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    • Page created, but author did not leave any comments.

      Anonymous

      v1, current

    • some minimum, just so that I can link to it

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • Started this having heard someone mention it.

      v1, current

    • a stub, just so that the link works

      v1, current

    • Add missing separation condition that Stone uses in his proof. Explain why it’s necessary.

      diff, v3, current

    • Fixed a link to point to his departmental ’profile’.

      diff, v4, current

    • Mention that Hausdorff is not required in the first Michael theorem (almost all standard sources, including Michael himself, impose it).

      diff, v10, current

    • Created a hyperlinked table of contents.

      v1, current

    • Corrected a serious mistake in the definition of a normal cover: normal covers can be star-refined an arbitrary number of times, not just once.

      Added a section on locales.

      diff, v4, current

    • have added pointer to

      • A.Yu. Korchin, D. Van Neck, M. Waroquier, Electromagnetic interaction in chiral quantum hadrodynamics and decay of vector and axial-vector mesons, Phys.Rev. C67 (2003) 015207 (arXiv:nucl-th/0302042)

      diff, v16, current

    • Changed phrasing; I hope I didn’t misinterpret

      Anonymous

      diff, v7, current

    • After a shamefully long time, I am working some more on cartesian bicategory; I have added some material on the locally cartesian structure, on the essential uniqueness of a cartesian structure on a bicategory, and a beginning of a section on the “Frobenius conditions”.

      I also inserted a little promissory note acknowledging that it really would be better to deal with framed cartesian bicategories, by tweaking the definition a little. It would require a certain amount of rewriting (which makes me believe that I had better do it sooner than later).

      A few days ago here at the nForum, I outlined a context where these Frobenius conditions imply “Frobenius reciprocity” (in response to a query of David Corfield). I want to see whether I can write out or at least sketch a proof in the context of a cartesian bicategory satisfying the Frobenius conditions, and see what else might be said on the relationship between the two Frobenii.

    • Started a page on the geometric theory capturing the concept of a continuous flat functor.

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • for completeness of the list of strange mesons

      v1, current

    • This page has been ‘mucked up’. The table has been destroyed giving just a block of text. I could roll back but thought it better for people to see the mess!

    • starting something – this should be the last of the types if light mesons in the list

      v1, current

    • I made some changes to bivector. While the idea section is correct (and should be strictly adhered to!) but the previous definition is wrong in general! The previous definition is consistent and used in wikipedia but it misses both the direct relation of bivectors, trivectors and general polyvectors to determinants as well as the standard nontrivial usage of bivectors in analytic geometry wher bivectors define equivalence classes of parallelograms and in particular with a point in space given define an affine plane. If we adhere to wikipedia and not to standard treatments in geometry (e.g. M M Postnikov, Analytic geometry) then we miss the nontriviality of the notion of bivector and its meaning which is more precise than that of a general element in the second exterior power.

      Bivector in a vector space VV is not any element in the second exterior power, but a DECOMPOSABLE vector in the second tensor power – in general dimension just such elements in Λ 2V\Lambda^2 V have the intended geometric meaning and define vector 2-subspaces and of course affine 2-subspaces if a point in the 2-subspace is given. It is true that every bivector in 2-d or in 3-d space is decomposable, but in dimension 4 this is already not true. Thus the bivectors form a vector space just in the dimensions up to 33. Similarly, trivectors form a vector space just in the dimensions up to 44. In the context of differential graded algebras, polyvector fields are usually taken as arbitrary elements in the exterior powers of vector fields.

    • Created page on Paul Fendley for linking from Fusion Categories

      v1, current

    • Hello, I added some words about an ambiguity I found in the literature about the name “dilatino”. If I am wrong, I’d be thankful to the one who makes me notice the mistake!

      diff, v2, current

    • need the link, but nothing here yet

      v1, current

    • stated the definition Δf=⋆d⋆df\Delta f = \star d \star d f and spelled out how this gives the usual component formula:

      ⋆d⋆df =⋆d⋆(∂ jf)dx j =⋆d(1(D−1)!|det((g ij))|g ij(∂ jf)ε ik 2⋯k Ddx k 2∧⋯∧dx k D) =⋆∂ k 1(1(D−1)!|det((g ij))|g ij(∂ jf)ε ik 2⋯k Ddx k 1∧dx k 2∧⋯∧dx k D) =|det((g ij))|1D!(D−1)!ε l 1l 2⋯l Dg l 1k 1g l 2k 2⋯g l Dk Dε ik 2⋯k D⏟=det((g ij) −1)δ i k 1∂ k 1(|det((g ij))|g ij(∂ jf)) =1|det((g ij))|δ i k 1∂ k 1(|det((g ij))|g ij(∂ jf)) =1|det((g ij))|∂ i(|det((g ij))|g ij(∂ jf)) \begin{aligned} \star d \star d f & = \star d \star (\partial_j f) d x^j \\ & = \star d \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \star \partial_{ \color{magenta} k_1} \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{magenta} k_1 } \wedge d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \underset{ = \det\big( (g_{i j})^{-1} \big) \delta^{ \color{magenta} k_1 }_i }{ \underbrace{ \tfrac{1}{ { \color{orange} D! } { \color{green} (D-1)! } } \epsilon_{ \color{orange} l_1 l_2 \cdots l_D } g^{ { \color{orange} l_1 } { \color{magenta} k_1 } } g^{ { \color{orange} l_2 } { \color{green} k_2 } } \cdots g^{ { \color{orange} l_D} { \color{green} k_D } } \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } } } \, \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \delta^{ \color{magenta} k_1 }_i \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \partial_{i} \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \end{aligned}

      diff, v7, current

    • expanded brane

      first a little remark on what D-branes are abstractly, in reply to an MO-question, then something on fundamental branes, going along with the discussion on the Café

    • brief category:people entry in order to make hyperlinks work

      v1, current

    • a stub, for the moment just so as to complete a pattern of entries, but I added pointer to

      v1, current

    • Added to BF-theory the reference that right now I am believing is the earliest one:

      Gary Horowitz, Exactly soluable diffeomorphism invariant theories Commun. Math. Phys. 125, 417-437 (1989)

      But maybe I am wrong. Does anyone have an earlier one? I saw pointers to A. Schwarz articles from the late 70s, but I am not sure if he really considered BF as such.

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