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.
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SUSY QM as point-particle limit of Superstrings
There is a pretty clear resemblance between the Lagrangians for SUSY QM (1-dim susy sigma model) and various superstring theories (2-dim susy sigma models). Again intuitively, one should expect the ...
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How the vacuum expectation value of $\sigma$ is related to the tadpole diagram?
In Peskin & Schroeder's book, they introduce the linear sigma model with a Lagrangian (chapter 11.1,p349) $$ L = \frac{1}{2} (\partial_\mu \phi^i)^2 + \frac{1}{2} \mu^2 (\phi^i)^2 - \frac{\lambda}{...
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$A$-twist and the stress-energy tensor
This is a horribly long question about the very basic thing: the definition of the $A$-twist (of, say, a SUSY-sigma model). There are two parts. In the first one, I am just asking what this definition ...
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On the "Charge fractionalization"
In the "Mirror symmetry" book by Hori, Vafa, et al. (which is available on the web), the observation is made (starting at 15.5.2, and directly up to 15.5.3) which can be summarized as ...
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Help with doing a Feynman 1-loop integral related to $\langle T_{++}\rangle$ in string theory
The goal is to compute the 1-loop integral, which is given equal to: $$\int{\frac{d^2l}{2\pi}\frac{l_{+}(l_++q_+)}{l^2(l+q)^2}}=-\frac{1}{4}\frac{q_+}{q_{-}}.\tag{3.31}$$ The above integral represents ...
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Parametrization of Spontaneous Symmetry Breaking
I am currently studying the spontaneous symmetry breaking part in Schwartz's Quantum Field Theory and the Standard Model. I am solving the Problem 28.1. In the main text, Schwartz presents the example ...
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$O(N)$ vector model and large $N$
I'm familiar with QFT at the level of QFT by Ryder. I have also consulted many books including QFT by Fradkin which includes $O(N)$ vector model, particularly in $\phi^4$, and nonlinear sigma model, ...
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Sigma-Omega model on curved space-time
I'm trying to get the equations of motion for the scalar meson $\sigma$, vector meson field $\omega_{\mu}$ and finally for the nucleons $\Psi=(\Psi_n,\Psi_p)^{T}$ in the sigma omega model on a curved ...
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Are the one-loop beta functions in bosonic string theory written in terms of bare or renormalized background fields?
Given a bosonic string theory defined by the action $$\tag1 S = \frac{1}{4\pi \alpha'}\int_\Sigma \! \mathrm{d}^2 \sigma \, \sqrt{|g|} \, \left[ G_{\mu\nu} \partial_\alpha X^\mu \partial_\beta X^\nu ...
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Quantization of string via topological twist
Polyakov action of a bosonic string propagating in Minkowskian spacetime is: $$S[\gamma, X] = \frac{T}{2}\int \mathrm{d}^{2}\sigma{\sqrt{-\gamma}}\gamma^{ab}\partial _{a}X^{\mu}(\sigma)\partial_{b}X^{\...
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Background field Method in non-linear $\sigma$-model: Covariantized Taylor series of geodesic between fields
In this paper, the authors try to develop an expansion of the non-linear $\sigma$-model action with covariant terms called the Background field method. We have a "fixed" background field $\...
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Cancelling one-loop divergences in non-linear sigma model expansion term
In the appendix A of this paper by Braaten et al., the authors try to compute the divergences of two integrals that come from an expansion of an action $I$ in $\langle e^{iI} \rangle$, via dimensional ...
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What does the WZ term in a WZW action means for string theory on group manifolds?
Let $G$ be a semi-simple Lie group. By Cartan's criterion its Killing form $B(X,Y)$ on $\frak g$ is non-degenerate. We can use it to define an inner product on the whole group by left translation $${\...
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Non-linear sigma model quantization
Given the following lagrangian for the non-linear sigma model: $$ \mathcal{L}=\frac{1}{2}\sum_{a,b}\partial_\mu\phi^a\partial^\mu\phi^b f_{ab}(\phi) $$ where $f_{ab}(\phi)$ is a matrix function. My ...
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Why $C^{\infty}(\Sigma, \mathbb{R}^D)$ instead of $\text{Emb}(\Sigma, \mathbb{R}^D)$ in string theory $\sigma$-model?
In most String Theory textbooks, e.g. Polchinski, Blumenhagen et. al., GSW, Becker & Schwarz, Zwiebach, the dynamics of the string is firstly motivated geometrically by the Nambu-Goto action $S_{...
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