CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en-ca"> <head> <title>nForum - brane scan</title> <link rel="shortcut icon" href="/themes/nforum/styles/nforum/favicon.ico" /> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <link rel="stylesheet" type="text/css" href="/extensions/TagThis/theme/tagthis.css" /> <link rel="stylesheet" type="text/css" href="/themes/nforum/styles/nforum/vanilla.css" media="screen" /> <link rel="stylesheet" type="text/css" href="/themes/nforum/styles/nforum/vanilla.print.css" media="print" /> <link rel="stylesheet" type="text/css" href="/extensions/GuestPost/style.css" /> <link rel="stylesheet" type="text/css" href="/extensions/GuestSignIn/style.css" /> <link rel="stylesheet" type="text/css" href="/extensions/OpenID/css/openid.css" /> <link rel="stylesheet" type="text/css" href="/extensions/PreviewPost/preview.css" /> <script type="text/javascript" src="/js/global.js"></script> <script type="text/javascript" src="/js/vanilla.js"></script> <script type="text/javascript" src="/js/ajax.js"></script> <script type="text/javascript" src="/js/ac.js"></script> <script type="text/javascript" src="/extensions/JQuery/jquery-1.4.2.min.js"></script> <script type="text/javascript" src="/extensions/OpenID/js/openid-jquery.js"></script> <script type="text/javascript" src="/extensions/OpenID/js/openid-en.js"></script> <script type="text/javascript" src="/extensions/MarkdownItex/itex.js"></script> <script type="text/javascript" src="/extensions/MembersList/library/tablesort.js"></script> <script type="text/javascript" src="/extensions/MembersList/library/paginate.js"></script> <script type="text/javascript" src="/extensions/PreviewPost/preview.js"></script> <script type="text/javascript" src="/extensions/CustomStyles/functions.js"></script> <script type="text/javascript" src="/js/prototype.js"></script> <script type="text/javascript" src="/js/scriptaculous.js"></script> <script type="text/javascript" src="/extensions/Notify/functions.js"></script> <link rel="alternate" type="application/rss+xml" href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=RSS2&DiscussionID=4929&FeedTitle=Discussion+Feed+%28brane+scan%29" title="Discussion Feed (RSS2)" /> <link rel="alternate" type="application/atom+xml" href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=ATOM&DiscussionID=4929&FeedTitle=Discussion+Feed+%28brane+scan%29" title="Discussion Feed (ATOM)" /> <script type="text/javascript"> ( function($) { $(document).ready(function() { openid.init('openid_identifier',true); openid.setFormID('frmSignInOpenID'); }); }) (jQuery ); </script><script type="text/x-mathjax-config">MathJax.Hub.Config({TeX: {extensions: ["AMScd.js"]}});</script><script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=MML_HTMLorMML-full"></script></head> <body id="CommentsPage" > <div id="SiteContainer"><div id="Session">Not signed in (<a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F4929%2F">Sign In</a>)</div><div id="Header"> <a id="pgtop"></a> <span id="logo"></span> <h1> nForum </h1> <div id="TagLine">A discussion forum about contributions to the <a href="https://ncatlab.org/">nLab wiki</a> and related areas of mathematics, physics, and philosophy.</div><ul><li><a href="/extension.php?PostBackAction=HomeCat" >Home</a></li><li class="TabOn"><a href="/" >Discussions</a></li><li><a href="/categories.php" >Categories</a></li><li><a href="/search.php" >Search</a></li><li><a href="https://ncatlab.org/nlab/show/HomePage" >nLab</a></li><li><a href="https://ncatlab.org/nlabmeta/show/Welcome+to+the+nForum" >Help</a></li></ul> </div><div id="Body"><div id="Panel"><ul> <li> <h2>Discussion Feed</h2> <ul><li> <a href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=RSS2&DiscussionID=4929&FeedTitle=Discussion+Feed+%28brane+scan%29" title="Subscribe to this feed..." class="RSS2">RSS2</a></li><li> <a href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=ATOM&DiscussionID=4929&FeedTitle=Discussion+Feed+%28brane+scan%29" title="Subscribe to this feed..." class="ATOM">ATOM</a></li></ul> </li> </ul><div id="GuestSignIn"> <h2>Not signed in</h2> <p>Want to take part in these discussions? Sign in if you have an account, or apply for one below</p> <fieldset><form id="frmSignInUser" method="post" action="https://nforum.ncatlab.org/people/"> <div> <input name="PostBackAction" value="SignIn" type="hidden" /> <input name="ReturnUrl" value="http://nforum.ncatlab.org/discussion/4929/" type="hidden" /> <ul> <li><label for="txtUsername">Username</label> <input id="txtUsername" name="Username" value="" class="Input" maxlength="20" type="text" /> </li> <li><label for="txtPassword">Password</label> <input id="txtPassword" name="Password" value="" class="Input" type="password" /> </li> <li id="RememberMe"><label for="RememberMeID"> <input name="RememberMe" value="1" id="RememberMeID" type="checkbox" checked="checked" /> Remember me</label> </li> <li> <input name="userSignIn" value="Sign In" class="Button" type="submit" /> </li> </ul> </div> </form> </fieldset> <fieldset><form id="frmSignInOpenID" method="post" action="https://nforum.ncatlab.org/people/"> <input name="PostBackAction" value="SignIn" type="hidden" /> <div> <ul> <li>Sign in using OpenID <div id="openid_choice"> <div id="openid_btns"></div> </div> <div id="openid_input_area"> <input id="openid_identifier" name="openid_identifier" type="text" value="http://" /> </div> </li> <li id="RememberMe"><label for="RememberMeID"> <input name="RememberMe" value="1" id="RememberMeID" type="checkbox" checked="checked" /> Remember me</label> </li> <li><input name="openidSignIn" value="Sign In" class="Button" type="submit" /></li> </ul> </div></form></fieldset> <ul class="MembershipOptionLinks"> <li class="ForgotPasswordLink"><a href="https://nforum.ncatlab.org/people/?PostBackAction=PasswordRequestForm">Forgot your password?</a></li> <li class="ApplyForMembershipLink"><a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">Apply for membership</a></li> </ul></div><h2>Discussion Tag Cloud</h2><div id="TagCloud"><span style="font-size:200%"><a href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Topics&Tag=cohomology" class="TagLink">cohomology</a></span> <span style="font-size:111%"><a href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Topics&Tag=string-theory" class="TagLink">string-theory</a></span> <span style="font-size:79%"><a href="https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Topics&Tag=supergeometry" class="TagLink">supergeometry</a></span></div><p id="AboutVanilla"><a href="http://getvanilla.com">Vanilla 1.1.10</a> is a product of <a href="http://lussumo.com">Lussumo</a>. More Information: <a href="http://lussumo.com/docs">Documentation</a>, <a href="http://lussumo.com/community">Community Support</a>.</p></div> <div id="Content"><div id="NoticeCollector" class="Notices"><div class="Notice"><strong>Welcome to nForum</strong> <br />If you want to take part in these discussions either <a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F4929%2F">sign in now</a> (if you have an account), <a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">apply for one now</a> (if you don't).</div></div><div class="ContentInfo Top"> <h1><a href="https://nforum.ncatlab.org/5/"><a href="https://nforum.ncatlab.org/18/">nLab</a> > </a> <a href="https://nforum.ncatlab.org/5/">Latest Changes</a>: brane scan</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 22 of 22</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div> <div id="ContentBody"> <script type="text/javascript"> //<![CDATA[ function toggle_source(id) { var mysrc = document.getElementById("CommentBody_" + id).firstChild; if (mysrc.className == "source") { if (mysrc.style.display == "none") { mysrc.style.display = "block"; } else { mysrc.style.display = "none"; } } } var commentIds = new Array(0); function hide_sources() { for (i = 0; i < commentIds.length; i++) { var myself = document.getElementById("Source" + commentIds[i]); var mycmt = document.getElementById("CommentBody_" + commentIds[i]); if (mycmt.firstChild.className != "source") { myself.style.display = "none"; } } } window.onload = hide_sources; //]]> </script> <ol id="Comments"><li id="Comment_39707"> <a id="Item_1"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>1.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>May 15th 2013</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=39707#Comment_39707">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_39707"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I gave the [[brane scan]] table a genuine $n$Lab incarnation and included it at _[[Green-Schwarz action functional]]_ and at _[[brane]]_.</code></div><div> <p>I gave the <a href="https://ncatlab.org/nlab/show/brane scan">brane scan</a> table a genuine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>Lab incarnation and included it at <em><a href="https://ncatlab.org/nlab/show/Green-Schwarz action functional">Green-Schwarz action functional</a></em> and at <em><a href="https://ncatlab.org/nlab/show/brane">brane</a></em>.</p> </div> </div> </li><li id="Comment_39735" class="Alternate"> <a id="Item_2"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>2.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/24/">Todd_Trimble</a></li> <li><span>CommentTime</span>May 16th 2013</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=39735#Comment_39735">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_39735"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/24/">Todd_Trimble</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I guess the pun is intentional? (sounds like brain scan)</code></div><div> <p>I guess the pun is intentional? (sounds like brain scan)</p> </div> </div> </li><li id="Comment_39737"> <a id="Item_3"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>3.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>May 16th 2013</li><li><em>(edited May 16th 2013)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=39737#Comment_39737">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_39737"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I think so. That's what it is being called since the 90s.</code></div><div> <p>I think so. That’s what it is being called since the 90s.</p> </div> </div> </li><li id="Comment_39792" class="Alternate"> <a id="Item_4"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>4.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>May 21st 2013</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=39792#Comment_39792">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_39792"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>added to the _[[brane scan]]_ now the type IIA D-branes and their characterizing super-$L_\infty$-algebra cocycles on the [[type II supergravity Lie 2-algebra]].</code></div><div> <p>added to the <em><a href="https://ncatlab.org/nlab/show/brane scan">brane scan</a></em> now the type IIA D-branes and their characterizing super-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra cocycles on the <a href="https://ncatlab.org/nlab/show/type II supergravity Lie 2-algebra">type II supergravity Lie 2-algebra</a>.</p> </div> </div> </li><li id="Comment_90413"> <a id="Item_5"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>5.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Mar 12th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=90413#Comment_90413">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_90413"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>hyperlinked one more item in the table: the _[[super 4-brane in 9d]]_ <a href="https://ncatlab.org/nlab/revision/diff/brane+scan/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/brane+scan/15">v15</a>, <a href="https://ncatlab.org/nlab/show/brane+scan">current</a></code></div><div> <p>hyperlinked one more item in the table: the <em><a href="https://ncatlab.org/nlab/show/super 4-brane in 9d">super 4-brane in 9d</a></em></p> <p><a href="https://ncatlab.org/nlab/revision/diff/brane+scan/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/brane+scan/15">v15</a>, <a href="https://ncatlab.org/nlab/show/brane+scan">current</a></p> </div> </div> </li><li id="Comment_97109" class="Alternate"> <a id="Item_6"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>6.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Dec 23rd 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=97109#Comment_97109">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_97109"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I have hyperlinked two more of the remaining entries in the brane scan: * [[super 2-brane in 5d]] and * [[super 3-brane in 8d]] Am creating the respective entries now... <a href="https://ncatlab.org/nlab/revision/diff/brane+scan/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/brane+scan/16">v16</a>, <a href="https://ncatlab.org/nlab/show/brane+scan">current</a></code></div><div> <p>I have hyperlinked two more of the remaining entries in the brane scan:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/super 2-brane in 5d">super 2-brane in 5d</a></li> </ul> <p>and</p> <ul> <li><a href="https://ncatlab.org/nlab/show/super 3-brane in 8d">super 3-brane in 8d</a></li> </ul> <p>Am creating the respective entries now…</p> <p><a href="https://ncatlab.org/nlab/revision/diff/brane+scan/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/brane+scan/16">v16</a>, <a href="https://ncatlab.org/nlab/show/brane+scan">current</a></p> </div> </div> </li><li id="Comment_112620"> <a id="Item_7"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>7.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Sep 3rd 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112620#Comment_112620">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112620"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>What are pullbacks of D-branes supposed to correspond to? Say pb(d1brane->stringIIB<-d3brane) ?</code></div><div> <p>What are pullbacks of D-branes supposed to correspond to? Say pb(d1brane->stringIIB<-d3brane) ?</p> </div> </div> </li><li id="Comment_112621" class="Alternate"> <a id="Item_8"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>8.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Sep 3rd 2023</li><li><em>(edited Sep 3rd 2023)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112621#Comment_112621">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112621"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>As you know (probably that's why you are asking) generally pullbacks of this form may serve in correspondences/dualities between one theory and the other. For instance, if instead of D-branes we consider M2-branes, then the pullback in question features in the higher M-theoretic analog of T-duality (Prop. 4.4 on [p. 32 here](https://arxiv.org/pdf/1803.05634.pdf#page=32)), at least rationally (further exposition of this example is in [arXiv:1805.00233](https://arxiv.org/abs/1805.00233), although the pullback diagram is not displayed there): $$ \array{ \mathfrak{m}2\mathfrak{brane} &&&& \mathfrak{m}2\mathfrak{brane} \\ & \searrow && \swarrow \\ && \mathbb{R}^{10,1\vert\mathbf{32}} } $$ But this requires that there are further cocycles satisfying conditions on the two sides of the pullback (the M5-banes in the above example) which should not be the case for D-branes. (This last statement, that the D-brane extensions carry no further invariant extensions in the brane scan, remains without rigorous proof; but it would mean another brane species on which D-branes can end like M2s end on M5s, and folklore does not expect this to exist.) So I don't know that there is any significance to the fiber products of D-brane cocycles. But there is much room left to explore the idea of the brane bouquet further.</code></div><div> <p>As you know (probably that’s why you are asking) generally pullbacks of this form may serve in correspondences/dualities between one theory and the other.</p> <p>For instance, if instead of D-branes we consider M2-branes, then the pullback in question features in the higher M-theoretic analog of T-duality (Prop. 4.4 on <a href="https://arxiv.org/pdf/1803.05634.pdf#page=32">p. 32 here</a>), at least rationally (further exposition of this example is in <a href="https://arxiv.org/abs/1805.00233">arXiv:1805.00233</a>, although the pullback diagram is not displayed there):</p> <math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mrow><mtable><mtr><mtd><mi>&mfr;</mi><mn>2</mn><mi>&bfr;&rfr;&afr;&nfr;&efr;</mi></mtd> <mtd/> <mtd/> <mtd/> <mtd><mi>&mfr;</mi><mn>2</mn><mi>&bfr;&rfr;&afr;&nfr;&efr;</mi></mtd></mtr> <mtr><mtd/> <mtd><mo>&searr;</mo></mtd> <mtd/> <mtd><mo>&swarr;</mo></mtd></mtr> <mtr><mtd/> <mtd/> <mtd><msup><mi>&Ropf;</mi> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mn>32</mn></mstyle></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathfrak{m}2\mathfrak{brane} &&&& \mathfrak{m}2\mathfrak{brane} \\ & \searrow && \swarrow \\ && \mathbb{R}^{10,1\vert\mathbf{32}} } </annotation></semantics></math> <p>But this requires that there are further cocycles satisfying conditions on the two sides of the pullback (the M5-banes in the above example) which should not be the case for D-branes.</p> <p>(This last statement, that the D-brane extensions carry no further invariant extensions in the brane scan, remains without rigorous proof; but it would mean another brane species on which D-branes can end like M2s end on M5s, and folklore does not expect this to exist.)</p> <p>So I don’t know that there is any significance to the fiber products of D-brane cocycles. But there is much room left to explore the idea of the brane bouquet further.</p> </div> </div> </li><li id="Comment_112623"> <a id="Item_9"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>9.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Sep 3rd 2023</li><li><em>(edited Sep 3rd 2023)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112623#Comment_112623">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112623"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Brane molecule. <a href="https://ncatlab.org/nlab/revision/diff/brane+scan/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/brane+scan/17">v17</a>, <a href="https://ncatlab.org/nlab/show/brane+scan">current</a></code></div><div> <p>Brane molecule.</p> <p><a href="https://ncatlab.org/nlab/revision/diff/brane+scan/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/brane+scan/17">v17</a>, <a href="https://ncatlab.org/nlab/show/brane+scan">current</a></p> </div> </div> </li><li id="Comment_112624" class="Alternate"> <a id="Item_10"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>10.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Sep 3rd 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112624#Comment_112624">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112624"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>You want to expand on that?</code></div><div> <p>You want to expand on that?</p> </div> </div> </li><li id="Comment_112625"> <a id="Item_11"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>11.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Sep 3rd 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112625#Comment_112625">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112625"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Seems like the info about the brane scan/bouquet is scattered amongst different pages, e.g. [[brane scan]], [[brane]] (which seems to have brane scan embedded on it), [[table of branes]], and [brane bouquet](https://ncatlab.org/schreiber/show/The+brane+bouquet), which is actually part of your personal pages. Ideally, there should be a single entry with all this info but on the other hand [[brane scan]] is probably better reserved for the *old* brane scan. Any suggestions?</code></div><div> <p>Seems like the info about the brane scan/bouquet is scattered amongst different pages, e.g. <a href="https://ncatlab.org/nlab/show/brane scan">brane scan</a>, <a href="https://ncatlab.org/nlab/show/brane">brane</a> (which seems to have brane scan embedded on it), <a href="https://ncatlab.org/nlab/show/table of branes">table of branes</a>, and <a href="https://ncatlab.org/schreiber/show/The+brane+bouquet">brane bouquet</a>, which is actually part of your personal pages. Ideally, there should be a single entry with all this info but on the other hand <a href="https://ncatlab.org/nlab/show/brane scan">brane scan</a> is probably better reserved for the <em>old</em> brane scan. Any suggestions?</p> </div> </div> </li><li id="Comment_112626" class="Alternate"> <a id="Item_12"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>12.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Sep 3rd 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112626#Comment_112626">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112626"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>You could start a page *[[brane bouquet]]* on the main web, that would be useful.</code></div><div> <p>You could start a page <em><a href="https://ncatlab.org/nlab/show/brane bouquet">brane bouquet</a></em> on the main web, that would be useful.</p> </div> </div> </li><li id="Comment_112709"> <a id="Item_13"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>13.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Sep 7th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112709#Comment_112709">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112709"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Will do. In [M-theory from the Superpoint](https://arxiv.org/abs/1702.01774), what is the precise role of considering extensions invariant with respect to the ‘simple external automorphisms’ inside the automorphism group of g? Is this what allows us to get extensions that basically correspond to the coset space (in this case Minkowski spaces as coset space of, at the level of algebras, the super-Poincare algebra modulo Lorentz algebra)? What if one drops this assumption, can one get the super-Poincare algebra instead? The reason I ask is the following. So far in these super- extensions of the Poincare algebra, we consider the anti-commutator of the supercharges to be proportional to momentum generator. In this case we can talk about R^{d-1,1|N} where the bracket is this pairing. However, in [The twelve dimensional super (2+2)-brane](https://arxiv.org/abs/hep-th/9612008), the main point is that one needs to consider different super- extensions of the Poincare algebra. In particular, the extensions that turns out to be the correct/useful one is that for which the anticommutator of the supercharges is not proportional to momentum but is a linear combination of momenta and rotations. So if I want my algebra extensions/cocycles to involve rotations, then it seems it cannot be an maximally invariant extension of some Minkowski space?</code></div><div> <p>Will do.</p> <p>In <a href="https://arxiv.org/abs/1702.01774">M-theory from the Superpoint</a>, what is the precise role of considering extensions invariant with respect to the ‘simple external automorphisms’ inside the automorphism group of g? Is this what allows us to get extensions that basically correspond to the coset space (in this case Minkowski spaces as coset space of, at the level of algebras, the super-Poincare algebra modulo Lorentz algebra)? What if one drops this assumption, can one get the super-Poincare algebra instead?</p> <p>The reason I ask is the following. So far in these super- extensions of the Poincare algebra, we consider the anti-commutator of the supercharges to be proportional to momentum generator. In this case we can talk about R^{d-1,1|N} where the bracket is this pairing. However, in <a href="https://arxiv.org/abs/hep-th/9612008">The twelve dimensional super (2+2)-brane</a>, the main point is that one needs to consider different super- extensions of the Poincare algebra. In particular, the extensions that turns out to be the correct/useful one is that for which the anticommutator of the supercharges is not proportional to momentum but is a linear combination of momenta and rotations. So if I want my algebra extensions/cocycles to involve rotations, then it seems it cannot be an maximally invariant extension of some Minkowski space?</p> </div> </div> </li><li id="Comment_112710" class="Alternate"> <a id="Item_14"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>14.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Sep 7th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112710#Comment_112710">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112710"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>The motivation for restricting to the "external" automorphisms is meant to be illuminated by "Example 2" on that same [p. 9](https://arxiv.org/pdf/1702.01774.pdf#page=9): The "internal" ones are the "R-symmetries" -- in particular, "external"/"internal" is in the physics sense of "spacetime symmetry"/"internal DOFs symmetry". Since we are after the (ordinary) spacetime symmetry algebras, this makes it natural to discard all the internal symmetry and focus on the external one. But in retrospect the definition is really justified by the result it yields. This whole business of emergence of super-spacetimes here clearly feels just like the tip of an iceberg that remains to be uncovered. So if you want to look at variations and see what these yield, that could be most interesting.</code></div><div> <p>The motivation for restricting to the “external” automorphisms is meant to be illuminated by “Example 2” on that same <a href="https://arxiv.org/pdf/1702.01774.pdf#page=9">p. 9</a>:</p> <p>The “internal” ones are the “R-symmetries” – in particular, “external”/”internal” is in the physics sense of “spacetime symmetry”/”internal DOFs symmetry”.</p> <p>Since we are after the (ordinary) spacetime symmetry algebras, this makes it natural to discard all the internal symmetry and focus on the external one.</p> <p>But in retrospect the definition is really justified by the result it yields.</p> <p>This whole business of emergence of super-spacetimes here clearly feels just like the tip of an iceberg that remains to be uncovered. So if you want to look at variations and see what these yield, that could be most interesting.</p> </div> </div> </li><li id="Comment_112713"> <a id="Item_15"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>15.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Sep 7th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112713#Comment_112713">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112713"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Just a very basic question. In the Outlook section where you describe the (unique) central extension of the superpoint R^{0,0|1}, how does one see that the resulting extension is R^{1,0|1} and not R^{0,1|1}? Does this mean R^{0,1|1} is itself the root of a different bouquet that is not the one emanating from the superpoint?</code></div><div> <p>Just a very basic question. In the Outlook section where you describe the (unique) central extension of the superpoint R^{0,0|1}, how does one see that the resulting extension is R^{1,0|1} and not R^{0,1|1}? Does this mean R^{0,1|1} is itself the root of a different bouquet that is not the one emanating from the superpoint?</p> </div> </div> </li><li id="Comment_112716" class="Alternate"> <a id="Item_16"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>16.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Sep 8th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112716#Comment_112716">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112716"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>It is $\mathbb{R}^{0, 1\vert 1}$. Maybe I was careless in dropping the "0" in that Outlook section, or maybe I was meaning to indicate that in this degenerate case the distinction is not relevant. In previous lecture notes [here](https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#ExtensionFromSuperpoint) I did write $\mathbb{R}^{0, 1\vert 1}$. This is the super Lie algebra of supersymmetric quantum mechanics, where the single supercharge squares to the Hamiltonian, hence to the time-translation generator. On the other hand, if you'd thought of it as a supersymmetric 1-spatial translation algebra it would look no different, whence one could argue that keeping the extra "0" around is meaningless.</code></div><div> <p>It is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{0, 1\vert 1}</annotation></semantics></math>. Maybe I was careless in dropping the “0” in that Outlook section, or maybe I was meaning to indicate that in this degenerate case the distinction is not relevant.</p> <p>In previous lecture notes <a href="https://ncatlab.org/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes#ExtensionFromSuperpoint">here</a> I did write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{0, 1\vert 1}</annotation></semantics></math>.</p> <p>This is the super Lie algebra of supersymmetric quantum mechanics, where the single supercharge squares to the Hamiltonian, hence to the time-translation generator.</p> <p>On the other hand, if you’d thought of it as a supersymmetric 1-spatial translation algebra it would look no different, whence one could argue that keeping the extra “0” around is meaningless.</p> </div> </div> </li><li id="Comment_112719"> <a id="Item_17"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>17.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Sep 8th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=112719#Comment_112719">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_112719"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Sounds good, thanks Urs.</code></div><div> <p>Sounds good, thanks Urs.</p> </div> </div> </li><li id="Comment_113350" class="Alternate"> <a id="Item_18"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>18.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Oct 14th 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=113350#Comment_113350">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_113350"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>In [1308.5264](https://arxiv.org/abs/1308.5264) we talk of *the* heterotic string $L_{\infty}$-algebra $\mathfrak{string}_{\text{het}}$. Where does the discussion of $E_8\times E_8$ and $\text{Spin}(32)/Z_2$ enter the picture?</code></div><div> <p>In <a href="https://arxiv.org/abs/1308.5264">1308.5264</a> we talk of <em>the</em> heterotic string <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_{\infty}</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>&sfr;&tfr;&rfr;&ifr;&nfr;&gfr;</mi> <mtext>het</mtext></msub></mrow><annotation encoding="application/x-tex">\mathfrak{string}_{\text{het}}</annotation></semantics></math>. Where does the discussion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8\times E_8</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mtext>Spin</mtext><mo stretchy="false">(</mo><mn>32</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>Z</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\text{Spin}(32)/Z_2</annotation></semantics></math> enter the picture?</p> </div> </div> </li><li id="Comment_113351"> <a id="Item_19"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>19.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Oct 14th 2023</li><li><em>(edited Oct 14th 2023)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=113351#Comment_113351">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_113351"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Yes, the brane scan, as is, only produces the heterotic string in its trivial background fields (flat space, no background gauge fields). This is actually no different from the other strings in the scan, only that for these it's not as notable, since they have no nonabelian background fields anyways. The superspace formulation of heterotic sugra with background fields, from which the super $L_\infty$-algebra of the heterotic string *would* be derivable if it were there, is in the references listed [here](https://ncatlab.org/nlab/show/heterotic%20string%20theory#SuperspaceFormulation). We once tried but never managed to connect this to the brane bouquet. The closest we came to seeing the heterotic string is in [[schreiber:Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation|arXiv:2008.08544]], where it certainly shows up in some guise, but rather more indirectly than one may have expected. Incidentally, something similar happens in the $E_{10}$/$E_{11}$-approach to M-theory. There, the heterotic string also does not show up. "Well, too bad for the heterotic string, then!", as Hermann Nicolai once said ([here](https://golem.ph.utexas.edu/category/2006/11/nicolai_on_e10_and_supergravit.html#c006336)). :-)</code></div><div> <p>Yes, the brane scan, as is, only produces the heterotic string in its trivial background fields (flat space, no background gauge fields). This is actually no different from the other strings in the scan, only that for these it’s not as notable, since they have no nonabelian background fields anyways.</p> <p>The superspace formulation of heterotic sugra with background fields, from which the super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra of the heterotic string <em>would</em> be derivable if it were there, is in the references listed <a href="https://ncatlab.org/nlab/show/heterotic%20string%20theory#SuperspaceFormulation">here</a>. We once tried but never managed to connect this to the brane bouquet.</p> <p>The closest we came to seeing the heterotic string is in <a href="https://ncatlab.org/schreiber/show/Twistorial Cohomotopy implies Green-Schwarz anomaly cancellation">arXiv:2008.08544</a>, where it certainly shows up in some guise, but rather more indirectly than one may have expected.</p> <p>Incidentally, something similar happens in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi> <mn>10</mn></msub></mrow><annotation encoding="application/x-tex">E_{10}</annotation></semantics></math>/<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi> <mn>11</mn></msub></mrow><annotation encoding="application/x-tex">E_{11}</annotation></semantics></math>-approach to M-theory. There, the heterotic string also does not show up.</p> <p>“Well, too bad for the heterotic string, then!”, as Hermann Nicolai once said (<a href="https://golem.ph.utexas.edu/category/2006/11/nicolai_on_e10_and_supergravit.html#c006336">here</a>). :-)</p> </div> </div> </li><li id="Comment_113353" class="Alternate"> <a id="Item_20"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>20.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Oct 14th 2023</li><li><em>(edited Oct 14th 2023)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=113353#Comment_113353">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_113353"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>But isn't it weird that $\mathfrak{string}_{\text{het}}$ just appears from a cocycle in $\mathbb{R}^{9,1\vert N=1}$ without any reference to the left-moving bosonic string modes? For instance, $\mathfrak{string}_{\text{IIB}}$ arises from an extension of $\mathbb{R}^{9,1\vert 16+\overline{16}}$, which we can regard as a pullback of a span where one leg is $\mathbb{R}^{9,1\vert 16}$ and the other is $\mathbb{R}^{9,1\vert \overline{16}}$. Can one read this as a left-moving, right-moving statement? If so, then shouldn't the full heterotic algebra arise as a similar pullback where now one leg is $\mathbb{R}^{9,1\vert 16}$ and the other is $\mathbb{R}^{25,1\vert 1}$? There some of the stuff appearing [here](https://math.ucr.edu/home/baez/octonions/node19.html) would appear, I'd think.</code></div><div> <p>But isn’t it weird that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>&sfr;&tfr;&rfr;&ifr;&nfr;&gfr;</mi> <mtext>het</mtext></msub></mrow><annotation encoding="application/x-tex">\mathfrak{string}_{\text{het}}</annotation></semantics></math> just appears from a cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mi>N</mi><mo>=</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{9,1\vert N=1}</annotation></semantics></math> without any reference to the left-moving bosonic string modes? For instance, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>&sfr;&tfr;&rfr;&ifr;&nfr;&gfr;</mi> <mtext>IIB</mtext></msub></mrow><annotation encoding="application/x-tex">\mathfrak{string}_{\text{IIB}}</annotation></semantics></math> arises from an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>16</mn><mo>+</mo><mover><mn>16</mn><mo>¯</mo></mover></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{9,1\vert 16+\overline{16}}</annotation></semantics></math>, which we can regard as a pullback of a span where one leg is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>16</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{9,1\vert 16}</annotation></semantics></math> and the other is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mover><mn>16</mn><mo>¯</mo></mover></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{9,1\vert \overline{16}}</annotation></semantics></math>. Can one read this as a left-moving, right-moving statement? If so, then shouldn’t the full heterotic algebra arise as a similar pullback where now one leg is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>9</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>16</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{9,1\vert 16}</annotation></semantics></math> and the other is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>25</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{25,1\vert 1}</annotation></semantics></math>? There some of the stuff appearing <a href="https://math.ucr.edu/home/baez/octonions/node19.html">here</a> would appear, I’d think.</p> </div> </div> </li><li id="Comment_113355"> <a id="Item_21"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>21.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/4/">Urs</a></li> <li><span>CommentTime</span>Oct 15th 2023</li><li><em>(edited Oct 15th 2023)</em></li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=113355#Comment_113355">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_113355"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/4/">Urs</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>I see where you are coming from, but you are thinking of the [[NSR superstring]]-type worldsheet formulation, while the brane scan sees the [[Green-Schwarz superstring]] formulation with target space supersymmetry. Specifically those cocycles in the brane scan are WZ-terms made entirely from target space super-fields. In this target space formulation, the left-moving worldsheet bosons of the NSR-type formulation of the heterotic string appear as the non-abelian target space background field, and hence don't appear if no such background field is considered. You can see this on the first pages of [Shapiro & Taylor 1987 ](https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#ShapiroTaylor87). Of course in this case the GS-form of the heterotic string essentially coincides with that of the Type-I string, and maybe that's a more suggestive way of speaking about it. But if you feel adventurous, maybe you can discover a new rule and a previously unrecognized branch of the bouquet.</code></div><div> <p>I see where you are coming from, but you are thinking of the <a href="https://ncatlab.org/nlab/show/NSR superstring">NSR superstring</a>-type worldsheet formulation, while the brane scan sees the <a href="https://ncatlab.org/nlab/show/Green-Schwarz superstring">Green-Schwarz superstring</a> formulation with target space supersymmetry.</p> <p>Specifically those cocycles in the brane scan are WZ-terms made entirely from target space super-fields. In this target space formulation, the left-moving worldsheet bosons of the NSR-type formulation of the heterotic string appear as the non-abelian target space background field, and hence don’t appear if no such background field is considered.</p> <p>You can see this on the first pages of <a href="https://ncatlab.org/nlab/show/Green-Schwarz+action+functional#ShapiroTaylor87">Shapiro & Taylor 1987 </a>.</p> <p>Of course in this case the GS-form of the heterotic string essentially coincides with that of the Type-I string, and maybe that’s a more suggestive way of speaking about it.</p> <p>But if you feel adventurous, maybe you can discover a new rule and a previously unrecognized branch of the bouquet.</p> </div> </div> </li><li id="Comment_113783" class="Alternate"> <a id="Item_22"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>22.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></li> <li><span>CommentTime</span>Nov 2nd 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/4929/brane-scan/?Focus=113783#Comment_113783">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_113783"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/3179/">perezl.alonso</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>If the content of [[non-Lorentzian type II string theory]] is meaningful then one should be able to use say [1806.01115](https://arxiv.org/abs/1806.01115) to get some of those things starting from M theory. In particular, (72) is where we specify the circle fiber is space-like and not time-like, right? Then to get $IIA_E$ one first should express $\mathbb{R}^{10,1\vert \mathbf{32}}$ as an extension of some Euclidean $\mathbb{R}^{10,0\vert ?}$ (with some ? spin representation). Of course, the ideal situation would be to first discover such Euclidean space itself as some extension, this is where I think something like the split- or [para-algebras](https://ncatlab.org/nlab/show/composition%20algebra#para_and_nonunital_composition_algebras) become relevant. In fact, I would think these bring up another branch stemming from $\mathbb{R}^{3,1}$, since only for the real numbers their para- variant is isomorphic to itself, but not for complex numbers and beyond.</code></div><div> <p>If the content of <a href="https://ncatlab.org/nlab/show/non-Lorentzian type II string theory">non-Lorentzian type II string theory</a> is meaningful then one should be able to use say <a href="https://arxiv.org/abs/1806.01115">1806.01115</a> to get some of those things starting from M theory. In particular, (72) is where we specify the circle fiber is space-like and not time-like, right? Then to get <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>IIA</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">IIA_E</annotation></semantics></math> one first should express <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mn>32</mn></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{10,1\vert \mathbf{32}}</annotation></semantics></math> as an extension of some Euclidean <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>10</mn><mo>,</mo><mn>0</mn><mo stretchy="false">|</mo><mo>?</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{10,0\vert ?}</annotation></semantics></math> (with some ? spin representation). Of course, the ideal situation would be to first discover such Euclidean space itself as some extension, this is where I think something like the split- or <a href="https://ncatlab.org/nlab/show/composition%20algebra#para_and_nonunital_composition_algebras">para-algebras</a> become relevant. In fact, I would think these bring up another branch stemming from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>&Ropf;</mi> <mrow><mn>3</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{3,1}</annotation></semantics></math>, since only for the real numbers their para- variant is isomorphic to itself, but not for complex numbers and beyond.</p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 22 of 22</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>