José Figueroa-O'Farrill
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Registered User
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I am a mathematical physicist at the Maxwell Institute for Mathematical Sciences and the
School of Mathematics of The University of Edinburgh, in sunny Scotland. I am a founding member of the Edinburgh Mathematical Physics Group and regular contributor to its blog.
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May 8 |
answered | BRST cohomology defintion |
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May 8 |
comment |
What happens to Virasoro at c=25? The answer to S. Sra's question, however random, is "yes". The state space of a bosonic string can be identified with the semi-infinite cohomology of $Vir_c$ relative its centre. The usual complex is obtained by tensoring a Virasoro module (not necessarily irreducible) with $c=26$ with the module of semi-infinite forms. In fact, the duality which answers André's question was first noticed by Feigin in the paper where he defined semi-infinite cohomology. |
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May 8 |
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What happens to Virasoro at c=25? Ben, thanks for the clarification. In that case, then at the top of page 236 in this paper of Feigin and Fuks (the English summary of their longer paper in one of my previous comments) link.springer.com/content/pdf/… you will find the statement that at the level of Verma modules, $(h,c)$ and $(-1-h, 26-c)$ are anti-equivalent. This is just a symmetry of the determinant formula of the Shapovalov form. |
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May 7 |
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What happens to Virasoro at c=25? André, just to make sure I understand. What do you call the universal central extension of the Lie algebra of diffeomorphisms of the circle? Is that what you have termed "the Virasoro algebra" in your previous comment? or is it what you call $Vir_c$? I agree that, contrary to the oversimplification in my previous comment, there are representations of the universal central extension of $\mathfrak{diff}(S^1)$ where the centre does not act like a scalar multiple. |
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May 7 |
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BRST cohomology defintion Hi Jim. Alas, there was a time when in the Physics literature every differential complex was called a BRST complex :( Which BRST cohomology are you interested in? The original of Becchi, Rouet, Stora and Tyutin? |
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May 7 |
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What happens to Virasoro at c=25? The relevant paper is this one: mathnet.ru/php/…, but the PDF there is in Russian. It should be possible to find the English translation. |
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May 7 |
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What happens to Virasoro at c=25? It's been a while and I don't have the relevant papers with me, but I think that the story at $c=25$ is dual to the story at $c=1$. There is a duality between modules with central charge $c$ and central charge $26-c$, in that the embedding diagrams between Verma modules at central charge $26-c$ are obtained from those at central charge $c$ by reversing arrows. |
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May 7 |
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What happens to Virasoro at c=25? Why is not $Vir_c$ a Lie algebra? It is the universal central extension of the Lie algebra of diffeomorphisms of the circle. Of course $c$ is not a number, but the central element, which does act by a number $c$ in any irreducible module. |
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May 5 |
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Clifford algebra on almost product structure @Hassan Jolany, Yazdegerd III's comment is correct, subject to conventions. Recall that Clifford algebras are associated to real vector spaces with quadratic forms. If you take the vector space to be one-dimensional and the quadratic form to be $q(x) = - x^2$, then the resulting Clifford algebra is isomorphic as a real associative algebra to the one you have written in your first comment above. |
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Apr 25 |
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Truncation of BG? I took the liberty to editing it: I changed a "<" to \lt. It seems to have done the trick. |
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Apr 25 |
revised |
Truncation of BG? changed "<" to \lt |
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Apr 18 |
accepted | Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold |
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Apr 16 |
revised |
Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold Added explicit references to Besse's book. |
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Apr 16 |
answered | Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold |
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Apr 13 |
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Endomorphisms in Category O and Schubert Classes Hi George! I fixed the brace problem by using \lbrace and \rbrace instead. |
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Apr 13 |
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Endomorphisms in Category O and Schubert Classes Fixed braces by using \lbrace and \rbrace instead. |
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Apr 8 |
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A question from Otto Forster’s book on Riemann surfaces It's not a missing dollar sign. Actually I can fix it by escaping (with a backslash) two of the underscores, but I do NOT understand why that ought to work, hence I'm leaving the question as is. Someone more familiar with the inner workings of jsMath should look at it. |
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Mar 14 |
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Stokes theorem for manifolds without orientation? Depending on what you intend to do, you could pull-back the forms to the oriented double-cover, where Stokes's theorem applies. |
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Mar 11 |
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maxwell’s equations and hodge theory Actually it's the fieldstrength which is harmonic. Maxwell's equations in vacuo are $$dF =0 \qquad\text{and}\qquad d\star F = 0$, where $F$ is the fieldstrength 2-form. |
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Mar 8 |
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Are all null curves of a Lorentzian metric extrema? Clearly not every null curve is a null geodesic. |
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Mar 8 |
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Are all null curves of a Lorentzian metric extrema? A null geodesic is a geodesic whose velocity is lightlike. Not just any curve with lightlike velocity is a null geodesic. |
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Mar 4 |
awarded | ● Enlightened |
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Mar 4 |
awarded | ● Nice Answer |
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Feb 27 |
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Building a representation out of a generalized Verma module (cont'd) So what exactly do you need the explicit submodule for? If you want to gain intuition about this construction, perhaps you should look at how it works with a simple Lie algebra of small dimension, such as $\mathfrak{su}(2)$. |
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Feb 27 |
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Building a representation out of a generalized Verma module The submodule in question might depend on the actual affine algebra that you are dealing with and in the actual highest weight. I am not sure that you can be very explicit in the general case. Already for finite simple Lie algebras, where the same construction applies, I am not aware of any explicit formula in general, but others will surely correct me if I'm wrong. Having said that, it is clear that the Verma module will have a maximal proper submodule and that the quotient, by definition, will be irreducible and of highest weight. |
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Feb 11 |
awarded | ● Nice Answer |
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Feb 10 |
awarded | ● Nice Answer |
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Feb 4 |
awarded | ● Nice Answer |
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Dec 31 |
awarded | ● Popular Question |
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Dec 18 |
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Triality of Spin(8) I took the liberty of editing the answer to get the braces and matrix to work. The solution was to use \lbrace and \rbrace instead of \{ and \}, respectively; and also to use \cr instead of \\. |
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Dec 18 |
revised |
Triality of Spin(8) got the matrix and braces to work. |
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Dec 12 |
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Explicit Computations of Examples in Spin Geometry Take a look at "Twistors and Killing spinors on Riemannian manifolds", by Baum, Friedrich, Grunewald and Kath. It has many examples. |
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Dec 10 |
awarded | ● Good Answer |
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Dec 2 |
awarded | ● Nice Answer |
