{"id":5520,"date":"2020-01-17T08:23:39","date_gmt":"2020-01-17T08:23:39","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=5520"},"modified":"2020-01-17T08:57:06","modified_gmt":"2020-01-17T08:57:06","slug":"riemannian-q-manifolds-and-their-modular-class","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2020\/01\/17\/riemannian-q-manifolds-and-their-modular-class\/","title":{"rendered":"Riemannian Q-manifolds and their modular class"},"content":{"rendered":"\n\n\n
\"board\"<\/td>\nIn a preprint Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields<\/a>, I examine the notion of a supermanifold equipped with an even Riemannian metric and an odd Killing vector field that is also homological. <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Background<\/strong>
\nIn a previous post<\/a>, I briefly disscussed the notion of a Q-manifold and their modular classes. This itself was based on my paper Modular classes of Q-manifolds: a review and some applications<\/a>, in which I review the notion of the modular class of a Q-manifold (see [1]) and present several illustrative examples.<\/p>\n

Q-manifolds have become an important part of mathematical physics due to their prominence in the AKSZ-formalism in topological quantum field theory<\/a> and the conceptionally neat formalism they provide for describe Lie algebroids<\/a> and Courant algebroids<\/a>, as well as various generalisations thereof. <\/p>\n

The preprint Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields<\/a> is a direct continuation in which I present a class of rather natural Q-manifolds that have vanishing modular class, I call theses Riemannian Q-manifolds<\/em>.<\/p>\n

Riemannian Q-manifolds<\/strong>
\nA Riemannian supermanifold we understand as a supermanifold<\/a> equipped with an even Riemannian metric<\/a>. These are, of course, a natural generalisation of pesudo-Riemannian manifolds<\/a>, which are well studied objects. There are a few ways of interpreting the metric, such as the formal definition as a pairing of vector fields<\/a>, but more informally a Riemannian metric defines a distance between “near-by points”. When dealing with supermanifolds this needs to be taken with a pinch of salt as supermanifolds are not truly defined via their points – we are really dealing with noncommutative algebraic geometry<\/a>, though the noncommutativity is rather mild. There is also the issue of even and odd Riemannian metrics on supermanifolds, but here I will only deal with even ones. <\/p>\n

Infinitesimal symmetries of a Riemannian supermanifold or Riemannian manifold<\/a> are generated by Killing vector fields<\/a>. Loosley, Killing vector fields give you “directions” in which the Riemannian metric does not change. <\/p>\n

On supermanifolds, we have even and odd vector fields, this is very natural as everything on a supermanifold is \\(\\mathbb{Z}_2\\)-graded. The simplest way to describe this is via their Lie bracket. So suppose that we have an even vector field \\(X\\), then the Lie bracket<\/a> with itself vanishes automatically as<\/p>\n

\\( [X,X] = X \\circ X – X \\circ X =0\\).<\/p>\n

However, the definition of the Lie bracket for odd vector fields gives<\/p>\n

\\( [Q,Q] = Q \\circ Q + Q \\circ Q\\).<\/p>\n

Either we have \\([Q,Q] = P\\), for some even vector field \\(P\\), which is the “supersymmetry algebra”, or \\([Q,Q]=0\\). The latter is the definition of a homological vector field. Note that this is a non-trivial condition and is only non-trivially possible on a supermanifold. <\/p>\n

A Riemannian Q-manifold, I define as a Riemannian supermanifold together with an odd killing vector field that is also a homological vector field. <\/p>\n

The Canonical Volume and Modular Class<\/strong>
\nIn almost exactly the same way as on a Riemannian manifold, Riemannian supermanifolds come equipped with a canonical Berezin volume<\/a>. Without any details, a Berezin volume is something you can integrate on a supermanifold. It turns out, just as in the classical case, that the Berezin volume does not change in the direction of a Killing vector field. <\/p>\n

The vanishing of the modular class of a Q-manifold tells us that there is a Berezin volume on that supermanifold that does not change in the direction of the homological vector field. On a Riemannian Q-manifold we have exactly such a homological vector field and so the modular class of a Riemannian Q-manifold vanishes. We call such Q-manifolds unimodular. <\/p>\n

Final Remarks<\/strong>
\nWith a bit of thought, one can quickly convince one’s self that the modular class of a Riemannian Q-manifold is vanishing. The aim of the paper is to show this very explicitly as a class of new examples of unimodular Q-manifolds. I give explicit examples thoughout the paper.<\/p>\n

References<\/strong>
\n[1] Lyakhovich, S.L., Mosman, E.A., Sharapov, A.A., Characteristic classes of Q-manifolds: classification and applications, J. Geom. Phys. 60 (2010), no. 5, 729\u2013759.<\/p>\n","protected":false},"excerpt":{"rendered":"

In a preprint Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields, I examine the notion of a supermanifold equipped with an even Riemannian metric and an odd Killing vector field that is also homological. Background In a previous post, I briefly disscussed the notion of a Q-manifold and their modular … Continue reading Riemannian Q-manifolds and their modular class<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":7,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20,13],"tags":[],"class_list":["post-5520","post","type-post","status-publish","format-standard","hentry","category-post-doc-luxembourg","category-research-work"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5520","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/comments?post=5520"}],"version-history":[{"count":23,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5520\/revisions"}],"predecessor-version":[{"id":5544,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/5520\/revisions\/5544"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=5520"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=5520"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=5520"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}