction, they ignore the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\ell^3</annotation></semantics></math>$ -correction but then “absorb a numerical factor” in $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">\ell^6</annotation></semantics></math>$ (below (4.3)). So at this point the two $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math>$ -s in the two prefactors are not even the same anymore.

So what I did in the entry was that I took for the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\ell^3</annotation></semantics></math>$ -term the prefactor that I know from other reasons to be there. Then the $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>ℓ</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">\ell^6</annotation></semantics></math>$ term is correct for some rescaling of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math>$ . Of course there should really be a comment on that step in the entry.

And it would be good to sort out the actual relative factor between these two correction terms. If you are interested in digging into this, please let me know.
- CommentRowNumber6.
- CommentAuthorDavidRoberts
- CommentTimeNov 11th 2019
- PermaLink
Author: DavidRoberts
Format: MarkdownItexSure, I'd be interested to have a look. Teaching is over, so have a bit more time now.

Sure, I’d be interested to have a look. Teaching is over, so have a bit more time now.
- CommentRowNumber7.
- CommentAuthorUrs
- CommentTimeMar 29th 2021
- PermaLink
Author: Urs
Format: MarkdownItexadded pointer to: * [[Paul Howe]], [[Dimitrios Tsimpis]], Around (56) in: _On higher-order corrections in M theory_, JHEP 0309 (2003) 038 ([arXiv:hep-th/0305129](https://arxiv.org/abs/hep-th/0305129)) <a href="https://ncatlab.org/nlab/revision/diff/I8/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/I8/15">v15</a>, <a href="https://ncatlab.org/nlab/show/I8">current</a>
added pointer to:
- Paul Howe, Dimitrios Tsimpis, Around (56) in: On higher-order corrections in M theory, JHEP 0309 (2003) 038 (arXiv:hep-th/0305129)
diff, v15, current

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nLab > Latest Changes: I8