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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.

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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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Non-Linear Sigma Model

I read C. Mudry's book (chapter 3, p. 72, eq. 3.1a (the definition) and eq. 3.2a) and have several quaestions (1) He defines NL$\sigma$M through the following partition function: $$\mathcal{Z}(N,\...
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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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Doubts about renormalization of the coupling $t_0$ in the large N limit of the non-linear $O(N)$ sigma model

Another follow up to my previous question involving the $O(N)$ non-linear sigma model. I am trying to understand the way the coupling is renormalized in order to absorb the divergence in the equation ...
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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 ...
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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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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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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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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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Obtaining Euler-Lagrange equation from action with constraint - Witten's topological sigma model

The action of Witten's topological sigma model (defined on a worldsheet, $\Sigma$, with target space an almost complex manifold denoted $X$) takes the form $$ S=\int d^2\sigma\big(-\frac{1}{4}H^{\...
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Transformation of field operator under SU(2) x SU(2)

I am a little confused on how field operators transform under compound symmetry groups. The following text is copied from Michael Dine Supersymmetry and String Theory As an example, relevant both ...
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Dimensional reduction of Rozansky-Witten theory

Rozansky-Witten theory is a 3d topological sigma model which is used to study topological invariants of 3-manifolds. In what follows, $X$ will denote its target space. In a question posted here - ...
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Orbifolds for non-conformal theories

By now, a lot is known about orbifolds of 1+1-dimensional CFTs. Consider starting out with an arbitrary 'seed' CFT, $\mathcal{C}$, and take $N$ copies, $\mathcal{C}^{\otimes N}$ and then quotient by ...
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The decoupling limit of the linear sigma model

I'm reading Schwartz's Quantum field theory and the standard model. I'm stuck at page 566. He says that when we take the decoupling limit, that's keeping $F_{\pi}$ fixed while sending $m$ and $\lambda$...
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Parametrization pion fields

The exercise 5.8 of the book "Gauge theory of elementary particle physics - Problems and Solutions" by Ta-Pei Cheng and Ling-Fong Li has a point: Let us consider an isospin one pion field $\mathbf{\...
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Tadpoles in sigma models

In some QFTs, the tadpoles are not taken into account, since they vanish due to certain symmetries of the theory. Peskin and Schröder address this issue in QED around Equation (10.5) of their book; ...
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What is a nonlinear manifold?

The Wikipedia article defines a non-linear sigma model as model for a scalar field $\Sigma$ which takes on values in a nonlinear manifold called the target manifold $T$. What is the definition of a ...
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Geometric origin of the Kahler potential for SUSY $\sigma$-models

I was reading Cecotti's book on Supersymmetric Field Theories, and there is a statement that confuses me. It is proven that if one considers a $\sigma$-model $\phi^i:~\Sigma\to\mathcal{M}$ with ...
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Form of scalar potential in SUSY/SUGRA $\sigma$-models

In supersymmetry or supergravity, textbooks always show that one can define a Kähler potential $K=K(\phi^i,(\phi^i)^\ast)$ and an holomorphic superpotential $W=W(\phi_i)$ such that the scalar ...
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Particle moving on group manifold

I'm trying to learn about particles / strings moving on group manifolds and am looking for a reference which introduces this idea. For example, in this paper the Lagrangian for a particle moving on $...
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Covariant renormalization of nonlinear sigma model

On page 160-161 in Tong's notes on string, he calculates one-loop divergent diagram and finds that we need a counter term of $\frac{\alpha'}{3\epsilon}\mathcal{R}_{\mu\nu}\partial Y^{\mu}\partial Y^{\...
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Question about the linear sigma-model

Suppose the linear sigma-model lagrangian: $$ L = \bar{N}(i\gamma_{\mu}\partial^{\mu}-g_A \phi)N + |\partial_{\mu}\phi|^{2} - V(|\phi|) - c\sigma , $$ where $$ N = \begin{pmatrix} p \\ n\end{pmatrix}, ...
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Why T-duality only work when the background has isometries?

I have been studying from some textbooks and papers about the T-dality topic. In particular for the Buscher rules it seems that they claim that in order to have T-duality in certain direction we need ...
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T-duality between $E_8 \times E_8$ and $\text{Spin(}32)/\mathbb{Z}_2$ heterotic strings at the $\sigma$-model level

I would like to understand how T-duality between the heterotic $E_8 \times E_8$ (HE) and heterotic $\textrm{Spin}(32)/\mathbb{Z}_2$ (HO) theories works, at the level of the worldsheet $\sigma$-model. ...
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Are the pions really all that light?

I'm studying the sigma model where the pions are identified as the (pseudo) Nambu-Goldstone bosons of chiral symmetry breaking ("pseudo" from mild isospin symmetry violation). This argument usually ...
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Fields in the action of the Non-linear Sigma Model (WZW)

I am trying to understand the action of the nonlinear sigma model in the context of understanding WZW-models. On Wikipedia, its action is given as $S_k\left(\gamma\right)=-\frac{k}{8\pi}\int_{S^2}\...
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Reparametrization invariance in scalar QFT: What does it mean, exactly?

In the Cecotti's book "Supersymmetric Field Theories" he wrote " Physical quantities are independent of the fields we use to parametrize the configuration, that is, observables are invariant under ...
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Symmetry of the Polyakov action?

Let us look at the Polyakov action for a string moving in a spacetime with metric $g_{\mu \nu}(X)$:$$S_P = -{1\over{4\pi \alpha'}} \int d^2 \sigma \sqrt{-\gamma} \gamma^{ab} \partial_a X^\mu \...
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Background independent string theory

I don't really understand what one actually means when one says about doing string theory in a background independent way. Apparently B. Sathiapalan is the only person (as far as I know from ...
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Riemannian generalization/adaption of the Hubbard–Stratonovich transformation

I'd like to write the Hubbard–Stratonovich (HS) transformation of a scalar function on a Riemannian manifold. This transformation is quite simple in Euclidean space. One can consider it as a Fourier ...
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Projective superspace: why extra bosonic coordinates

I'm studying the projective superspace formalism for N = 4 supersymmetric $\sigma$-models in two dimensions. My question is: why do we need the extra bosonic coordinates for the manifest action? I ...
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Renormalization of Auxiliary Fields

I have the following non-linear sigma model (the base space $\mathcal{M}$ is Euclidean): $$ \mathcal{L}=\dfrac{1}{2\alpha}\int_{\mathcal{M}}\mathrm{d}^2\sigma\ \partial^2X^{\mu}\partial^2X_{\mu} $$ ...
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How can we see that a 4D N = 2 sigma model will yield a 3D N = 4 sigma model when compactified on a circle?

I have a question about sigma models in 3D. If we have $\mathcal{N}=2$ field theory on $\mathbb{R}^4$ and compactify it on $\mathbb{R}^3 \times S^1_R$ (in which $S^1_R$ is a circle of radius $R$) we ...