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Questions tagged [sigma-models]

A σ-model, generically, is a spinless quantum field theory with an appropriate group symmetry structure. Normally, it serves as an effective theory of pseudoscalar mesons rising out of chiral symmetry breaking in QCD, and a scalar σ whose v.e.v. controls PCAC; this is the linear model. The nonlinear σ model has this σ field frozen to its v.e.v. and thus absent from the spectrum.

A σ-model, generically, is a spinless quantum field theory with an appropriate group symmetry structure. Normally, it serves as an effective theory of pseudoscalar mesons rising out of chiral symmetry breaking in QCD, and a scalar σ whose v.e.v. controls PCAC; this is the linear model. The nonlinear σ model has this σ field frozen to its v.e.v. and thus absent from the spectrum.

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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 with the following question: My aim is to relate…
user5831
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$\operatorname{O}(N)$ sigma model at large $N$

I would like to better understand the main principles of large-$N$ expansion in quantum field theory. To this end, I decided to consider a simple toy model with lagrangian (from Wikipedia) $ \mathcal{L} = \frac{1}{2}(\partial_{\mu}…
user43283
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Sigma Models on Riemann Surfaces

I'm interested in knowing whether sigma models with an $n$-sheeted Riemann surface as the target space have been considered in the literature. To be explicit, these would have the action \begin{align*} S=\frac{1}{2}\int d^2x\, \left(\partial_a…
Matthew
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Effect of linear terms on a QFT

I was told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \begin{equation} {\cal L} = \frac{1}{2} \partial _\mu \phi _i…
JeffDror
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Gauge fields in Polyakov's treatment of renormalization for nonlinear sigma model

I am deriving the results of renormalization for nonlinear sigma model using Polyakov approach. I am closely following chapter 2 of Polyakov's book--- ``Gauge fields and strings''. The action for the nonlinear sigma model (NLSM)…
maxr
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Are all pseudoscalars secretly Goldstone bosons?

A pseudoscalar Goldstone boson, $\pi(x)$, is protected by a shift symmetry: it shows up with a derivative in its interaction terms in a Lagrangian. As a pseudoscalar, we may also write it with the usual $i\gamma^5$ interaction. There are thus two…
Henry Deith
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Canonical quantization in supersymmetric quantum mechanics

Suppose you have a theory of maps $\phi: {\cal T} \to M$ with $M$ some Riemannian manifold, Lagrangian $$L~=~ \frac12 g_{ij}\dot\phi^i\dot\phi^j + \frac{i}{2}g_{ij}(\overline{\psi}^i D_t\psi^j-D_t\overline{\psi}^i\psi^j)…
jj_p
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Do strings propagate through spacetime or do they make spacetime?

In the beginning of string theory textbooks, strings are said to live in a background "target" spacetime. They then propagate through this spacetime. Strings also have a spin 2 ("graviton") mode, and can scatter off of each other. So in one sense,…
user1379857
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Boundary conditions on bosons and fermions in computing Partition function/Index Path integrals

Consider the path integral computation for the partition function: $$Z=Tr\space [\exp(-\beta H)]=\int_{AP} D\bar{\psi}D\psi ~Dx~\exp(-S_E)\tag{10.125},$$ and that for the Index (Mirror Symmetry, (10.125) and (10.126)): $$Tr[(-1)^F\exp(-\beta…
Kong
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Impostor Higgs?

I recently came across this article, published in the respectable European Physical Journal A. (Apparently, there isn't any corresponding arXiv article for this, so I'm sorry if everyone isn't able to access the article.) Here's the possibility that…
299792458
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sigma model on $S^1 \times S^3$

In arXiv:1207.3497 - 4D partition function on $S^1 \times S^3$ and 2D Yang-Mills with nonzero area, Yuji Tachikawa explains the partition function for an 4d $\mathcal{N}=2$ sigma model on $S^3 \times S^1$ with target $T^*G_{\mathbb{C}}=G \times…
john mangual
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"Bad" behavior of propagator in $O(N)$ model

In Polyakov's book about gauge fields & strings, in chapter devoted to non-linear sigma model he emphasizes problem with large $N$ expansion of this model. Lagrangian of 2D model is $$\frac{1}{2g^2}(\partial_{\mu}{\bf n})^2$$ with constraint ${\bf…
Artem Alexandrov
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Spontaneously broken linear sigma model in Peskin & Schroeder: where is the miracle?

P&S spend almost 12 pages discussing the renormalisability of the spontaneously broken linear sigma model and give a detailed calculation of the cancellation of divergences at one-loop level and call this a miracle. Now I think the only thing they…
Oбжорoв
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How to perform Wilsonian RG for the $O(N)$ nonlinear sigma model?

I am interested in deriving the $\beta$ function and anomalous dimension $\gamma$ of the $O(N)$ nonlinear $\sigma$ model, in particular defined by the action $S = \frac{1}{2g} \int d^2 x (\partial_{\mu}\vec{n}(x))^2$ with the constraint…
Abhi Sarma
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2D CFT from sigma models

$X$ is a closed manifold with a positive-definite metric $g$. $M_2$ is a 2D oriented closed manifold with a positive-definite metric $G$ and a compatible volume form $\omega$. We can then consider the following Euclidean path integral for a…
Leo
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