Questions tagged [mathematical-physics]

DO NOT USE THIS TAG just because your question involves math! If your question is on simplification of a mathematical expression, please ask it at math.stackexchange.com. Mathematical physics is the mathematically rigorous study of the foundations of physics, and the application of advanced mathematical methods to problems in physics. Examples include partial differential equations (PDEs), variational calculus, functional analysis, and potential theory.

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128
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0answers
6k views

Linear sigma models and integrable systems

I'm a mathematician who recently became very interested in questions related to mathematical physics but somehow, I faced difficulties in penetrating the literature... I'd highly appreciate any help ...
47
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0answers
2k views

Can Lee-Yang zeros theorem account for triple point phase transition?

Now the prominent Lee-Yang theorem (or Physical Review 87, 410, 1952) has almost become a standard ingredient of any comprehensive statistical mechanics textbook. If the volume tends to infinity, ...
27
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0answers
452 views

Minimal strings and topological strings

In http://arxiv.org/abs/hep-th/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free ...
24
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2answers
750 views

Is there a physical interpretation to invariant random matrix ensembles?

Disclaimer. I am a graduate student in pure mathematics, so my knowledge of physics more advanced than basic 1st/2nd year undergraduate physics is very limited. I welcome corrections on any ...
22
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0answers
942 views

TQFTs and Feynman motives

Questions Is a topological quantum field theory metrizable? Or else a TQFT coming from a subfactor? For a given metric, are there always renormalization and Feynman diagrams? Is there always a Feynman ...
22
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0answers
953 views

p-Adic String Theory and the String-orientation of Topological Modular Forms (tmf)

I am going to ask a question, at the end below, on whether anyone has tried to make more explicit what should be a close relation between p-adic string theory and the refinement of the superstring ...
21
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0answers
524 views

Hypersingular Boundary Operator in Physics

This has been a question I've been asking myself for quite some time now. Is there a physical Interpretation of the Hypersingular Boundary Operator? First, let me give some motivation why I think ...
20
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0answers
578 views

Electric charges on compact four-manifolds

Textbook wisdom in electromagnetism tells you that there is no total electric charge on a compact manifold. For example, consider space-time of the form $\mathbb{R} \times M_3$ where the first factor ...
12
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0answers
301 views

How can we deduce that a hydrogen atom is stable in relativistic QED?

Consider relativistic quantum electrodynamics (QED) with three quantum fields: the electromagnetic field $A_\mu$, one fermion field $\psi$ for electrons/positrons, and one fermion field $\psi'$ for ...
11
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0answers
385 views

Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions

I am unfamiliar with moduli spaces and ergodic theory which appear to be essential in Maryam Mirzakhani's math contributions which won her the Fields Medal. However, I am well conversant in essential ...
10
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0answers
240 views

How are local observables encoded in this formulation of quantum field theory?

I've recently begun trying to understand a formulation of quantum field theory as a functor from a category of spacetimes-with-boundaries (bordisms) to a category of Hilbert spaces, as reviewed in ...
9
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0answers
130 views

With a local anomaly, is the determinant of the Dirac operator still a section of a complex line bundle?

In the literature about anomalies in quantum field theory, the determinant of the Dirac operator plays an important role. The Dirac operator may depend on some background data, and the subject of ...
9
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0answers
370 views

Haag's theorem in a box

Haag's theorem states that the interaction picture does not exist in a rigorous way in relativistic quantum field theory. Some ways in which the interaction picture can fail are that the eigenstates ...
9
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2answers
596 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
8
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585 views

Level-rank duality in WZW models and CS theories

Cross-posting from Physics Overflow: https://www.physicsoverflow.org/41281/level-rank-duality-in-wzw-models-and-cs-theories I know that the classical level-rank duality in the $\widehat{\mathfrak{sl}}...
8
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0answers
186 views

Metric transformation, polygons and gravitons

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
8
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414 views

Objective time derivative that is not a Lie derivative

Summary Led by an interest into the concept of "Material Objectivity", I am asking myself: Are there objective time rates that are not Lie derivatives? The long read I am trying to understand the ...
8
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0answers
222 views

Wightman axioms always imply triviality in 4D?

Someone mentioned to me in passing that it had been proven that the Wightman axioms are over-restrictive in four dimensions and provably always result in trivial correlators (or maybe a trivial S-...
8
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0answers
212 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
8
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0answers
187 views

Topology-dependent groud state degeneracy of $B \wedge F + B \wedge B$ and $B \wedge F + B \wedge B \wedge B$

There are some examples of topological BF theory with extra terms allow it still being topological. See this Ref. paper In 4d (3+1D), we have the trace of: $$ \int\frac{k}{2\pi}\text{Tr}[B \wedge F + ...
8
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0answers
320 views

Coleman-Mandula theorem in mathematical language

Every supersymmetry text starts off mentioning the Coleman-Mandula theorem. Often it is introduced using rather colloquial terminology. I was wondering if anyone knew a precise mathematical ...
8
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0answers
218 views

The Integral Trick and An Equality in Nakajima's Lecture

In Nekrasov et al's series papers MNS, they calculate such kinds of integral $$ \frac{E_1 E_2}{N(2\pi i)^N(E_1+E_2) }\oint d\phi_1 \wedge d\phi_2\wedge \ldots\wedge d\phi_N \prod_{i<N} (-\phi_i) \...
7
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0answers
107 views

Which Osterwalder-Schrader axiom does a tree-level QFT violate?

(I think this question is really about any truncation of the perturbation series, but I want to avoid having to think hard, so I'll talk about tree-level only to ask the question.) Let's start with ...
7
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0answers
137 views

Where do theta terms live?

Consider a gauge theory with group $G$. The canonical kinetic term for the gauge field is $F\wedge\star F$ and, depending on the dimensionality of spacetime, there are other allowed terms, such as ...
7
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0answers
86 views

Can you do gauge theories over topological groups?

Quantum gauge theories involve (functional) integration over a Lie group. Is there any meaningful generalisation to (non-manifold) topological groups? Consider for example the Whitehead tower $$ \...
7
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0answers
516 views

Is there a Virasoro group?

On page 14 of the survey article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive, the authors show that smooth selfmaps of the circle form a Lie group corresponding ...
7
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0answers
65 views

When does the correlator of a string of fields and the current vanish “sufficiently fast” at infinity and Ward's identity?

One consequence of the Ward identity (cf. Di Francesco et al) is that it means variation of correlators under infinitesimal transformation is zero. This can be seen by integrating the ward identity, ...
7
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0answers
356 views

explicit matrix elements for a representation decomposed into subgroup by branching rules

I'm looking for a way to construct a representation for a simple Lie group such that one particular subgroup is manifest. I learned the branching rules from Cahn, Georgi and Slansky, but I'm still not ...
7
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0answers
95 views

What is the importance of studying degeneration on $M_g$

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$. It seems to be important in physics to study ...
7
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0answers
399 views

1-form formulation of quantized electromagnetism

In a perpetual round of reformulations, I've put quantized electromagnetism into a 1-form notation. I'm looking for references that do anything similar, both to avoid reinventing the wheel and perhaps ...
6
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0answers
86 views

Anomalies in QFT: why do we require smooth dependence on the background fields?

If $D$ is the Dirac operator for some dynamic spinor fields in a background gauge field $A$, then the partition function is supposed to be $\mathrm{det}(D)$. But if the coupling to the gauge field is ...
6
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0answers
82 views

Are Chern-Simons theories classified by bordism groups?

For a long time it was thought that anomalies for a group $G$ were classified by $H^n(BG)$, although it is now understood that they are in fact classified by $\Omega^n(BG)$. On the other hand, ...
6
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0answers
72 views

Lost reference: Kähler gravity in six dimensions and three dimensional $SL(2,\mathbb{C})$ Chern-Simons theory

I've noticed that several references take for a fact that by studying Kähler gravity on a Calabi-Yau threefold one can demostrate that any lagrangian submanifold embedded in the threefold posees three ...
6
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0answers
302 views

What is density matrix in QFT?

In quantum mechanics exist fundamental object Density matrix. (See for introduction last chapter in Principles of Quantum Mechanics by David Skinner). Density matrix nesesary to describe systems even ...
6
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0answers
140 views

Intuitive/Physical reason why fields are distributions

I read in Urs Schreiber's notes on mathematical QFT that the infinities in the standard approach to QFT appear because the product between operator-valued field distributions is not always well ...
6
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0answers
198 views

How can I get the Seiberg-Witten curve from M-theory?

I know that we can use the SU($n$) 6d (2,0) SCFT, the M5-brane world-volume theory to get $\mathcal{N}=2$ theories. Still, e.g. reading Tachikawa's "Supersymmetric dynamics for pedestrians" I cannot ...
6
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0answers
124 views

BF theory without perturbation theory (spin network appoach)

The BF model given by the action $$S = \int \mathrm{tr}(B \wedge F)$$ with a 2-form field $B$ and fibre bundle curvature tensor $F$ will have the following partition function after integration over $...
6
votes
0answers
1k views

What is a branch cut singularity in QFT?

Peskin & Schröder say on page 216: The poles in $p^2$ come only from one-particle intermediate states, while multiparticle intermediate states give weaker branch cut singularities. In order to ...
6
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0answers
421 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
6
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0answers
165 views

What is the physical interpretation of the automorphism on bounded operators induced by an S matrix?

In a QFT, the S-matrix $S$ is a unitary operator, that fixes the vacuum and commutes with the unitary operators implementing the action of the Poincare group on an appropriate Hilbert space $H$. ...
6
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0answers
969 views

What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
6
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0answers
270 views

Level quantization of 7d $SO(N)$ Chern-Simons action

In 3d, one can write down the $SO(N)$ Chern-Simons action to be $$S(A)=\frac{k}{192\pi}\int_{M}\text{Tr}(A d A +\frac{2}{3}A^3),$$ where $A$ is an $SO(N)$ connection. The level quantization can be ...
6
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0answers
317 views

Topological Quantum Field Theories

I've asked this on Math.SE, but with no avail. So, I decided to ask it here. I was wondering about the following after reading the Wikipedia article on TQFTs. It is said that TQFTs have vanishing ...
6
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0answers
263 views

Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I'm a mathematician with only the basic knowledge of Physics, so my question may be trivial: in this case, mercy me. :-) Let $\Omega \subseteq \mathbb{R}^N$ be a domain and let $V,m:\Omega \to \...
6
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0answers
238 views

K3 gravitational instanton

Could you please recommend a sufficiently elementary introduction to K3 gravitational instanton in general relativity and the problem of finding its explicit form? Under 'sufficiently elementary' I ...
5
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0answers
47 views

Physics behind the Kobayashi-Hitchin correspondence

Let $X$ be a $d$-dimensional Kähler manifold with Kähler matric $\omega$. Let's consider the following setups: Suppose $E \rightarrow X$ is hermitian vector bundle with hermitian connection $A$. In ...
5
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0answers
133 views

Physicist path integral and cylinder set measures

Path integral via discretization So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is just what I understood!). Let a quantum system ...
5
votes
0answers
450 views

Are there any open problems in mathematics whose resolutions would have important physical implications?

Are there any open problems in pure mathematics whose resolutions would resolve a well-formulated open physics question, whose solution could (at least in principle) explain an experimentally ...
5
votes
0answers
226 views

What are the applications of hyperbolic $3$-manifold theory to cosmology?

I am a pure mathematician specialized in hyperbolic $3$-manifold topology. That has been an incredibly active field of research in the past few decades due to the seminal work of Thurston, as many of ...
5
votes
4answers
496 views

Combining rotations for orbital and rotational motion of a planet

If I have a planet orbiting the Sun (assuming circular orbit) at angular velocity $\Omega$ and rotating about its axis at $\omega$. I also have a normal to the surface of the planet $\vec{n}_{\rm surf}...

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