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.

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24 votes
1 answer
585 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 $...
150 votes
26 answers
47k views

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

There have been similar questions on mathoverflow, but the answers always gave some advanced introduction to the mathematics of quantum field theory, or string theory and so forth. While those may be ...
1 vote
0 answers
119 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 ...
2 votes
1 answer
138 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? (...
9 votes
0 answers
330 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 $P\rightarrow M$ ...
3 votes
0 answers
252 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. ...
8 votes
0 answers
272 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-...
5 votes
1 answer
354 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
1 answer
395 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
1 answer
197 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
0 answers
212 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 $\...
4 votes
0 answers
229 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://...
8 votes
0 answers
158 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-...
5 votes
0 answers
123 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
1 answer
307 views

Combinatorial Pin structure

David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the ...
9 votes
0 answers
267 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 ...
7 votes
0 answers
373 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 ...
18 votes
0 answers
295 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 ...
3 votes
2 answers
406 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 ...
14 votes
2 answers
694 views

Is a unitary Hamiltonian TQFT the same as a unitary axiomatic TQFT?

Introduction Axiomatic TQFTs An axiomatic $n$-dimensional TQFT is a symmetric monoidal functor $\mathcal{Z}\colon \operatorname{Bord}_n \to \operatorname{Hilb}$ from $n$-dimensional oriented ...
5 votes
0 answers
293 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....
9 votes
1 answer
399 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 ...
4 votes
1 answer
291 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 ...
6 votes
1 answer
304 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
0 answers
262 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 ...
6 votes
0 answers
293 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 ...
1 vote
0 answers
515 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{...
3 votes
1 answer
232 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}) := \...
12 votes
1 answer
566 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 ...
4 votes
1 answer
251 views

Krichever-Novikov-Dubrovin description for not-algebraic spectral curve

Non-algebraic curves play an increasing role in string theory, sometimes they are known to be related to the integrable systems of the KP/Toda type. Are there any investigated examples of the ...
4 votes
0 answers
325 views

The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) operator on $\mathbb{C}[x,y]$: $\tilde{D}_{X}(f)=...
3 votes
0 answers
127 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 ...
0 votes
3 answers
368 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
0 answers
72 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 ...
0 votes
0 answers
235 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 +\...
0 votes
1 answer
154 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
0 answers
133 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
0 answers
123 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 ...
5 votes
1 answer
291 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 ...
5 votes
1 answer
291 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 ...
8 votes
1 answer
603 views

Phase transitions between Category Theories

question: What are the mathematical theories suitable to describe the "continuous Phase transitions between Category Theories"? The phase transitions mean that in terms of the quantum statistical ...
38 votes
4 answers
4k 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 ...
1 vote
0 answers
61 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
0 answers
208 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. ...
5 votes
0 answers
100 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_{...
17 votes
3 answers
4k views

Is there a Poincare-Hopf Index theorem for non compact manifolds?

Does Poincare-Hopf index theorem generalizes in any way to non compact manifolds ? In particular, I am interested in the case of a smooth vector field on a cylinder $\mathbb{T}_1\times\mathbb{R}$? If ...
4 votes
0 answers
321 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 ...
5 votes
1 answer
3k views

Intuition behind the "Lapse Function"

I came across the following definite of the Lapse Function: $N=\sqrt{\frac{1}{2}g(L,\overline{L})}$ where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...
0 votes
0 answers
117 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/...
1 vote
0 answers
78 views

Finite sum of spherical Bessel functions

In L.G. Afanasyev, A.V. Tarasov, Breakup of relativistic pi+pi- atoms in matter, Phys. At. Nucl. 59 (1996) 2130 the following identity is given for the spherical Bessel functions $j_n(z)=\sqrt{\frac{\...

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