{"id":3491,"date":"2013-04-23T07:08:58","date_gmt":"2013-04-23T06:08:58","guid":{"rendered":"http:\/\/blogs.scienceforums.net\/ajb\/?p=3491"},"modified":"2013-04-23T07:08:58","modified_gmt":"2013-04-23T06:08:58","slug":"compatible-homological-vector-fields","status":"publish","type":"post","link":"http:\/\/blogs.scienceforums.net\/ajb\/2013\/04\/23\/compatible-homological-vector-fields\/","title":{"rendered":"Compatible homological vector fields"},"content":{"rendered":"\n\n\n
\"\"<\/td>\n\n

In an earlier post, here<\/a>, I showed that the homological condition on an odd vector field \\(Q \\in Vect(M)\\), on a supermanifold \\(M\\), that is \\(2Q^{2}= [Q,Q]=0\\), is precisely the condition that \\(\\gamma^{*}x^{A} = x^{A}(\\tau)\\), where \\(\\gamma \\in \\underline{Map}(\\mathbb{R}^{0|1}, M)\\), be an integral curve of \\(Q\\). <\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

A very natural question to answer is what is the geometric interpretation of a pair of mutually commuting homological vector fields<\/em>?<\/p>\n

Suppose we have two odd vector fields \\(Q_{1}\\) and \\(Q_{2}\\) on a supermanifold \\(M\\). Then we insist that any linear combination of the two also be a homological vector field, say \\(Q = a Q_{1} + b Q_{2}\\), where \\(a,b \\in \\mathbb{R}\\). It is easy to verify that this forces the conditions<\/p>\n

\\([Q_{1}, Q_{1}]= 0 \\), $latex[Q_{2}, Q_{2}]= 0 $ and \\([Q_{1}, Q_{2}]= 0 \\).<\/p>\n

That is, both our original odd vector fields must be homological and they mutually commute. Such a pair of homological vector fields are said to be compatible. So far this is all algebraic.<\/p>\n

Applications of pairs, and indeed larger sets of compatible vector fields, include the description of n-fold Lie algebroids [1,3] and Q-algebroids [2].<\/p>\n

The geometric interpretation<\/strong>
\nBased on the earlier discussion about integrability of odd flows, a pair of compatible homological vector fields should have something to do with an odd flow. We would like to interpret the compatibility of a pair of homological vector fields as the integrability of the flow of \\(\\tau = \\tau_{1} + \\tau_{2}\\). <\/em>Indeed this is the case;<\/p>\n

Consider \\(\\gamma^{*}_{\\tau_{1} + \\tau_{2}}(x^{A}) = x^{A}(\\tau_{1} + \\tau_{2}) = x^{A}(\\tau_{1}, \\tau_{2})\\), remembering that we define the flow via a Taylor expansion in the “odd time”. Expanding this out we get<\/p>\n

\\( x^{A}(\\tau_{1}, \\tau_{2}) = x^{A} + \\tau_{1}\\psi_{1}^{A} + \\tau_{2}\\psi_{2}^{A} + \\tau_{1} \\tau_{2}X^{A}\\).<\/p>\n

Now we examine the flow equations with respect to each “odd time”. We do not assume any<\/em> conditions on the odd vector fields \\(Q_{1}\\) and \\(Q_{2}\\) at this stage.<\/p>\n

\\(\\frac{\\partial x^{A}}{\\partial \\tau_{1}} = \\psi_{1}^{A} + \\tau_{2}X^{A} = Q_{1}^{A}(x(\\tau_{1}, \\tau_{2}))\\)
\n\\(= Q_{1}^{A}(x) + \\tau_{1}\\psi^{B}_{1} \\frac{\\partial Q^{A}_{1}(x)}{\\partial x^{B}} + \\tau_{2}\\psi^{B}_{2} \\frac{\\partial Q^{A}_{1}(x)}{\\partial x^{B}} + \\tau_{1}\\tau_{2}X^{B} \\frac{\\partial Q^{A}_{1}(x)}{\\partial x^{B}}\\),<\/p>\n

and
\n\\(\\frac{\\partial x^{A}}{\\partial \\tau_{2}} = \\psi_{2}^{A} {-} \\tau_{1}X^{A} = Q_{2}^{A}(x(\\tau_{1}, \\tau_{2}))\\)
\n\\(= Q_{2}^{A}(x) + \\tau_{1}\\psi^{B}_{1} \\frac{\\partial Q^{A}_{2}(x)}{\\partial x^{B}} + \\tau_{2}\\psi^{B}_{2} \\frac{\\partial Q^{A}_{2}(x)}{\\partial x^{B}} + \\tau_{1}\\tau_{2}X^{B} \\frac{\\partial Q^{A}_{2}(x)}{\\partial x^{B}}\\).<\/p>\n

Then equating coefficients in order of \\(\\tau_{1}\\) and \\(\\tau_{2}\\) we arrive at three types of equations<\/p>\n

i) \\(\\psi_{1}^{A} = Q_{1}^{A}\\), \\(\\psi^{B}_{1} \\frac{\\partial Q_{1}^{A}}{\\partial x^{B}}=0\\) and \\(\\psi_{2}^{A} = Q_{2}^{A}\\), \\(\\psi^{B}_{2} \\frac{\\partial Q_{2}^{A}}{\\partial x^{B}}=0\\).<\/p>\n

ii) \\(X^{A} = \\psi^{B}_{2} \\frac{\\partial Q_{1}^{A}}{\\partial x^{B}}\\) and \\(X^{A} = {-}\\psi^{B}_{1} \\frac{\\partial Q_{2}^{A}}{\\partial x^{B}}\\).<\/p>\n

iii) \\(X^{B} \\frac{\\partial Q_{1}^{A}}{\\partial x^{B}} =0\\) and \\(X^{B} \\frac{\\partial Q_{2}^{A}}{\\partial x^{B}} =0\\).<\/p>\n

It is now easy to see that;<\/p>\n

i) implies that \\([Q_{1}, Q_{1}] =0 \\) and \\([Q_{2}, Q_{2}] =0 \\) meaning we have a pair of homological vector fields.<\/p>\n

ii) implies that \\([Q_{1}, Q_{2}]=0\\), that is they are mutually commuting, or in other words compatible.<\/p>\n

iii) is rather redundant and follows from the first two conditions.<\/p>\n

Thus our geometric interpretation was right.<\/p>\n

References<\/strong>
\n[1] Janusz Grabowski and Mikolaj Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys<\/em>. 59<\/strong>(2009), 1285-1305.<\/p>\n

[2]Rajan Amit Mehta, Q-algebroids and their cohomology, Journal of Symplectic Geometry<\/em> 7<\/strong> (2009), no. 3, 263-293.<\/p>\n

[3] Theodore Th. Voronov, Q-Manifolds and Mackenzie Theory, Commun. Math. Phys.<\/em> 2012; 315:279-310.<\/p>\n","protected":false},"excerpt":{"rendered":"

In an earlier post, here, I showed that the homological condition on an odd vector field \\(Q \\in Vect(M)\\), on a supermanifold \\(M\\), that is \\(2Q^{2}= [Q,Q]=0\\), is precisely the condition that \\(\\gamma^{*}x^{A} = x^{A}(\\tau)\\), where \\(\\gamma \\in \\underline{Map}(\\mathbb{R}^{0|1}, M)\\), be an integral curve of \\(Q\\). A very natural question to answer is what is … Continue reading Compatible homological vector fields<\/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":[6,13],"tags":[],"class_list":["post-3491","post","type-post","status-publish","format-standard","hentry","category-general-mathematics","category-research-work"],"_links":{"self":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/3491","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=3491"}],"version-history":[{"count":0,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/posts\/3491\/revisions"}],"wp:attachment":[{"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/media?parent=3491"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/categories?post=3491"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/blogs.scienceforums.net\/ajb\/wp-json\/wp\/v2\/tags?post=3491"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}