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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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
$${\...
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Open problems in sigma models
Im a mathematician which also learn physics, Ive read several papers on sigma models, and it was quite interesting . So my question is where can I read about open problems of sigma models?
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Non-linear sigma model quantization
Given the following lagrangian for the non-linear sigma model:
$$
\mathcal{L}=\frac{1}{2}\sum_{a,b}\partial_\mu\phi^a\partial^\mu\phi^b f_{ab}(\phi)
$$
where $f_{ab}(\phi)$ is a matrix function.
My ...
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Why $C^{\infty}(\Sigma, \mathbb{R}^D)$ instead of $\text{Emb}(\Sigma, \mathbb{R}^D)$ in string theory $\sigma$-model?
In most String Theory textbooks, e.g. Polchinski, Blumenhagen et. al., GSW, Becker & Schwarz, Zwiebach, the dynamics of the string is firstly motivated geometrically by the Nambu-Goto action $S_{...
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Path Integral Formalism and Two-Point Function
I'm given an action,
$$A[\vec{S}] = \frac{\Theta}{2} \int_{-\infty}^{\infty} dt \left(\frac{d\vec{S}}{dt}\right)^2, \tag{1}$$
with $$\vec{S}^2 = 1.\tag{2}$$ I'm asked to calculate the two point ...
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Decoupling in the Linear Sigma Model
In Schwartz's 'QFT and The Standard Model' the Lagrangian for the linear sigma model is given by:
$$L=\frac{1}{2}(\partial_\mu\sigma)^2+(\sqrt\frac{2m^2}{\lambda}+\frac{1}{\sqrt 2}\sigma(x))^2\frac{1}{...
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Sigma models as topological quantum field theories
I'm wondering how sigma models are supposed to define TQFTs. Suppose I want to consider a 2D TQFT with target $X$ (see page 15 of https://www.ams.org/bookstore/pspdf/ulect-72-intro.pdf)*. According ...
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Chiral Lagrangian Field
In the chiral model $SU(N)_R × SU(N)_L$ with gauged Left-handed $SU(N)$, we take as the field the $SU(N)_L$-valued $\Sigma (x)$, defined as
$$\Sigma(x) = \exp\big( \frac{2i}{v} \chi^a(x)T^a\big).$$
...
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One-loop Effective potential, Higgs VEV and renormalization condition
On Peskin & Schroeder's QFT, chapter 11.4, the book discusses the computation of the effective action of linear sigma model.
I am troubled for the relation between renormalization condition and ...
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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 ...
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Peskin and Schroeder, Linear sigma model, renormalized perturbation theory
On Peskin & Schroeder's QFT pages 353-355, the book uses the Linear sigma model to illustrate the renormalization and symmetry.
We can write the Lagrangian of Linear sigma model with
$$
\begin{...
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Reference request for QFT $SO(3)$ non-linear sigma model
I was wondering if anyone has a reference that could help me understand quantum field theories that have a nonlinear configuration space. For example, from classical mechanics if we have a three-...
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Extra term in $2+\epsilon$ expansion of sigma model
I'm working through David Tong's notes on Statistical Field Theory, in particular the $2+\epsilon$ expansion of the sigma model with free energy
$$F[\vec{n}]=\int d^dx \frac{1}{2e^2}\nabla\vec{n}\cdot\...
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Is alpha prime ($\alpha'$) exactness the same as exactness in conformal perturbation theory?
I am very confused by these two different notions of exactness. I will avoid equations because I think my problem is conceptual.
Take the world-sheet CFT and deform it with a marginal operator. In ...
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Metric structure on a group
I have a question about metric structure on a group manifold $G$. Imagine we have a sigma model, i.e. a map $g: \Sigma\rightarrow G$ from some 2D source to our group. One can define the left/right ...
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Coefficient of effective chiral Lagrangian of $\pi\pi$ scattering
I have been suffering from the coefficient in the expansion of chiral lagrangian.
Consider $$L=\frac{F^{2}}{4} \rm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U),$$
where $$U=\exp(i\frac{\phi}{F}).$$ ...
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Is the Nambu-Goto action defined only for the torus?
For simplicity, I will use the Nambu-Goto action, but the following question would probably be the same for the Polyakov action.
According to David Tong's lecture notes on string theory, the Nambu-...
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Propagation of a wavefunction on a Riemannian sigma model
I have a question about Riemannian sigma model, in particular how wavefunctions propagate. Here the Riemannian sigma model refers to the one introduced in 10.4.1 and 10.4.2 of the book $\ulcorner$ K. ...
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Feynman diagram for two-dimensional nonlinear sigma model with antisymmetric coupling
I am studying the review by C. Calland and L. Thorlacius (link: https://www.damtp.cam.ac.uk/user/tong/string/sigma.pdf) on sigma models in string theory. In particular I am trying to evaluate the ...
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Component field expansion of $(1,1)$ supersymmetric Polyakov action
I am working on the renormalizibility of two-dimensional nonlinear sigma models in string theory and particularly the $(1,1)$ supersymmetric extension of it. The following is based on the review by C. ...
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Background field expansion of supersymmetric string action
For a reasearch project I am studying the paper by L. Alvarez-Gaumé, D. Freedman and S. Mukhi called "The Background Field Method and the UV Structure of the Supersymmetric Nonlinear $\sigma$ ...
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Help with variation of the 3-dimensional $\sigma$-model action
Consider the following action
$$S=\int\mathrm{d}^3x\sqrt{h}\left[R^{(3)}-\frac{1}{4}\mathrm{Tr}\left(\chi^{-1}\chi_{,i}\chi^{-1}\chi^{,i}\right)\right]$$
where $h$ is the determinant of the 3-...
user161157
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Geometry of Landau-Ginzburg models?
A non-linear sigma model, whose action is generically given by
$$S_\text{NLSM}=\int d^dx\, g_{ij}\partial_\mu\phi^i\partial^\mu\phi^j$$
has a natural geometric interpretation that $\phi$ is a map $\...
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$C$, $P$, $T$ symmetry of $O(3)$ non-linear sigma model
Consider the $O(3)$ nonlinear sigma model with topological theta term in 1+1 D: $$\mathcal{L}=|d\textbf{n}|^{2}+\frac{i\theta}{8\pi}\textbf{n}\cdot(d\textbf{n}\times d\textbf{n}).$$
The time reversal ...
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Difference between moduli spaces of supersymmetric vs non-supersymmetric theories?
I have a basic conceptual question regarding the difference between moduli spaces in supersymmetric vs non-supersymmetric theories.
In usual non-supersymmetric theories, the existence of flat ...
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Gauge orbit orthogonality in a gauged linear sigma model on $\mathbb{C}P^{N}$
I am back with another question from the book Mirror Symmetry, this time from Section $15.1.1$.
Consider the gauged linear sigma model for $N$ complex scalar fields and the Lagrangian:
$$
L=-\sum_{i=1}...
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Computing the spectrum of a lagrangian in field theory
I have the following lagrangian density:
$$L = \bar{\psi}i \gamma^\mu \partial_\mu \psi
- g\bar{\psi}(\sigma + i\gamma^5\pi)\psi +
\frac{1}{2}(\partial_\mu \sigma)^2+
\frac{1}{2}(\partial_\mu \pi)^2
-...
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RG flow of 4d Nonlinear Sigma model with $SU(n)$ target space
Let's consider the 4d Nonlinear Sigma model with $SU(n)$ target space, without a topological term. The Lagrangian is
$$\frac{f^2}{16}\mathrm{Tr}(\partial_{\mu}U^{-1} \partial^{\mu}U)$$
where $U$ is a $...
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One-loop approximation in chiral sigma models
Consider a principal chiral model $$S[g] = \frac{1}{4\pi\lambda^2}\int(|g^{-1}dg|^2) = \frac{1}{4\pi\lambda^2}\int(g^{-1}dg\wedge\star g^{-1}dg) ,\tag{2.1}$$ where $g:\Sigma \rightarrow G $ is the so ...
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Relation between *sigma algebra* and *sigma models*
Is there connection between sigma models in physics and sigma algebra in the probability theory?
Background: I have never had to study the former, but I am somewhat familiar with the sigma-algebra in ...
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Degrees of freedom in a Kahlerian NLSM
Lagrangian for $d=1$ $\mathcal{N}=4$ SUSY model on a $n$-complex dimensional Kahlerian target space is given as (see p.213, eqn. (10.251) in the Mirror Symmetry book (pdf))
$$\begin{equation}
L= ...
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In string theory path integral, what happens if I fix worldsheet metric?
In string theory worldsheet path integral, integral is done over all possible topologies, metric and coordinates.
And I was wondering if there is something in string theory similar to quantum field ...
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Renormalization of linear sigma model fixing the Vacuum Expectation Value (VEV)
In the linear sigma model
$$ \mathcal L = (\partial_{\mu} \phi^i)^2 + \frac 12 \mu_0 (\phi^i)^2 + \frac{\lambda_0}{4} ((\phi^i)^2)^2 ,$$
the symmetry is broken around the vacuum expectation value (VEV)...
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Slow and fast variables in Non-linear Sigma model
I am following Peskin Section 13.3, where they solve the nonlinear sigma model using Polyakov method. This system has Lagrangian
\begin{equation}
\mathcal{L}=\frac{1}{2g^2}|\partial_\mu\vec{n}|^2,\tag{...
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SUSY sigma model in QM, bosonic sector?
The bosonic sigma model in ordinary QM (i.e. a 'free' particle trapped on a curved manifold $\mathcal{M}$), has a Hamiltonian which is just the negative Laplacian on $\mathcal{M}$.
For any $\mathcal{...
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Can we tell the difference between a scalar field and a non-linear sigma model?
Suppose a $U(1)$ non-linear sigma model field $\Sigma(x)$ take values on a circle. But if this circle is very large and the value don't vary so much, shouldn't this be almost identical to a scalar ...
user84158
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"Bad" behavior of propagator in $O(N)$ model
In Polyakov's book about gauge fields & strings, in chapter devoted to non-linear sigma model he emphasizes problem with large $N$ expansion of this model. Lagrangian of 2D model is
$$\frac{1}{2g^...
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Non-linear sigma-models on curved worldsheet
I am studying nonlinear sigma-models and topological twists using E.Witten's article "Mirror manifolds and topological field theory" (https://arxiv.org/abs/hep-th/9112056), as well as "Mirror symmetry"...
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Holographic duals of (super)gravity sigma models
Consider a (super)gravity theory on asymptotically AdS spacetime $N$ with fixed conformal boundary $\partial N$ coupled to scalars $\phi_i$ taking values in a manifold $M$, possibly in addition to ...
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Quantum corrections to metric on non-linear sigma model target space
I am trying to make sense of what physicists mean when they talk of quantum corrections to the metric on the target spaces of nonlinear sigma models, for example [GHL99].
First some quick notation. ...
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Can we ignore the scalar field (dilaton) term in the Polyakov sigma-model action when deriving the classical equations of motion?
I have the full Polyakov sigma model action:
\begin{equation}
\begin{split}
&S=S_P + S_B + S_\Phi = \\
&- {1 \over 4 \pi \alpha'} \Big[ \int_\Sigma d^2\sigma \sqrt{-g} g^{ab} \partial_a X^\mu ...
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Momentum Operator on a Riemannian Manifold
Consider a non-linear sigma model on a Riemannian manifold with metric $g_{ij}$ with the action
$$S= \frac{1}{2} \int dt g_{ij}(X) \frac{dX^i}{dt} \frac{dX^j}{dt}.$$
The momentum operator is $$P_i= \...
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Simplest model in field theory which leads to a pseudo-Goldstone boson
What can be a simple (if not simplest) continuum field theory model that gives rise to a pseudo Goldstone boson (doesn't matter if it is a toy model)? For example, I would be very happy if one can ...
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QCD flavor gauging?
I ran across this old post:
What the heck is the sigma (f0) 600?
The question author is explaining his understanding of the spectrum of QCD in terms of various interesting things like chiral symmetry ...
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Do strings propagate through spacetime or do they make spacetime?
In the beginning of string theory textbooks, strings are said to live in a background "target" spacetime. They then propagate through this spacetime. Strings also have a spin 2 ("...
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Meson scattering amplitude in the linear sigma model
I am trying to calculate scattering amplitudes with linear sigma model Lagrangian, given as
$$\mathcal L= \frac{1}{2}(\partial_{\mu}\sigma)^2+\frac{1}{2}(\partial_{\mu}\vec{\pi})^2-\mathcal U(\sigma,\...
