Mathematical methods in classical mechanics, classical and quantum field theory, quantum mechanics, statistical mechanics, condensed matter, nuclear and atomic physics.
24
votes
1answer
299 views
Has the $E_8$-based generating function for squares numbers been proven?
In his 2004 paper Conformal Field Theory and Torsion Elements of the Bloch Group, Nahm explains a physical argument due to Kadem, Klassen, McCoy, and Melzer for the following remarkable identity. Let $...
1
vote
0answers
76 views
Determine all possible magnetic monopole of gauge theories
In Wikipedia, it states about the magnetic monopole of the gauge theory is determined by the fact:
This argument for monopoles is a restatement of the lasso argument for a pure U(1) theory. It ...
8
votes
0answers
155 views
Twisted Chern-Simons, and Twisted Wess-Zumino Term
I am asking this question about Chern-Simons theory from the paper "Quantum Field Theory and Jones Polynomial" by Edward Witten.
Let $M$ be a closed three dimensional manifold, and $P\rightarrow M$ ...
2
votes
0answers
39 views
Resource request: Moyal $\star$-product based calculations
I already asked two questions about the Moyal $\star$-product here and here but I think I'll have a lot more similar questions, so I'm wondering if anyone can help me with finding some good resources.
...
4
votes
1answer
66 views
Moyal $\star$-product of $\star$-exponentials
Definitions and assumptions
On a 2n-dimensional phase-space with coordinates $x$ and $p$, the Moyal product can be written explicitly as
$$g(x,p) \star h(x,p) = g(x,p) e^{\frac{i}{2}\left( \...
8
votes
1answer
152 views
Moyal $\star$-product inverse?
On a 2n-dimensional phase-space with coordinates $x$ and $p$, the Moyal product can be written explicitly as
$$g(x,p) \star h(x,p) = g(x,p) e^{\frac{i}{2}\left( \overleftarrow{\partial_x} \cdot \...
6
votes
1answer
87 views
Relationship between irreducible representations of the Schur covering group and elements of $H^2(G,U(1))$
Let $G$ be a finite group and let $D(g)$ be a projective representation of $G$ i.e.
\begin{equation}
D(g) D(h) = e^{i \omega(g,h)} D(gh)
\end{equation}
These can be classified by the equivalence ...
2
votes
0answers
131 views
Define a (lattice) Yang-Mills theory on $\mathbb{T}^4$ v.s. $\mathbb{R}^4$
Pure Yang-Mills theory (YM) can be easily defined on $\mathbb{T}^4$ on a periodic lattice, using the Wilson lattice gauge theory approach. In reality, we know some of these mathematical results on $\...
-2
votes
0answers
98 views
Oscillating particle motion
Suppose I have a particle with an initial position at $X_0 =0$. Its velocity is given by the following:
\begin{align*}
\dot{x} = \begin{cases} 1 & \text{if } x = 0 \\
-1 & \text{ otherwise }\...
3
votes
0answers
77 views
Virtual fundamental class of Moduli space of stable maps in genus 1
What is the virtual fundamental class of $\overline{M}_{1,n}(\mathbb{P}^2,d)$? In general the virtual fundamental class is difficult to compute I guess. But if you look at Proposition 2.5 of https://...
9
votes
0answers
130 views
Define the 3d Chern-Simons TQFT on a discrete simplicial complex
Question: What is the challenge and the current status to define the 3d Chern-Simons(-Witten) (CSW) theory on a simplicial complex or on a discrete lattice? (Or is there a no-go or an obstruction ...
7
votes
0answers
71 views
On constructible Hall algebra and instantons
I heard in a talk by Yan Soibelman that by starting with a quiver $Q$ with a set of vertices $I$ we can either symmetrize or anti-symmetrize its Euler-Ringel form $\chi_Q$. He claims that anti-...
6
votes
1answer
479 views
(In)stability of a two-dimensional dynamical system
Consider the following system of coupled differential equations
$$
\dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\
\dot{x}_2(t) = -\gamma x_2(t) - \cos(\...
4
votes
2answers
220 views
Weyl's Branching Rule for $SU(N)$-Setting
On the Wikipedia page for restricted representations
https://en.wikipedia.org/wiki/Restricted_representation
there is presented a number of explicit "branching rules". In particular, there is the ...
6
votes
0answers
68 views
Degenerations of $OG_{+}(5,10)$ over the Kapustin-Witten $P^1$ and tetragonal genus 7 curves
If $X_{i,j}$ is a $2 \times 4$ matrix and $Y_{j,k}$ is a $4 \times 2$ matrix,
then there is a $P^1$ family of ideals defined by the four equations $XY = 0$ and the six equations $s \cdot minors(X) + t ...
7
votes
0answers
146 views
A robust version of Schur's lemma?
Does a robust version of Schur's lemma exist? Specifically, I was wondering about something like this:
Let $B$ be a bounded operator over a vector space $V$, with underlying field $\mathbb{C}$ and ...
4
votes
0answers
66 views
Modular $S$-matrix for an extended affine Lie algebra
This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.
In this paper, the authors ...
6
votes
0answers
133 views
$U(1)$ v.s. $SU(N)$ v.s. $SO(N)$ instantons
I am interested in knowing the details of the comparison between $U(1)$, $SU(N)$ and $SO(N)$ instantons for their gauge theories in 4 spacetime dimensions., in terms of:
Chern class (1st, 2nd), and
...
12
votes
0answers
161 views
Profiles of very high dimensional functions
This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...
5
votes
0answers
117 views
Bosonic topological orders and unitary fully dualizable fully extended TQFT
I would like to ask if the following statement can be true:
bosonic topological orders in $n$-dimensional space-time 1-to-1 correspond to unitary fully dualizable fully extended TQFT in $n$-dimensions....
8
votes
1answer
193 views
Young tableaux for exceptional Lie algebras
Irreducible representations for the $A$-series Lie algebras are labelled Young diagrams, with a basis of each given by Young tableaux. Moreover, analogues exist for the $B,C$, and $D$ series.
Does ...
6
votes
1answer
170 views
Branching from $E(6)$ to $SO(10) \times U(1)$
In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...
4
votes
0answers
106 views
Instanton configurations of self-dual and anti-self-dual instantons interplay
Yang-Mills gauge theory is given by the action
$$ S_\text{YM}[A] = \int_M\mathrm{Tr}_\mathfrak{g}(F\wedge \star F)$$
whose Euler-Lagrange equations are the classical equations of motion. The classical ...
4
votes
1answer
181 views
Symplectic forms and sign of eigenvalues
This question has come out while reading J. Moser "New Aspects in the Theory of Stability
of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
1
vote
0answers
81 views
Conventions / Normalizations of Yang-Mills Field Theories
Let the spacetime be 4-dimensional.
In the usual Maxwell theory of Abelian gauge fields $A$, where field strength $F=dA$ one considers the Maxwell action written as
$$
S_{Maxwell}\equiv\int
-\frac{...
5
votes
0answers
162 views
Hints on an expository article about Kardar-Parisi-Zhang (KPZ)
It seems the KPZ is the next big thing in mathematical physics and probability. The skeletal idea is probably that while classical averages are in the Gaussian universality class, lots of other ...
9
votes
1answer
191 views
Compactification of 6d (2, 0) SCFT on 4-manifolds
This question is about the 6d (2, 0) superconformal field theory (also called 'theory X' by some people). This SCFT, which can be considered as a relative quantum field theory (see here for a ...
7
votes
0answers
162 views
The Fock Space vs the Hilbert space in the context of Quantum Field Theory
Mathematically the definitions are as follows : if $H_n$ is a $n-$dimensional complex Hilbert space then its two different corresponding ``Fock space"(s) are often denoted as $F_{1}$ and $F_{-1}$ ...
3
votes
0answers
90 views
Asymptotic Expansion of Seiberg-Witten Differential?
Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by
\begin{equation}
\mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow ...
7
votes
1answer
165 views
What is a formal definition of a Fermionic quantum field?
I could not locate a definition of Fermionic quantum field (like for an electron!) in even Kevin Costello's book, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.4961&rep=rep1&...
0
votes
3answers
128 views
Single quantum particle entropy
Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...
2
votes
0answers
64 views
About anti-commutation of gauge charged Fermionic quantum fields
[Please correct anything I might say wrong in what follows!]
For everything that follows I am thinking in the context of a supersymmetric QFT. Hence I guess everytime I say "spacetime" it needs to be ...
3
votes
2answers
306 views
Can one calculate the following operator? [closed]
Summary
I recently defined some numbers which obey multiplication but not addition. To my surprise after some heuristic manipulations (ignoring convergence), it seems I can express the creation and ...
0
votes
0answers
65 views
Spectrum of a Hamiltonian on the real line
Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
3
votes
1answer
157 views
Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$?
Nakajima & Yoshioka [1] showed that
\begin{equation}
F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q}) = \sum_{n = 1}^\infty \mathbf{q}^nF^{inst}_n(\epsilon_1,\epsilon_2,\mathbf{a}) := \...
0
votes
1answer
62 views
Convergence of an integral with respect to the Wiener measure
Most probably this question should be well studied in the theory of stochastic processes, but I am not educated in that area. Sorry if this question is too elementary.
Let $V\colon \mathbb{R}\to \...
4
votes
0answers
92 views
Langlands dual and integrable representations
Assume I successfully classified the integrable representations of a certain semi-simple Lie group $G$. Given this information, what do I know about the integrable representations of $G^\vee$, the ...
3
votes
0answers
89 views
Other than Brownian motion, when else is it possible to define “normalized weighted infinite dimensional Lebesgue measure”?
In this article Sourav Chatterjee poses the question, how do we define the measure:
$$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$
The $Z$ here is an infinite normalizing ...
6
votes
0answers
105 views
Why are Levin-Wen/Turaev-Viro models said to be non-chiral?
I'd like to bring together the following two notions of "non-chiral":
On the abstract algebraic side, a modular fusion category describing the anyon content of some physical system is said to be non-...
3
votes
1answer
81 views
Quantum tunneling on the line with non-symmetric double well potential
Consider the Schroedinger equation on the line
$$i\frac{\partial \Psi(x,t)}{\partial t}=[-\frac{d^2}{dx^2}+V(x)]\Psi(x,t),$$
where one assumes that $V(x)\to +\infty$ as $|x|\to +\infty$, and $V$ has ...
27
votes
5answers
2k views
Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?
I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free.
The idea is to use ...
4
votes
1answer
88 views
Some examples of vertex algebra modules
Recently I'm learning the vertex modules. In the paper, there are a lot of abstract theory about the module theory,for instance the $C_{2}-$cofinite conditions and associated variety. I hope to find ...
5
votes
1answer
81 views
An extension of the Izergin-Korepin determinant to the eight-vertex model
In the six-vertex model, edges in a square lattice are oriented so that the in-degree of each vertex is exactly two. This gives six types of allowable vertices:
$$\begin{array}{cccccc}
\begin{...
2
votes
1answer
104 views
Two questions on Zuber's “KdV and W-flows”
I'm having difficulty following computations in the paper "KdV and W-flows" by Zuber.
On pg. 2, what would be the conserved quantity $I_4$, related to the conservation laws of the KdV hierarchy? (...
1
vote
0answers
44 views
Realizing $N$-body Hamiltonian operator from $2$-body operator
Let $N\in\mathbb{N}$, and consider the formal $N$-body Schrodinger operator
$$\sum_{j=1}^{N}-\partial_{x_{j}}^{2}+2c\sum_{1\leq j_{1}<j_{2}\leq N}\delta(X_{j_{1}}-X_{j_{2}}), \tag{1}$$
where $c\in\...
4
votes
0answers
162 views
Open-closed string correspondence
Recently, after many years of searching for the right source, I came across the excellent lecture by Aspinwall, "Some Applications of Commutative Algebra
to String Theory", in Eisenbud's Festschrift. ...
29
votes
4answers
2k views
A family of polynomials whose zeros all lie on the unit circle
I had posted the following problem on stack exchange before.
Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the ...
4
votes
0answers
67 views
Paving property
In their famed paper (https://arxiv.org/abs/math-ph/0011053), Bourgain and Goldstein conjecture what they call the paving property:
Let $H_{jk}=\delta_{j,k+1}+\delta_{j,k-1}+v(\theta+j\omega)\delta_{...
3
votes
0answers
150 views
Topological field theories and their path integrals
Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten ...
0
votes
0answers
89 views
Barnes's double $\Gamma$-function, $\gamma_{h}(0) = -\frac{1}{12}$?
I apologize if I may have gotten the name of the function incorrectly. The function $\gamma_h(x)$ is defined in Appendix A of the paper SEIBERG-WITTEN THEORY AND RANDOM PARTITIONS, https://arxiv.org/...
