User j verma - MathOverflow

Equivariant $\hat{A}$ - genus of a spin manifold

2013-02-20T04:47:18Z

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 :

$Ind_{D}(X)= \int_{M} Ch(L,X)\hat{A}(M,X)$

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,

$\hat{A}(M,X) = det^{1/2} ( \frac{(T + R)/2}{sinh (T + R)/2} )$

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.

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.

Answer by J Verma for Video lectures of mathematics courses available online for free

2011-02-05T18:46:36Z

Here are some of my favorites :

Sidney Coleman's Quantum Field Theory
Shiraz Minwalla's String Theory
MIT OCW
Videos to short courses at some workshops can be found at IAS and MSRI

Answer by J Verma for Mathematical "urban legends"

2011-01-25T02:05:35Z

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.

Hamiltonian Reduction and Affine quotient

2011-11-22T22:38:58Z

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 :

$X = \mu^{-1}(0)// G = \text{Spec}\mathbb{C}[\mu^{-1}(0)]^{G}$

This is an algebraic Variety (may be singular).

The Hamiltonian reduction of $V$ is defined to be

$Y = \mu^{-1}(0)/ G$.

Q : When are these two notions same ?

Configuration space of points in Euclidean Space with fixed distances

2011-11-22T22:16:36Z

What can we say about the configuration space of $n$ points in $\mathbb{R}^{m}$ with fixed distances between the points ?

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 $SO(3)$.

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.

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.)

What is known about these problems in general for any $n,m$ ?

Answer by J Verma for Are quivers useful outside of Representation Theory?

2011-10-12T18:30:21Z

In addition to being a nice example for abelian, $A_{\infty}$ and Calabi-Yau categories, and being a prototypical example for Generalized Donaldson - Thomas Invariants and the Wall Crossing Phenomenon, 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,

Most prominent is Algebraic Geometry, particularly Moduli problems and GIT (read motivations in Reinke's article)and Video lectures on quivers by Reineke at Newton Institute, Cambridge.

Recently, a correspondence has been proposed between Gromov - Witten invariants and Quivers. (Pandharipande - Gross).

Also in physics applications in String Theory, Supersymmetry, Black Holes and Particle physics.

Relation with quantum dilogrithm, number theory, and cluster algebras, read e.g. this review by Keller.

Also through the work of Nakajima, there is relation to Instantons of Yang-Mills theory, Hitchin Moduli spaces and the theory of Hyperkahler manifolds.

Donaldson-Thomas Invariants in Physics

2011-09-15T05:17:12Z

First of all, I am sorry for there are a bunch of questions (though all related)and may not be well framed.

What are the DT invariants in physics. When one is computing DT invariants for a Calabi-Yau manifold, what is one computing in physics?

What about the generalized (motivic) version?

Also what does the Gromov-Witten/DT correspondence (MNOP) say in terms of physics, are there (strong) physical reasons to believe such a correspondence.

Please suggest some useful references. Thanks.

Answer by J Verma for Book on mixed Hodge structures?

2011-08-29T00:11:03Z

Mixed Hodge Structures, Peters and Steenbrink

Hirzebruch surfaces

2011-06-16T19:30:11Z

I am sorry for too naive and stupid question,

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 $SO(3)$.

I have to admit that my knowledge of Hirzebruch surfaces is limited to the few lines in the wikipedia page.

vector multiplet/hypermultiplet moduli space of String Theory

2011-06-14T03:48:07Z

What is vector multiplet and hypermultiplet moduli space associated to IIA/B string theory (or in general to a N = 2 Supersymmetric theory) ?

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.

Details (I am aware of the space & time constraints) and references will be greatly appreciated.

equivariant index of Dirac Operator on $S^{2}$

2011-04-14T15:45:38Z

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.

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 $S^{2}$ ?

What is the general expression for equivariant index of $D$ ?

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 $S^{2}$.

What can we say about the product of n $S^{2}$'s.

Thanks

notion of stability in a category

2011-04-12T01:36:51Z

This is a question in general sense, but answers about specific examples are also welcome.

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.

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.

What is the motivation to define stability on a category in general.

Answer by J Verma for (Preferably rare) Audio/Video recordings of famous mathematicians?

2011-03-15T18:45:21Z

Here's is one of Einstein

Heuristic behind $A_{\infty}$ - algebras

2011-03-06T18:36:07Z

How to think about the $A_{\infty}$-algebras ?

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.

Thanks

Answer by J Verma for Examples in mirror symmetry that can be understood.

2011-03-06T04:21:25Z

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.

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.

Answer by J Verma for Biography of Felix Hausdorff

2011-03-01T21:12:25Z

one place I know is the website of Hausdorff Institute website, particularly this short biography(62 pages).

Axiomatic treatment of A-model and B-model in sense of Atiyah

2011-03-01T17:58:41Z

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.

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.

Answer by J Verma for Elementary mathematical books

2011-02-22T22:43:53Z

Unfortunately for Galois theory there isn't anything suitable for high school students, but the nice introduction is here Galois theory for beginners, John Stillwell, in addition to historic essay in the introduction to the book by Edwards.

for arithmetics, I think the books by Alan Baker (Theory of numbers) and by G.H. Hardy might be helpful.

for geometry/topology this essay by S.S. Chern can provide some motivation for the subject.

Answer by J Verma for What is the shortest Ph.D. thesis?

2011-02-08T17:37:46Z

I believe the shortest PhD thesis is of Burt Totaro "Milnor K-theory is the simplest part of algebraic K-theory", 12 pages.

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.

Burt Totaro's webpage at Cambridge

Answer by J Verma for What is the motivation for a vertex algebra?

2011-02-01T22:28:16Z

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.

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.

Answer by J Verma for Tips on Writing a CV for a Conference

2011-01-29T04:16:58Z

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.

birational equivalence and mirror CYs

2010-11-22T20:57:21Z

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.

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.

Answer by J Verma for Topology of black holes

2011-01-19T03:17:26Z

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.

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

Answer by J Verma for A soft introduction to physics for mathematicians who don't know the first thing about physics

2011-01-07T18:43:17Z

Let me suggest a reading plan :

The Feynman's books are a pleasure to read and provide great insights into basic physics.
For classical mechanics, "Mathematical methods of Classical mechanics" by V.I. Arnold and "Mechanics" by Landau-Lifshitz.
for quantum mechanics, "Mathematical foundations of quantum mechanics" - Von Neumann. another nice book on quantum mechanics is by R.Shanker.
for statistical physics, the two volumes by Landau are my favorites.
For General Relativity, R.M. Wald and more mathematically inclined is "Large Scale structures of Space time" by Hawking-Ellis.
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.
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.
Ivan Mirkovic has nice lecture notes here http://www.math.umass.edu/~mirkovic/ the notes on string theory. It also has discussion on classical and quantum mechanics.

is complex moduli space of a Calabi - Yau Kahler

2011-01-04T20:37:48Z

The complex moduli space of a Calabi-Yau manifold is a complex manifold (BTT). Is it also Kahler ?

Wall Crossing in Physics and Mathematics

2011-01-03T07:14:02Z

This question is motivated by the current interest of Mathematics and Physics community in Wall Crossing. My questions are :

What is wall crossing in Physics, what are the reasons for current interest in it.
What is wall crossing in terms of mathematics, what is the reason for interest, is it just physics or some mathematical motivation.

thanks.

triangulated/derived categories in Physics and algebraic geometry

2010-12-30T06:35:35Z

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.

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.

I am not sure whether to make it community wiki, feel free to do so. Thanks.

stability conditions in the sense of Kontsevich-Soibelman

2010-12-27T21:16:05Z

What are the stability conditions in the sense of Kontsevich-Soibelman.

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 :

What are the Kontsevich-Soibelman Stability conditions ?
How is it related to Bridgeland's Stability (or Douglas' $\pi$ - stability on D-branes) ?
Why do we need to consider Kontsevich-Soibelman stability.

I've to admit my ignorance of the field. Please suggest some references. Thanks.

topological B model

2010-11-29T06:43:44Z

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.

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.
Thanks.

what is large compex structure limit of CY moduli space

2010-11-22T20:36:39Z

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.

Comment by J Verma

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.

Comment by J Verma

2011-11-23T05:04:20Z

thanks for the answer.

Comment by J Verma

2011-11-23T02:40:22Z

non necessarily compact, but connected G. I've edited the question accordingly.

Comment by J Verma

2011-09-16T03:16:54Z

@ Vivek Thanks for the comment on DT/GW correspondence.

Comment by J Verma

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.

Comment by J Verma

2011-06-17T16:47:17Z

Thanks for the detailed answer.

Comment by J Verma

2011-06-16T19:49:25Z

Can you say more or suggest some reference about how $F_{2}$ is the compactification of $SO(3) \times \mathbb{R}_{+}$.

Comment by J Verma

2011-06-16T19:46:58Z

@ Dmitri - Thanks, this is exactly what I wanted.

Comment by J Verma

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

Comment by J Verma

2011-06-14T17:23:50Z

thanks Andy. So it seems that there is no mathematical definition of hypermultiplet moduli space.

Comment by J Verma

2011-04-29T18:45:51Z

+ 1 superb answer!

Comment by J Verma

2011-04-19T03:52:09Z

+1 Just Amazing...

Comment by J Verma

2011-04-16T21:26:01Z

V.I. Arnold - Mathematical Methods of Classical Mechanics.

Comment by J Verma

2011-04-16T20:23:42Z

thanks for your reply, very enlightening.

Comment by J Verma

2011-04-15T18:43:55Z

@ Sebastian - But a twisted Dirac Operator on a twisted spinor bundle can have non trivial kernel.