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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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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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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 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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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 ...