# 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 $\sigma$ ...
1 vote
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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-... 1 vote
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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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### 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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The Supersymmetric Index is known to have the following path integral representation (Mirror Symmetry, equation (10.126)):$$Tr[(-1)^Fexp(-\beta H)]=\int_{PBCs}DxD\bar{\psi}D{\psi}\exp(-S_E)\tag{10.126}... 6 votes 2 answers 372 views ### Definition of Sigma Model Path Integral All references I have consulted have been extremely sketchy about this point. The (2 dimensional) nonlinear sigma model in some Riemannian manifold M with metric g_{\mu\nu} has action$$S = \frac{... 869 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 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 ...
1 vote
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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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### 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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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 ...