User j verma - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:27:47Z http://mathoverflow.net/feeds/user/9534 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122376/equivariant-hata-genus-of-a-spin-manifold Equivariant $\hat{A}$ - genus of a spin manifold J Verma 2013-02-20T04:47:18Z 2013-02-20T04:47:18Z <p>I am trying to understand the Berline - Vergne Localization formula for the equivariant Index of the Dirac operator on a spin manifold M which states that the G - equivariant index of Dirac operator $Ind_{D}(X)$ where $X \in \mathfrak{g}$ is given by the following integral :</p> <p>$Ind_{D}(X)= \int_{M} Ch(L,X)\hat{A}(M,X)$</p> <p>Where Ch(L,X) is the G - equivariant Chern character of the twisting bundle L and $\hat{A}(M,X)$ is the G-equivariant A - genus of M. These terms are defined in e.g Berline - Getzler - Vergne (Heat Kernels and Dirac operators). For a compact Riemannian manifold $(M,g)$ acted by a compact Lie group preserving the metric, </p> <p>$\hat{A}(M,X) = det^{1/2} ( \frac{(T + R)/2}{sinh (T + R)/2} )$</p> <p>Where T is the endomorphis of the tangent bundle induced by the G - action and R is the curvature of the Levi - Civita connection on M. </p> <p>I'm having trouble in understanding these quantities especially the $\hat{A}(M,X)$ (the square root of the determinant of sinh is bothering me). How does one think of these classes. I would greatly appreciate if someone can explain these equivariant Clases, may be in a simple example say action of rotations about z- axis on sphere. Unfortunately in the book by Berline - Getzler - Vergne there are very few examples. Please provide any simple examples which can be helpful. </p> http://mathoverflow.net/questions/54430/video-lectures-of-mathematics-courses-available-online-for-free/54438#54438 Answer by J Verma for Video lectures of mathematics courses available online for free J Verma 2011-02-05T18:46:36Z 2013-01-06T23:01:12Z <p>Here are some of my favorites : </p> <ol> <li><p>Sidney Coleman's <a href="http://www.physics.harvard.edu/about/Phys253.html" rel="nofollow">Quantum Field Theory</a> </p></li> <li><p>Shiraz Minwalla's <a href="http://theory.tifr.res.in/~minwalla/" rel="nofollow">String Theory</a> </p></li> <li><p>MIT <a href="http://ocw.mit.edu/courses/audio-video-courses/#mathematics" rel="nofollow">OCW</a> </p></li> <li><p>Videos to short courses at some workshops can be found at <a href="http://video.ias.edu/sm" rel="nofollow">IAS</a> and <a href="http://www.msri.org/web/msri/online-videos" rel="nofollow">MSRI</a> </p></li> </ol> http://mathoverflow.net/questions/53122/mathematical-urban-legends/53169#53169 Answer by J Verma for Mathematical "urban legends" J Verma 2011-01-25T02:05:35Z 2012-01-14T18:38:07Z <p>I heard this story a couple of years back (not sure though if it is true): A young Japanese mathematician was giving a talk based on his results at Courant Institute. His work was built on the work of S.R.S Varadhan. But apparently during the talk Varadhan had his eyes closed and the speaker mistook it for him sleeping. He made a joke by saying somthing like "hopefully not everybody is sleeping". A few minutes later Varadhan open his eyes and said "consider this counterexample". But Varadhan liked the speaker's idea and invited him to spent some time at Courant institute. The correct result is now known as 'Speaker'-Varadhan theorem.</p> http://mathoverflow.net/questions/81662/hamiltonian-reduction-and-affine-quotient Hamiltonian Reduction and Affine quotient J Verma 2011-11-22T22:38:58Z 2011-11-23T04:10:48Z <p>Given a smooth affine symplectic variety $V$ with an action of a connected algebraic $G$. If $\mu$ is the moment map, the define the affine quotient to be :</p> <p>$X = \mu^{-1}(0)// G = \text{Spec}\mathbb{C}[\mu^{-1}(0)]^{G}$ </p> <p>This is an algebraic Variety (may be singular). </p> <p>The Hamiltonian reduction of $V$ is defined to be </p> <p>$Y = \mu^{-1}(0)/ G$. </p> <p>Q : When are these two notions same ? </p> http://mathoverflow.net/questions/81660/configuration-space-of-points-in-euclidean-space-with-fixed-distances Configuration space of points in Euclidean Space with fixed distances J Verma 2011-11-22T22:16:36Z 2011-11-22T23:15:02Z <p>What can we say about the configuration space of $n$ points in $\mathbb{R}^{m}$ with fixed distances between the points ? </p> <p>e.g. for 2 points in $\mathbb{R}^{3}$ with fixed distance between them,the configuration space is $S^{2}$ and for 3 points in $\mathbb{R}^{3}$ with known distances between them, the configuration space is $SO(3)$. </p> <p>A similar problem is to ask the configuration space of a point in $\mathbb{R}^{m}$ such that its distances from $n$ fixed points is known. </p> <p>e.g. Consider a point in $\mathbb{R}^{3}$ such that its distances from 2 fixed points is $r_{1}$ and $r_{2}$. Then the point can lie on a sphere of radius $r_{1}$ around point 1 and also on a sphere of radius $r_{2}$ around point 2. Thus the configuration space is the intersection of these 2 sphere, which generically is a circle. (Correct if I'm wrong.)</p> <p>What is known about these problems in general for any $n,m$ ? </p> http://mathoverflow.net/questions/77934/are-quivers-useful-outside-of-representation-theory/77943#77943 Answer by J Verma for Are quivers useful outside of Representation Theory? J Verma 2011-10-12T18:30:21Z 2011-10-12T18:30:21Z <p>In addition to being a nice example for abelian, $A_{\infty}$ and Calabi-Yau categories, and being a prototypical example for <a href="http://arxiv.org/abs/0811.2435" rel="nofollow">Generalized Donaldson - Thomas Invariants and the Wall Crossing Phenomenon</a>, the quivers have a lot of applications in variours different fields. Since the question is applications in addition to representation theory, I'm listing a few cases, </p> <p>Most prominent is Algebraic Geometry, particularly Moduli problems and GIT (read motivations in <a href="http://arxiv.org/abs/0802.2147" rel="nofollow">Reinke's article</a>)and Video lectures on quivers by Reineke at Newton Institute, Cambridge. </p> <p>Recently, a correspondence has been proposed between Gromov - Witten invariants and Quivers. (<a href="http://arxiv.org/abs/0909.5153" rel="nofollow">Pandharipande - Gross</a>).</p> <p>Also in physics applications in <a href="http://arxiv.org/abs/hep-th/9603167" rel="nofollow">String Theory</a>, <a href="http://arxiv.org/abs/hep-th/0006189" rel="nofollow">Supersymmetry</a>, <a href="http://arxiv.org/abs/hep-th/0206072" rel="nofollow">Black Holes</a> and <a href="http://arxiv.org/abs/hep-th/0007173" rel="nofollow">Particle physics</a>.</p> <p>Relation with quantum dilogrithm, number theory, and cluster algebras, read e.g. <a href="http://arxiv.org/abs/1102.4148" rel="nofollow">this</a> review by Keller.</p> <p>Also through the work of Nakajima, there is relation to Instantons of Yang-Mills theory, Hitchin Moduli spaces and the theory of Hyperkahler manifolds.</p> http://mathoverflow.net/questions/75482/donaldson-thomas-invariants-in-physics Donaldson-Thomas Invariants in Physics J Verma 2011-09-15T05:17:12Z 2011-09-15T19:02:07Z <p>First of all, I am sorry for there are a bunch of questions (though all related)and may not be well framed.</p> <p>What are the DT invariants in physics. When one is computing DT invariants for a Calabi-Yau manifold, what is one computing in physics?</p> <p>What about the generalized (motivic) version?</p> <p>Also what does the Gromov-Witten/DT correspondence (MNOP) say in terms of physics, are there (strong) physical reasons to believe such a correspondence. </p> <p>Please suggest some useful references. Thanks.</p> http://mathoverflow.net/questions/73924/book-on-mixed-hodge-structures/73925#73925 Answer by J Verma for Book on mixed Hodge structures? J Verma 2011-08-29T00:11:03Z 2011-08-29T00:11:03Z <p><a href="http://www.springer.com/mathematics/algebra/book/978-3-540-77015-2" rel="nofollow">Mixed Hodge Structures, Peters and Steenbrink</a></p> http://mathoverflow.net/questions/67992/hirzebruch-surfaces Hirzebruch surfaces J Verma 2011-06-16T19:30:11Z 2011-06-16T20:04:35Z <p>I am sorry for too naive and stupid question, </p> <p>How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Can F_{2} be realizable as the total space of a bundle over $\mathbb{R}_{+}$ with fibre $SO(3)$. </p> <p>I have to admit that my knowledge of Hirzebruch surfaces is limited to the few lines in the wikipedia page. </p> http://mathoverflow.net/questions/67722/vector-multiplet-hypermultiplet-moduli-space-of-string-theory vector multiplet/hypermultiplet moduli space of String Theory J Verma 2011-06-14T03:48:07Z 2011-06-14T06:42:30Z <p>What is vector multiplet and hypermultiplet moduli space associated to IIA/B string theory (or in general to a N = 2 Supersymmetric theory) ?</p> <p>The vector multiplet moduli space is special Kahler while hypermultiplet moduli is hyperkahler. It seems that the vector multiplet is the moduli of the Calabi - Yau on which the theory is compactified, though I am not very sure. </p> <p>Details (I am aware of the space &amp; time constraints) and references will be greatly appreciated. </p> http://mathoverflow.net/questions/61715/equivariant-index-of-dirac-operator-on-s2 equivariant index of Dirac Operator on $S^{2}$ J Verma 2011-04-14T15:45:38Z 2011-04-16T11:02:31Z <p>First, I have to admit that I don't have much knowledge of Spin Geometry and Index Theory, the question could be too simple or naive and secondly there may be too many questions. </p> <p>Let $D$ be the Dirac Operator for standard metric and $S$ be the Spin bundle on $S^{2}$.There is a unique Spin structure on $S^{2}$. How does $D$ look like, Can we write a general form for the harmonic spinors on $S^{2}$ ?</p> <p>What is the general expression for equivariant index of $D$ ? </p> <p>If $W \times S$ is the twisted spinor bundle and $D$ is the twisted Dirac operator, we can write the equivariant index as an integral over the fixed point manifold in terms of equivariant Chern Character of $W$ and $\hat{A}$ - genus of the fixed point manifold using the Aiyah-Segal-Singer theorem. What I am interested in is the final expression for $S^{2}$. </p> <p>What can we say about the product of n $S^{2}$'s.</p> <p>Thanks</p> http://mathoverflow.net/questions/61354/notion-of-stability-in-a-category notion of stability in a category J Verma 2011-04-12T01:36:51Z 2011-04-12T08:02:08Z <p>This is a question in general sense, but answers about specific examples are also welcome.</p> <p>Why do we need the notion of stability of objects in a category. If we've a subcategory of stable objects, what can we do with this subcategory. What is the general philosophy of the stability data. </p> <p>In case of vector bundles the theorem of Donaldson and of Ulhelbeck-Yau says that stable vector bundles give the solutions of the hermitian Yang-Mills equation, for triangulated category the stability conditions were an attempt to rigorize $\pi$ stability for D-branes. </p> <p>What is the motivation to define stability on a category in general. </p> http://mathoverflow.net/questions/58554/preferably-rare-audio-video-recordings-of-famous-mathematicians/58558#58558 Answer by J Verma for (Preferably rare) Audio/Video recordings of famous mathematicians? J Verma 2011-03-15T18:45:21Z 2011-03-15T18:45:21Z <p>Here's is one of <a href="http://www.youtube.com/watch?v=CC7Sg41Bp-U&amp;feature=rec-LGOUT-farside_rev-rn-5r-10-HM" rel="nofollow">Einstein</a></p> http://mathoverflow.net/questions/57589/heuristic-behind-a-infty-algebras Heuristic behind $A_{\infty}$ - algebras J Verma 2011-03-06T18:36:07Z 2011-03-07T15:53:04Z <p>How to think about the $A_{\infty}$-algebras ?</p> <p>I am looking at the Bernhard Keller's introduction, he says a few words about the topological origin (not in details) and motivates by stating two problems in homological algebra but what I am looking for is the intuitive idea and some prototypical examples to keep in mind and justify the axioms of $A_{\infty}$- algebra and their morphisms. </p> <p>Thanks</p> http://mathoverflow.net/questions/57520/examples-in-mirror-symmetry-that-can-be-understood/57533#57533 Answer by J Verma for Examples in mirror symmetry that can be understood. J Verma 2011-03-06T04:21:25Z 2011-03-06T04:21:25Z <p>You need the machinery of triangulated categories and homological algebra to understand the mirror symmetry as it stand today, Homological mirror symmetry. But one can get an idea of mirror symmetry without delving into these concepts,I am talking about the classical picture of mirror symmetry as noticed by Physicists. i.e. Mirror symmetry as an isomorphism between the complex and Kahler moduli spaces of Calabi-Yau 3-folds.(Beware : this was first definition of mirror symmetry and even here I am overlooking the subtleties involving the large complex structure limit.) e.g. Elliptic curve (Dijgraaf), Quintic (Greene, Plesser and Candelas et al).This may give an idea of Mirror symmetry but to understand this picture properly one need an understanding of the geometry of Calabi-Yau manifolds, Variations of mixed Hodge structures, quantum cohomology and GW invariants.</p> <p>Also there is a modern picture of mirror symmetry called SYZ conjecture which is more geometric and doesn't involve homological algebra and triangulated categories. But again you need the knowledge of geometry of special Lagrangian submanifolds of CY manifolds. </p> http://mathoverflow.net/questions/57031/biography-of-felix-hausdorff/57035#57035 Answer by J Verma for Biography of Felix Hausdorff J Verma 2011-03-01T21:12:25Z 2011-03-01T21:14:19Z <p>one place I know is the website of Hausdorff Institute website, particularly <a href="http://www.hausdorff-research-institute.uni-bonn.de/files/Felix-Hausdorff_en.pdf" rel="nofollow">this short biography</a>(62 pages).</p> http://mathoverflow.net/questions/57018/axiomatic-treatment-of-a-model-and-b-model-in-sense-of-atiyah Axiomatic treatment of A-model and B-model in sense of Atiyah J Verma 2011-03-01T17:58:41Z 2011-03-01T20:47:43Z <p>I am looking for an axiomatic construction of topological sigma models, A-model and B-model in the sense of Atiyah's TQFT axioms.The B-model on genus - $0$ cobordisms had been constructed by Barannikov and Kontsevich. </p> <p>I am aware of the work of Costello on higer genus B - model. But I am looking for a simple (if any) construction of genus $0$ A - model and B - model. </p> http://mathoverflow.net/questions/56314/elementary-mathematical-books/56334#56334 Answer by J Verma for Elementary mathematical books J Verma 2011-02-22T22:43:53Z 2011-02-22T22:43:53Z <p>Unfortunately for Galois theory there isn't anything suitable for high school students, but the nice introduction is here <a href="http://www.jstor.org/stable/2324574" rel="nofollow">Galois theory for beginners, John Stillwell</a>, in addition to historic essay in the introduction to the book by Edwards. </p> <p>for arithmetics, I think the books by Alan Baker (Theory of numbers) and by G.H. Hardy might be helpful.</p> <p>for geometry/topology <a href="http://www.jstor.org/stable/2324574" rel="nofollow">this essay by S.S. Chern</a> can provide some motivation for the subject.</p> http://mathoverflow.net/questions/54775/what-is-the-shortest-ph-d-thesis/54796#54796 Answer by J Verma for What is the shortest Ph.D. thesis? J Verma 2011-02-08T17:37:46Z 2011-02-08T21:14:04Z <p>I believe the shortest PhD thesis is of Burt Totaro "Milnor K-theory is the simplest part of algebraic K-theory", 12 pages.</p> <p>Milnor K-theory is the simplest part of algebraic K-theory, Ph.D. thesis, University of California, Berkeley, 1989; K-Theory 6 (1992), 177-189.</p> <p><a href="http://www.dpmms.cam.ac.uk/~bt219/papers.html" rel="nofollow">Burt Totaro's webpage at Cambridge</a></p> http://mathoverflow.net/questions/53988/what-is-the-motivation-for-a-vertex-algebra/54029#54029 Answer by J Verma for What is the motivation for a vertex algebra? J Verma 2011-02-01T22:28:16Z 2011-02-01T22:28:16Z <p>I'll take a more physical POV. Although, we don't have any mathematically concrete definition of QFT in general, but we can define 2d Conformal Field theory (this is attributed to infinite symmetries and exact solvability), other that CFTs we can also define Topological QFTs (Atiyah).Greame Segal proposed a geometric definition of CFT. In Conformal field theory we deal with vertex operators analog to operators in QFT, we can write a Taylor like expansion of two vertex operators, k.a Operator product expansions (OPEs) which gives the QFT analog of two fields interacting. All these notions are captured by the axiomatization based on Vertex Algebras. To see a better picture,look at the classic paper of Belavin, Polyakov, and Zamolodchikov, where an algebraic approach of CFT was proposed. </p> <p>Recently Kapustin and Orlov proposed a more general definition of Vertex algebras and they showed the relation between their algebraic definition and Segal's geometric one.</p> http://mathoverflow.net/questions/53680/tips-on-writing-a-cv-for-a-conference/53686#53686 Answer by J Verma for Tips on Writing a CV for a Conference J Verma 2011-01-29T04:16:58Z 2011-01-29T04:16:58Z <p>every thing looks fine.You have to convince the selection committee that you have adequate background for the conference and this conference is going to help you in your current and future mathematical projects (e.g.your thesis). So you can include the list of courses you have taken which are related to the theme of the project. And you can say something about your project related to the topic. </p> http://mathoverflow.net/questions/47000/birational-equivalence-and-mirror-cys birational equivalence and mirror CYs J Verma 2010-11-22T20:57:21Z 2011-01-21T18:11:17Z <p>If a CY X is a mirror to Y then any CY Z which is birational to X is also a Mirror of Y. This is the motivation for the Kawamata's "moveable Kahler cone" which includes the Kahler cones of all the CYs birational to X. Can you suggest a proof of this statement OR is there some reference where this has been proved.</p> <p>One way to look at it is in terms of the Bondal/Orlov conjecture about the isomorphism of the derived categories for birational CYs and Bridgeland's proof for dimension $\leq 3$, correct me if I am wrong. </p> http://mathoverflow.net/questions/52452/topology-of-black-holes/52464#52464 Answer by J Verma for Topology of black holes J Verma 2011-01-19T03:17:26Z 2011-01-19T13:21:09Z <p>Hawking's Theorem of Black Hole topology asserts that the in case of $4$d asymptotically flat stationary black holes satisfying the suitable energy condition (dominant energy condition), the cross sections of the evernt horizon are spherical. </p> <p>Galloway and Schoen extended this theorem to higher dimensions; they showed that the cross sections of event horizon (stationary case) and the outer (apparent) horizon (general case) are of positive Yamabe type. This paper can be found at Galloway's webpage www.math.miami.edu/~galloway/papers/220_2006_19_OnlinePDF.pdf</p> http://mathoverflow.net/questions/51395/a-soft-introduction-to-physics-for-mathematicians-who-dont-know-the-first-thing/51421#51421 Answer by J Verma for A soft introduction to physics for mathematicians who don't know the first thing about physics J Verma 2011-01-07T18:43:17Z 2011-01-07T18:43:17Z <p>Let me suggest a reading plan :</p> <ul> <li>The Feynman's books are a pleasure to read and provide great insights into basic physics. </li> <li>For classical mechanics, "Mathematical methods of Classical mechanics" by V.I. Arnold and "Mechanics" by Landau-Lifshitz.</li> <li>for quantum mechanics, "Mathematical foundations of quantum mechanics" - Von Neumann. another nice book on quantum mechanics is by R.Shanker.</li> <li>for statistical physics, the two volumes by Landau are my favorites. </li> <li>For General Relativity, R.M. Wald and more mathematically inclined is "Large Scale structures of Space time" by Hawking-Ellis.</li> <li>For QFT and String Theory, read AMS book "Quantum Fields and Strings for mathematicians". It contains beautiful lectures by experts in the field addressed to mathematicians.</li> <li>Another good book is Clay monograph "Mirror Symmetry" by Hori et al.It starts with classical mechanics, moves through quantum mechanics to QFT, String Theory.</li> <li>Ivan Mirkovic has nice lecture notes here <a href="http://www.math.umass.edu/~mirkovic/" rel="nofollow">http://www.math.umass.edu/~mirkovic/</a> the notes on string theory. It also has discussion on classical and quantum mechanics. </li> </ul> http://mathoverflow.net/questions/51155/is-complex-moduli-space-of-a-calabi-yau-kahler is complex moduli space of a Calabi - Yau Kahler J Verma 2011-01-04T20:37:48Z 2011-01-04T20:47:03Z <p>The complex moduli space of a Calabi-Yau manifold is a complex manifold (BTT). Is it also Kahler ?</p> http://mathoverflow.net/questions/50992/wall-crossing-in-physics-and-mathematics Wall Crossing in Physics and Mathematics J Verma 2011-01-03T07:14:02Z 2011-01-04T08:40:44Z <p>This question is motivated by the current interest of Mathematics and Physics community in Wall Crossing. My questions are :</p> <ol> <li><p>What is wall crossing in Physics, what are the reasons for current interest in it.</p></li> <li><p>What is wall crossing in terms of mathematics, what is the reason for interest, is it just physics or some mathematical motivation.</p></li> </ol> <p>thanks.</p> http://mathoverflow.net/questions/50702/triangulated-derived-categories-in-physics-and-algebraic-geometry triangulated/derived categories in Physics and algebraic geometry J Verma 2010-12-30T06:35:35Z 2010-12-30T06:53:43Z <p>Why do physicists care about the triangulated/derived categories? I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror symmetry and stability of D-branes. </p> <p>Also I would like to know some problems in algebraic geometry which we hope to solve using triangulated categories, e.g. the derived category of coherent sheaves on an algebraic variety is believed to capture the geometry of the variety.</p> <p>I am not sure whether to make it community wiki, feel free to do so. Thanks. </p> http://mathoverflow.net/questions/50513/stability-conditions-in-the-sense-of-kontsevich-soibelman stability conditions in the sense of Kontsevich-Soibelman J Verma 2010-12-27T21:16:05Z 2010-12-27T21:49:41Z <p>What are the stability conditions in the sense of Kontsevich-Soibelman. </p> <p>I am reading Bridgeland's stability conditions and I've heard people talking about the Kontsevich-Soibelman Stability. I would appreciate a brief introduction on this, in particular my questions are :</p> <ol> <li><p>What are the Kontsevich-Soibelman Stability conditions ?</p></li> <li><p>How is it related to Bridgeland's Stability (or Douglas' $\pi$ - stability on D-branes) ?</p></li> <li><p>Why do we need to consider Kontsevich-Soibelman stability.</p></li> </ol> <p>I've to admit my ignorance of the field. Please suggest some references. Thanks. </p> http://mathoverflow.net/questions/47647/topological-b-model topological B model J Verma 2010-11-29T06:43:44Z 2010-11-29T06:43:44Z <p>The topological A model was constructed by Witten in Comm. Math. Phys. Volume 118, Number 3 (1988), 411-449. I am looking for the original paper where topological B model was first introduced. I am reading another paper of Witten, Topological Field theory and mirror manifolds, where he discussed the B model, but I am looking for first construction of B model.</p> <p>Can you also suggest some other references about the modern treatment of B model. I know Kevin Costello has done some work in this direction.<br> Thanks.</p> http://mathoverflow.net/questions/46997/what-is-large-compex-structure-limit-of-cy-moduli-space what is large compex structure limit of CY moduli space J Verma 2010-11-22T20:36:39Z 2010-11-28T16:45:52Z <p>What is the Large Complex Structure limit(LCL) of complex moduli space of a Calabi-Yau 3-fold and why do we need to consider LCL in Mirror symmetry. </p> http://mathoverflow.net/questions/81660/configuration-space-of-points-in-euclidean-space-with-fixed-distances Comment by J Verma J Verma 2011-11-23T21:13:16Z 2011-11-23T21:13:16Z The setting I'm interested is slightly different from the polygon spaces, in this case every point is connected to every other by an edge, whereas in polygon spaces we've edges between consecutive points i.e. a 4-gon is a quadrilateral, but what I am interested in is a tetrahedron. After some googling it seems that what I'm looking for is configuration spaces of complete graphs. Also what I'm asking is - is it a well established theory e.g. Do we know the general configuration space for such problems. Like google tells me that for weighted graphs configuration space is some sphere. http://mathoverflow.net/questions/81662/hamiltonian-reduction-and-affine-quotient/81686#81686 Comment by J Verma J Verma 2011-11-23T05:04:20Z 2011-11-23T05:04:20Z thanks for the answer. http://mathoverflow.net/questions/81662/hamiltonian-reduction-and-affine-quotient Comment by J Verma J Verma 2011-11-23T02:40:22Z 2011-11-23T02:40:22Z non necessarily compact, but connected G. I've edited the question accordingly. http://mathoverflow.net/questions/75482/donaldson-thomas-invariants-in-physics/75493#75493 Comment by J Verma J Verma 2011-09-16T03:16:54Z 2011-09-16T03:16:54Z @ Vivek Thanks for the comment on DT/GW correspondence. http://mathoverflow.net/questions/75482/donaldson-thomas-invariants-in-physics/75493#75493 Comment by J Verma J Verma 2011-09-15T18:20:40Z 2011-09-15T18:20:40Z Thanks for the answer. I also think of BPS states as some representations of super-Poincare algebra. But whenever I tried to read a physics paper or a video lecture by a physicist, they talk about black holes which are BPS states, which I don't know how to think of. http://mathoverflow.net/questions/67992/hirzebruch-surfaces/67993#67993 Comment by J Verma J Verma 2011-06-17T16:47:17Z 2011-06-17T16:47:17Z Thanks for the detailed answer. http://mathoverflow.net/questions/67992/hirzebruch-surfaces/67993#67993 Comment by J Verma J Verma 2011-06-16T19:49:25Z 2011-06-16T19:49:25Z Can you say more or suggest some reference about how $F_{2}$ is the compactification of $SO(3) \times \mathbb{R}_{+}$. http://mathoverflow.net/questions/67992/hirzebruch-surfaces/67993#67993 Comment by J Verma J Verma 2011-06-16T19:46:58Z 2011-06-16T19:46:58Z @ Dmitri - Thanks, this is exactly what I wanted. http://mathoverflow.net/questions/67992/hirzebruch-surfaces Comment by J Verma J Verma 2011-06-16T19:39:54Z 2011-06-16T19:39:54Z @ Grigory M - Thanks for your reply at SE. I posted this here expecting a quicker response. If the base space if a compact space, say $[a,b]$ or $S^{1}$. Thanks http://mathoverflow.net/questions/67722/vector-multiplet-hypermultiplet-moduli-space-of-string-theory/67734#67734 Comment by J Verma J Verma 2011-06-14T17:23:50Z 2011-06-14T17:23:50Z thanks Andy. So it seems that there is no mathematical definition of hypermultiplet moduli space. http://mathoverflow.net/questions/50040/when-can-witten-esque-moduli-spaces-be-used-to-define-invariants-of-geometric-str/50069#50069 Comment by J Verma J Verma 2011-04-29T18:45:51Z 2011-04-29T18:45:51Z + 1 superb answer! http://mathoverflow.net/questions/62163/math-and-wormholes/62181#62181 Comment by J Verma J Verma 2011-04-19T03:52:09Z 2011-04-19T03:52:09Z +1 Just Amazing... http://mathoverflow.net/questions/61953/deeper-meanings-of-phase-space-any-books Comment by J Verma J Verma 2011-04-16T21:26:01Z 2011-04-16T21:26:01Z V.I. Arnold - Mathematical Methods of Classical Mechanics. http://mathoverflow.net/questions/61715/equivariant-index-of-dirac-operator-on-s2/61909#61909 Comment by J Verma J Verma 2011-04-16T20:23:42Z 2011-04-16T20:23:42Z thanks for your reply, very enlightening. http://mathoverflow.net/questions/61715/equivariant-index-of-dirac-operator-on-s2/61810#61810 Comment by J Verma J Verma 2011-04-15T18:43:55Z 2011-04-15T18:43:55Z @ Sebastian - But a twisted Dirac Operator on a twisted spinor bundle can have non trivial kernel.