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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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47 views

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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156 views

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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1answer
84 views

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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1answer
170 views

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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319 views

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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112 views

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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265 views

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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206 views

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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181 views

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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123 views

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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106 views

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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230 views

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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1answer
221 views

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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255 views

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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389 views

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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105 views

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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170 views

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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879 views

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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1answer
181 views

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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148 views

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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549 views

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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1answer
242 views

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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1answer
501 views

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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386 views

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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154 views

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 ...