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Questions tagged [mp.mathematical-physics]

Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.

3
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1answer
63 views

Incompressible Navier-Stokes equation with heat conduction

How does the incompressible Navier-Stokes system read with heat conduction? Where can I find an existence result for its weak solutions?
3
votes
1answer
77 views

Globalizing Feigin--Frenkel duality

Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{g}^L$ be its Langlands dual. Feigin--Frenkel duality says $$ W^k(\mathfrak{g})=W^{k_L}(\mathfrak{g}^L) $$ if $r'(k+h^{'})(k_L+h'_L)=1$, where ...
4
votes
0answers
135 views

Counting “motifs” with the same “energy”

This question is motivated by physics --- trying to understanding the so-called 'accidental' (i.e. non representation-theoretic) degeneracies that occur in the spectrum of the Haldane--Shastry spin ...
4
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0answers
91 views

Chern-Simons and framing dependence$.$

I posted this question to physics.SE last week (cf. here), but it got not attention. I hope it is not too trivial to post it here. According to ref.1, the correlation functions of a Chern-Simons ...
3
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0answers
126 views

Yang-Mills theory v.s. Kaluza–Klein theory: Classical actions

In general Yang-Mills theory [1] seems to be different from the dimensional reduced Kaluza–Klein theory. However, the historical account was that people tried to trace back the origin of non-Abelian ...
4
votes
1answer
577 views

Is this Hermite polynomial identity known?

In some physics related problem, I found out the curious identity $$\sum\limits_{n_1+n_2+n_3=n}\frac{n!}{n_1!\,n_2!\,n_3!}\,H_{2n_1}(x)\,H_{2n_2}(y)\,H_{2n_3}(z)=\frac{H_{2n+1}(r)}{2r},$$ where $H_n(x)...
6
votes
1answer
169 views

Navier-Stokes fluid dynamics, Einstein gravity and holography

There was some activity a while ago, like 10 years ago, string theoreists try to relate the fluid dynamics, for example, governed by Navier-Stokes equation, to the Einstein gravity, and its ...
5
votes
1answer
118 views

Anderson localization for fractional Laplacians

There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as $$ -\Delta+\lambda V $$ where $\Delta$ is the discrete ...
8
votes
3answers
312 views

Physical Applications of Locally Symmetric Spaces

Locally Symmetric Spaces are the basis of the Langlands program—a set of ambitious and interconnected conjectures connecting representation theory to number theory, firstly proposed in 1967 by Robert ...
8
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0answers
182 views

Yang-Mills theory with non-compact gauge groups G

Physicists are familiar working with Yang-Mills theory with compact and semi-simple gauge groups $G$ (Lie groups). However, it is not entirely clear the formulation of Yang-Mills theory with non-...
4
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0answers
54 views

GSO projection and $H^d(M, \mathbb{Z}_2)$

This follows up the comment which suggests that asking the later 2nd part of subquestion in "GSO (Gliozzi-Scherk-Olive) projection and its Mathematics" as a new different question GSO (Gliozzi-Scherk-...
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0answers
143 views

GSO (Gliozzi-Scherk-Olive) projection and its Mathematics?

GSO (Gliozzi-Scherk-Olive) projection is an ingredient used in constructing a consistent model in superstring theory. The projection is a selection of a subset of possible vertex operators in the ...
5
votes
2answers
109 views

Lifting a diffeomorphism into a spinor bundle automorphism

I know several papers that treat this, but it seems that most of these papers do things very differently with quite different conclusions, so I am confused. Basically, when one tries to do classical ...
9
votes
2answers
434 views

Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...
2
votes
0answers
118 views

Holomorphic map, Instantons of Complex Projective Space and Loop Group

It seems that holomorphic (or rational) maps play a crucial role to relate the following data: Instanton in 1-dimensional complex Projective Space $$\mathbb{P}^1$$ in a 2 dimensional (2d) spacetime. ...
4
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0answers
158 views

Chern-Simons theory with non-compact gauge groups G

This is related to a previous question, where a nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. There we ask the general ...
2
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0answers
110 views

Minkowski isometries

Consider theorem 1.7 from chapter III of 'Elementary differential geometry' by O'Neill. It says that: Theorem 1.7: If $\phi$ is an isometry of $E^3 $, then there exists a unique translation $T$ and a ...
3
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1answer
337 views

Does current follow the path(s) of least (total) resistance?

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $2$ electrodes $E_1$ and $E_2$ (...
6
votes
0answers
263 views

Witten zeta function v.s. Riemann zeta function

From a talk, we learned that The symplectic volume of the space $M$ of gauge equivalence classes of flat G connections is given by the “Witten zeta function”: where we sum over irreducible ...
4
votes
1answer
121 views

Nonlinear sigma models with non-compact groups / target spaces

A nonlinear σ model (NLSM) describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The target manifold T is equipped with a Riemannian metric g. Σ is ...
3
votes
1answer
351 views

Conserved Positive Charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
5
votes
1answer
223 views

Moduli space of flat connections over a Riemann surface

If I understand correctly, in the Refs below: We can see that the moduli space of SU($N$) flat connections over a torus, is equivalent to a complex projective space $\mathbb{P}^{N-1}$ Namely, $$ M_{\...
3
votes
1answer
78 views

Reference request for quantum Teichmuller space

I would like to ask for some detailed reference for quantum Teichmuller theory, better in a mathematical taste. I read a little bit on Kashaev's or Chekhov and Fock's, but find that I need to fill ...
7
votes
1answer
250 views

The Precise Meaning of the Moduli Space of Flat Connections?

Questions: I would like to have a precise description of the meanings of the Moduli Space of Flat Connections, such that it is understandable by mathematical physicists and physicists. For 3d Chern-...
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0answers
99 views

Solution of nonlinear heat equation decreases in time

Let $f\colon \mathbb{R} \to \mathbb{R}$ be a smooth decreasing function which is non-negative and bounded above, and consider the heat equation on a smooth bounded domain $$u_t - \Delta u = f(u)$$ ...
8
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0answers
100 views

Coleman–Mandula theorem and a mathematical proof

Coleman–Mandula theorem (by Sidney Coleman and Jeffrey Mandula) [1] is a no-go theorem in theoretical physics. It states that "space-time and internal symmetries cannot be combined in any but a ...
2
votes
1answer
80 views

stationary measure for linear cocycle(random transformation matrices)

Let $(M,\mathcal B, \mu)$ be a probability space which $M=\{A_{1},A_{2},...,A_{N}\}^{\mathbb{N}}$ ($A_{i} \in GL(d ,\mathbb{R})$) and $\mu=p^{\mathbb{N}}$. Let $F:M\times \mathbb R^d\to M\times \...
6
votes
1answer
85 views

Further Developments of Lieb-Schultz-Mattis theorem in Mathematics

The Lieb-Schultz-Mattis theorem [1] and its higher-dimensional generalizations [2] says that a translation-invariant lattice model of spin-1/2's cannot allow a non-degenerate ground state preserving ...
4
votes
0answers
137 views

Complex projective algebraic variety, moduli space of flat connections, and instantons

In Looijenga's work below, if I understand correctly, it shows that Statement 1: At an algebraic variety, the moduli space of SU($N$) flat connections on a 2-torus $T^2$ is given by the space of ...
4
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0answers
111 views

Dimensions of the instanton moduli space from Atiyah-Hitchin-Singer

Atiyah-Hitchin-Singer Ref 1 states that the number of virtual dimensions of the instanton moduli space for SU(N) Yang-Mills theory with topological charge $\mathcal{Q}$ over a manifold $X$ is given ...
8
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0answers
78 views

Borel-Ecalle re-summation and resurgence: Criteria and Results

This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\...
11
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0answers
472 views

The Grassmannian Gr(2,8) and an E7 surprise

Are there any mathematical explanations for the following surprising facts? $$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$ and $$\int_{Gr(2,6)} c_{\text{top}}...
3
votes
2answers
221 views

Casimir operator of a given Lie algebra and relation with its matrix representation

I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
17
votes
1answer
438 views

How is Chern-Simons theory related to Floer homology?

Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ ...
5
votes
0answers
221 views

Have complex manifolds with dual number structure on the holomorphic tangent bundle been studied?

If $M$ denotes a $2n$-dimensional real smooth manifold, then $M$ together with $J\in\operatorname{End}(TM)$ with $J^2 =\mathrm{id}$ (also called almost product structure on $M$) have been surveyed in ...
4
votes
1answer
158 views

The embedding of $\mathfrak{g}_2$ into $\mathfrak{b}_3$ ($\mathfrak{so}_7$) on Chevalley generators

Let $\mathfrak{g}_2$ / $\mathfrak{b}_3$ be the simple complex Lie algebra of type $\mathsf{G}_2$ / $\mathsf{B}_3$ (the latter is also known as $\mathfrak{so}_7$). How is the embedding $\mathfrak{g}...
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votes
0answers
31 views

Decoupling/Simplifying a long algebraic expression as a function of 5 parameters

I have the following variables defined by 5 parameters $(g_1, g_2, \kappa_1, \kappa_2, \Gamma)$: $$ u_1=36g_1^{2}(2p-3\kappa_2)+\big(36g_2^{2}+(2p-3\kappa_1)(2p-3\kappa_2)\big)\big(4p-12\kappa_+\big)\...
5
votes
1answer
171 views

Non-Hamiltonian actions in physics

I was reading the following article when I came across the interesting sentence "non-Hamiltonian [symplectic group] actions also occur in physics" I took a cursory look at the article cited but ...
3
votes
1answer
131 views

Gauge integral versus path integral

According to this paper, "The gauge integral [a.k.a. Henstock-Kurzweil integral] provides the only formal framework that is close to the original development of the Feynman path integral", and also "...
1
vote
1answer
45 views

Existence of bisolutions for wave operators on globally hyperbolic Lorentzian manifolds

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold with Lorentzian volume density $dV$ and $\Sigma$ a Cauchy hypersurface, i.e. each inextendible causal curve hits $\Sigma$ exactly once. ...
0
votes
1answer
123 views

Nonlocal integral

I have a little problem with the next integral, $$ \int{d^3{\bf r^{\prime}}\left[\frac{exp(-ar^{\prime})}{r^\prime}\right]u({\bf r}-{\bf r^\prime}})=\int{4\pi r^\prime dr^{\prime}exp(-ar^{\...
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vote
0answers
88 views

Euclidean volumes under the matrix norm of one-parameter subsets of unit balls of $2 \times 2$ matrices

Prove that the Hilbert-Schmidt volume (normalized to equal 1 at $\varepsilon=1$) of the subset of the unit ball in operator norm (http://mathworld.wolfram.com/OperatorNorm.html) of the $2\times 2$ ...
2
votes
1answer
160 views

PDE’s whose solutions can be presented using path integrals

It is well known that solutions of the Schroedinger equation and of the heat equation can be presented using path integrals: $$\psi(x,t)=\int K(x,t;y,0)\psi(y,0)dy,$$ where the kernel $K(x,t;y,0)$ is ...
2
votes
0answers
44 views

What restrictions on the form of an integral equation have a unique solution f=0?

We're stuck on the following question in a problem relevant to a physics paper on AdS/CFT that we are working on. Given a Fredholm equation of second kind with the form $f$+$\int_D K f\,dx = 0$, where ...
7
votes
3answers
395 views

Simple Subalgebras of Simple Lie Algebras

Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra ...
16
votes
2answers
394 views

What are some mathematical consequences of the study of 6D $\mathcal N = (2,0)$ SCFT?

Arguments made in physics apparently predict the existence of a family of six-dimensional $\mathcal N = (2,0)$ superconformal field theories (Wikipedia, nLab, PhysicsOverflow) sometimes called Theory $...
-4
votes
1answer
197 views

Borromean rings, Condorcet's paradox and Quantum chromodynamics [closed]

In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in ...
2
votes
1answer
108 views

$P(1)$ strange type classical Lie superalgebras

I am reading the book "Lie superalgebras and enveloping algebras" by Ian M. Musson. The strange type $P(n)$ series of Lie superalgebras are defined (§2.4.1, p. 17) only for $n \ge 2$ even though for $...
4
votes
0answers
70 views

Mathematical proof of Regge symmetry

In the representation theory of the group $SU_2$ a big role is played by so-called $6j-$symbols. Let me sketch its definition (some other interpretations could be found here). Denote a ...
3
votes
0answers
84 views

About mod 2 Index of Dirac Operators in 3D on Non-Orientable Manifold

I was reading Witten's paper "Fermion Path Integrals and Topological Phases" (https://arxiv.org/abs/1508.04715). He claimed that Indeed, on an orientable 3-manifold, the eigenvalues of the Dirac ...