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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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-...
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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 ...
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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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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,\...
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Partition function computation on a Riemannian Manifold

This discussion comes from Chapter 10, Mirror Symmetry. I am given a Riemannian manifold $M$, and a classical (Euclidean) theory: $$(1)\quad S_E={\int}_0^\beta d\tau\space\Bigl(\frac{1}{2}g_{ij}{\dot{...
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Mistake in Peskin & Schroeder, Renormalization of Linear Sigma Model?

In section 11.4 of Peskin & Schroeder's "Introduction to Quantum Field Theory", the authors calculate the effective potential of the linear sigma model to one-loop order: $$\begin{align*} V_{\...
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Path integral representation for the Supersymmetric Index

The Supersymmetric Index is known to have the following path integral representation (Mirror Symmetry, equation (10.126)):$$Tr[(-1)^Fexp(-\beta H)]=\int_{PBCs}DxD\bar{\psi}D{\psi}\exp(-S_E)\tag{10.126}...
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Definition of Sigma Model Path Integral

All references I have consulted have been extremely sketchy about this point. The (2 dimensional) nonlinear sigma model in some Riemannian manifold $M$ with metric $g_{\mu\nu}$ has action $$S = \frac{...
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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....
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Reality properties of auxiliary fields after Wick rotation

I was reading the treatment of the large $N$ limit of the Non-Linear Sigma Model (NLSM) in Peskin & Schroeder, Sec. 13.3, and I noticed something strange in the evaluation of the path-integral by ...
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Why is the auxiliary field saddle point space independent in sigma models?

I am followint https://www.sciencedirect.com/science/article/pii/0370157384900218. Consider the sigma model in 2 dimensional Euclidean space with action $$ S[\sigma^a,\alpha]=\frac{1}{2}\int d^2x\bigg[...
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What are the quantum consequences of the usual constraint used in sigma models?

Consider the sigma model in 2 dimensions with $N$ sigma fields $$ \mathcal{L}=\frac{N}{2f}\partial_{\mu}\sigma^a\partial^{\mu}\sigma^a. $$ We want these fields to obey the constraints $$ \sigma^a\...
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Why is it said that a light sigma is important for low energy hadron physics?

I'm studying linear sigma model. I understand it's a useful effective model for more than one theory with the same symmetries but I don't understand why it is desirable for sigma to be light. I read ...
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Conjugate Momenta in a Supersymmetric Sigma Model

Consider the following theory comprising of $n$ bosons and $n$ fermions (along with their conjugates) on a Riemannian Manifold, with arc length parameter $t$ (section 10.4.1, Mirror Symmetry by Vafa ...
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Whats the difference between a linear and non-linear sigma model?

Wikipedia says In physics, a sigma model is a physical system that is described by a Lagrangian density of the form: $L(\phi_1,...,\phi_n)=g_{ij} d\phi_i \wedge d\phi_j$ With Einsteins summation ...
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Auxiliary field path integral in non-linear sigma models

I am trying to understand the functional integral over the auxiliary field in the $\mathcal{N}=(2,2)$ supersymmetric non-linear sigma model, or NLSM (reviewed in Chapter 13 of Mirror Symmetry http://...
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Gauged Linear Sigma Model (GLSM) with target space $E^8$ gauge group

I just read a few reviews (and also Witten's original paper http://arxiv.org/abs/hep-th/9301042) about the GLSM (Gauged Linear Sigma Model) in (2,2) and (0,2) formulations. I have several natural ...
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How is Wilson Flow related to renormalization?

I'm looking at a couple papers on Wilson Flow on the lattice and I'm not getting the connection to renormalization entirely. In Luscher's paper on Wilson flow, he explains that the field ...
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Linear term in NLSM action, expansion in terms of geodesic tangent vectors

The current question has arised as I started reading the the book by Ketov. We define the NLSM action as (for simplicity, I also assume $g_{ab}=g_{ba}$ and also the vanishing torsion): \begin{...
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3 votes
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Linear terms in Wilson approach to renormalization

In Wilson's approach to renormalization we break up a field $\phi_0$ which includes modes up to some cutoff $\Lambda$ into two parts, $\phi_0=\phi+\tilde\phi,$ where $\phi$ only has modes up to some ...
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7 votes
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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 ...
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Question about gauge transformations of a Non-linear sigma model

Consider a set of scalar fields $\phi^i$ ($i = 1, 2, \ldots, N$) which we now would to couple to a set of gauge vector fields $A_\mu^A$ where $A = 1, 2, \ldots \text{dim}(G)$ ($G$ is generically a non-...
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How does spacetime metric become dynamical (gravity quantized) in string theory?

I asked a similar question in What do we mean by worldsheet metric fluctuating in string theory, when we have a "target manifold"?, but the question had my misunderstanding that in Polyakov ...
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Supersymmetry of action with constraints in terms of unconstrained fields - Witten's topological sigma model

The following is a continuation of this question. The action of Witten's topological sigma model (defined on a worldsheet, $\Sigma$, with target space an almost complex manifold denoted $X$) takes the ...
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What do we mean by worldsheet metric fluctuating in string theory, when we have a "target manifold"?

This is an elementary question in string theory. In Polyakov action, it is often explained that worldsheet metric is independent dynamic variable and target manifold often (though it does not have to ...
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15 votes
2 answers
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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 ...
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