nLab sigma-model

Idea

A $\sigma$-model is a particular kind of physical theory of certain fields. The basic data describing a specific $\sigma$-model is some kind of “space” $X$, in a category of “spaces” which includes smooth manifolds. We call $X$ the target space, and we define the “configuration space of fields” $Conf_\Sigma$ over a manifold $\Sigma$ to be the mapping space/mapping stack $Map(\Sigma, X)$. That is, a “configuration of fields” over a manifold $\Sigma$ is like an $X$-valued function on $\Sigma$.

We assign a dimension $n \in \mathbb{N}$ to our $\sigma$-model, take $dim \Sigma \leq n$ and assume that target space $X$ is equipped with a “circle n-bundle with connection”.

For $n = 1$ this is an ordinary circle bundle with connection and models a configuration of the electromagnetic field on $X$. To distinguish this “field” on $X$ from the fields on $\Sigma$ we speak of a background gauge field. (This remains fixed background data unless and until we pass to second quantization.) A field configuration $\Sigma \to X$ on $\Sigma$ models a trajectory of a charged particle subject to the forces exerted by this background field.

For $n = 2$, a circle $n$-bundle with connection is a circle 2-group principal 2-bundle or equivalently a bundle gerbe with connection. This models a “higher electromagnetic field”, called a Kalb-Ramond field. Now $\Sigma$ is taken to be 2-dimensional and a map $\Sigma \to X$ models the trajectory of a string on $X$, subject to forces exerted on it by this higher order field.

This pattern continues. In the next dimension a membrane with 3-dimensional worldvolume is charged under a circle 3-bundle with connection, for instance something called the supergravity C-field.

While one can speak of higher bundles in full generality and full analogy to ordinary principal bundles, it is useful to observe that any circle $n$-bundle is characterized by a classifying map $\alpha : X \to \mathbf{B}^n U(1)$ in our category of spaces, so we can just think about classifying maps instead. Here $U(1)$ is the circle group, and $\mathbf{B}^n$ denotes its $n$th delooping ; thus such a map is also a sort of cocycle in “smooth $n$th cohomology of $X$ with coefficients in $U(1)$”. The additional data of a connection refines this to a cocycle in the differential cohomology of $X$.

Such connection data $\nabla$ on a circle $n$-bundle defines – and is defined by – a notion of higher parallel transport over $n$-dimensional trajectories: for closed $n$-dimensional $\Sigma$ it defines a map $hol : (\gamma : \Sigma \to X) \mapsto \exp(i \int_\Sigma \gamma^*\nabla) \in U(1)$ that sends trajectories to elements in $U(1)$: the holonomy of $\nabla$ over $\Sigma$, given by integration of local data over $\Sigma$. The local data being integrated is called the Lagrangian of the $\sigma$-model. Its integral is called the action functional.

In the quantum $\sigma$-model one considers in turn the integral of the action functional over all of configuration space: the “path integral”. In the classical $\sigma$-model one considers only the critical locus of the action functional (where the rough idea is that the path integral to some approximation localizes around the critical locus). Points in this critical locus are said to be configurations that satisfy the “Euler-Lagrange equations of motion”. These are supposed to be the physically realized trajectories among all of them, in the classical approximation.

Finally, just like an ordinary circle group-principal bundle has an associated vector bundle once we fix a representation of $U(1)$ to be the fibers, any “circle n-bundle” has an associated “n-vector bundle” once we fix a “∞-representation” $\rho : \mathbf{B}^n U(1) \to n Vect$ on “n-vector spaces”. Just as for the ordinary $U(1)$, here we usually pick the canonical 1-dimensional such “representation”. Finally, we define bundles $V_\Sigma : Conf_\Sigma \to \mathcal{C}$ of “internal states” by transgression of these associated bundles.

The passage from principal ∞-bundles to associated ∞-bundles is necessary for the description of the quantum $\sigma$-model: it assigns in positive codimension spaces of sections of these associated bundles. For a 1-categorical description of the resulting QFT ordinary vector bundles (assigned in codimension 1) would suffice, but the $\sigma$-model should determine much more: an extended quantum field theory. This requires sections of higher vector bundles. For instance for $n = 2$ some boundary conditions of the $\sigma$-model are given by sections of the background 2-vector bundle: these are the twisted vector bundles known as the Chan-Paton bundles on the boundary-D-branes of the string. (…)

We now try to fill this with life by spelling out some standard examples. Further below we look at precise formalizations of the situation.

Terminology and history

In physics one tends to speak of a model if one specifies a particular quantum field theory for describing a particular situation, for instance by specifying a Lagrangian or local action functional on some configuration space. This is traditionally not meant in the mathematical sense of model of some theory. But in light of progress of mathematically formalizing quantum field theory (see FQFT and AQFT), it can with hindsight be interpreted in this way:

a $\sigma$-model is supposed to be a type of model for the theory called quantum field theory. This sounds like a tautology, but much effort in mathematical physics is devoted to eventually making this a precise statement. In special cases and toy examples this has been achieved, but for the examples that seem to be directly relevant for the phenomenological description of the observed world, lots of clues still seem to be missing.

As to the “$\sigma$” in “$\sigma$-model”: back in the 1960s people were interested in a hypothetical particle called the $\sigma$-particle. Murray Gell-Mann came up with a theory of them. It was called ‘the $\sigma$-model’. It was an old-fashioned field theory where the field took values in a vector space. Then someone came up with a modified version of the σ-model where the field took values in some other manifold and this was called ‘the nonlinear σ-model’.

While the parameter space (the domain space of the fields) of the original $\sigma$-models was supposed to be our spacetime and the target space was some abstract space, with the advent of string theory the nonlinear $\sigma$-models gained importance as quantum field theories whose target space is spacetime $X$ and whose parameter space is some low dimensional space, usually denoted $\Sigma$. A field configuration $\Sigma \to X$ is then interpreted as being the trajectory of an extended fundamental particle – a fundamental brane – in $X$, and the $\sigma$-model describes the quantum mechanics of that brane propagating in $X$.

In particular the quantum mechanics of a relativistic particle propagating on $X$ is described by a $\sigma$-model on the real line $\Sigma = \mathbb{R}$ – the worldline of the particle.

In string theory one considers 2-dimensional $\Sigma$ and thinks of maps $\Sigma \to X$ as being the worldsheets of the trajectory of a string propagating in spacetime.

In the context of 11-dimensional supergravity there is a $\sigma$-model with 3-dimensional $\Sigma$, describing the propagation of a membrane in spacetime.

Exposition of classical sigma-models

We survey, starting from the very basics, classical field theory aspects of $\sigma$-models that describe dynamics of particles, strings and branes on geometric target spaces.

The content of this section is at

• sigma-model – exposition of classical sigma-models

Exposition of higher gauge theories as $\sigma$-models

We discuss how gauge theories and their higher analogs are naturally regarded as $\sigma$-models.

The content of this section is at

• sigma-model – exposition of higher gauge theories as sigma-models.

Exposition of quantum $\sigma$-models

Above we have discussed some standard classical sigma-models and higher gauge theories as sigma-models, also mostly classically. Here we talk about the quantization of these models (or some of them) to QFTs: quantum $\sigma$-models .

The content of this section is at

• sigma-model – exposition of quantum sigma-models.

See there for discussion of string topology, Gromov-Witten theory, Chern-Simons theory.

Exposition of a general abstract formulation

We give a leisurely exposition of a general abstract formulation $\sigma$-models, aimed at readers with a background in category theory but trying to assume no other prerequisites.

What is called an $n$-dimensional $\sigma$-model is first of all an instance of an $n$-dimensional quantum field theory (to be explained). The distinctive feature of those quantum field theories that are $\sigma$-models is that

1. these arise from a simpler kind of field theory – called a classical field theory – by a process called quantization

2. moreover, this simpler kind of field theory encoded by geometric data in a nice way: it describes physical configuration spaces that are mapping spaces into a geometric space equipped with some differential geometric structure.

We give expositions of these items step-by-step:

1. Quantum field theory

2. Classical field theory

3. Quantization

4. Classical sigma-models

5. Quantum sigma-models

We draw from (FHLT, section 3).

The content of this section is at

• sigma-model – exposition of a general abstract formulation.

Exposition of second quantization of $\sigma$-models

We discuss second quantization in the context of $\sigma$-models.

The content of this section is at

• sigma-model – exposition of second quantization of sigma-models

Examples

Non-topological $\sigma$-models

• The canonical textbook example of a quantum mechanical system is of this form for $n=1$: A line bundle with connection $E \to X$ on a (pseudo-)Riemannian manifold $X$ induces the 1-dimensional quantum field theory which is the quantum mechanics of a point particle which propagates on $X$, subject to the forces of gravity (given by the pseudo-Riemannian metric on $X$) and electromagnetism (given by the line bundle with connection). The Hamilton operator encoding this quantum dynamics in this case is the Laplace-operator of $T X$ twisted by the line bundle $E$.

  For $X$ a spacetime this is called the relativistic particle.

  For $\Sigma$ or $X$ a supermanifold this is the superparticle.

• Generalizing in the above example the line bundle $E$ by an abelian bundle gerbe with a connection yields a background for a 2-dimensional $\sigma$-model which mayb be thought of as describing the propgation of a string. The best-studied version of this is the case where $X = G$ is a Lie group, in which case this $\sigma$-model is known as the Wess–Zumino–Witten model.

Topological $\sigma$-models

• Dijkgraaf-Witten theory is the (2+1)-dimensional $\sigma$-model induced from an abelian 2-gerbe on $\mathbf{B} G$, for $G$ a finite group.

• Chern-Simons theory is supposed to be analogously the $\sigma$-model induced from an abelian 2-gerbe with connection on $\mathbf{B}G$, but now for $G$ a Lie group.

• topological membrane

• the Poisson sigma-model is a model whose target is a Poisson Lie algebroid.

• in AKSZ theory this is generalized to a large class of sigma models with symplectic Lie n-algebroids as target.

• Rozansky–Witten theory is essentially the $\sigma$-model for $X$ a smooth projective variety.

• generally ∞-Chern-Simons theory is a $\sigma$-model with a smooth ∞-groupoid of ∞-connections.

Related concepts

• sigma model

  • worldvolume

  • target space

  • background gauge field

  • genus

• gauged linear sigma-model

• supersymmetric sigma-model

  • 2d (2,0)-supersymmetric sigma-model

• CP^N sigma-model

• brane

  • particle, superparticle

  • string

  • membrane

• homological mirror symmetry

  • A-model, B-model

  • Landau-Ginzburg model

References

General

The concept of sigma-models originates with the introduction of the σ-meson to chiral perturbation theory in

• Murray Gell-Mann, M. Lévy, The axial vector current in beta decay, Il Nuovo Cimento (1955-1965) volume 16, pages 705–726 (1960) (doi:10.1007/BF02859738)

A standard reference on 2-dimensional string $\sigma$-models is

• Pierre Deligne, Dan Freed, Classical field theory , chapter 5, page 211

  and

  Krzysztof Gawedzki, Lectures on conformal field theory , part 3, lecture 3

  in

  Pierre Deligne, Pavel Etingof, Dan Freed, L. Jeffrey,

David Kazhdan, John Morgan, D.R. Morrison and Edward Witten, eds. Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

General exposition of $n$-dimensional $\sigma$-models, including the fact that these are non-renormalizable for $n\geq 3$:

• Christopher Hull, Lectures on Non-Linear Sigma-Models and Strings, in Super field theories, eds. H. C. Lee, V. Elias, G. Kunstatter, R. B. Mann, K. S. Viswanathan, NATO Science Series 160, Springer (1987) 77–168 [doi:10.1007/978-1-4613-0913-0_4]

Further (and original) discussion of string sigma-models and their Ricci flow renormalization group flow is for instance in

• Daniel Friedan. Nonlinear models in $2+\epsilon$ dimensions. Annals of Physics, 163(2):318–419, 1980.

• Daniel Friedan. Nonlinear models in $2+\epsilon$ dimensions. Physical Review Letters, 45(13):1057, 1980.

• Arkady Tseytlin, Sigma model approach to string theory, Int. J. Mod. Phys. A 4, 1257 (1989).

• C. Callan, L. Thorlacius, Sigma models and string theory, Particles, Strings and Supernovae, Volumes I and II. Proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics, held June 6 - July 2, 1988, at Brown University, Providence, Rhode Island. Edited by A. Jevicki and C.-I. Tan. Published by World Scientific, New York, 1989, p.795 (pdf)

• Arkady Tseytlin, Sigma model approach to string theory effective actions with tachyons, J. Math.Phys.42:2854-2871 (2001) (arXiv:hep-th/0011033)

• Arkady Tseytlin, On sigma model RG flow, “central charge” action and Perelman’s entropy, Phys.Rev.D75:064024,2007 (arXiv:hep-th/0612296)

First indications on how to formalize $\sigma$-models in a higher categorical context were given in

• Dan Freed, Higher algebraic structures and quantization, (arXiv)

A grand picture developing this approach further is sketched in

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups, (arXiv)

A discussion of 2- or (2+1)-dimensional $\Sigma$-models whose target is an derived stack/infinity-stack is in

• David Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld centers in derived geometry, (arXiv) .

More discussion of the latter is at geometric infinity-function theory.

A discussion of $\sigma$-models of higher gauge theory type is at

• Urs Schreiber, infinity-Chern-Simons theory .

Concrete applications of $\sigma$-models with target stacks (typically smooth ones, hence smooth groupoids, and typically smooth gerbes among these) in string theory and supergravity are discussed in

• Tony Pantev, Eric Sharpe, String compactifications on Calabi-Yau stacks, Nucl.Phys. B733 (2006) 233-296, (arXiv:hep-th/0502044)

• Tony Pantev, Eric Sharpe, Gauged linear sigma-models for gerbes (and other toric stacks), Adv.Theor.Math.Phys. 10 (2006) 77-121 (arXiv:hep-th/0502053)

• S. Hellerman, Eric Sharpe, Sums over topological sectors and quantization of Fayet-Iliopoulos parameters, (arXiv:1012.5999)

with review in

• Eric Sharpe, Sigma models on gerbes, talk at Higher Structures in M-Theory, Durham, August 2018 (pdf slides, video recording)

Discussion of geometric Langlands duality in terms of 2d sigma-models on stacks (moduli stacks of Higgs bundles over a given algebraic curve) is in

• Edward Witten, section 6 of Mirror Symmetry, Hitchin’s Equations, And Langlands Duality, in Oscar Garcia-Prada, Jean Pierre Bourguignon, and Simon Salamon, The many faces of geoemtry: A tribute to Nigel Hitchin, Oxford Scholarship Online (arXiv:0802.0999)

See also:

• Rafał R. Suszek. Towards higher super-$\sigma$-model categories. (2023). (arXiv:2311.01768).

Exceptional-geometric brane sigma-models

On U-duality-covariant exceptional geometric super $p$-brane sigma-models (worldvolume exceptional field theory):

• Yuho Sakatani, Shozo Uehara, Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry, Phys. Rev. Lett. 117 191601 (2016) [doi:10.1103/PhysRevLett.117.191601, arXiv:1607.04265, talk slides]

• Yuho Sakatani, Shozo Uehara, Exceptional M-brane sigma models and $\eta$-symbols, Progress of Theoretical and Experimental Physics 2018 3 (2018) 033B05, [doi:10.1093/ptep/pty021, arXiv:1712.10316]

• David Osten: Currents, charges and algebras in exceptional generalised geometry, J. High Energ. Phys. 2021 70 (2021) [doi:10.1007/JHEP06(2021)070, arXiv:2103.03267]

• Machiko Hatsuda, Ondřej Hulík, William D. Linch, Warren D. Siegel, Di Wang, Yu-Ping Wang: $\mathcal{A}$-theory: A brane world-volume theory with manifest U-duality, J. High Energ. Phys. 2023 87 (2023) [doi:10.1007/JHEP10(2023)087, arXiv:2307.04934]

• David Osten: On exceptional QP-manifolds, J. High Energ. Phys. 2024 28 (2024) [doi:10.1007/JHEP01(2024)028, arXiv:2306.11093]

• David Osten: On the universal exceptional structure of world-volume theories in string and M-theory, Physics Letters B 855 (2024) 138814 [doi:10.1016/j.physletb.2024.138814, arXiv:2402.10269]

• Machiko Hatsuda, Ondřej Hulík, William D. Linch, Warren D. Siegel, Di Wang, Yu-Ping Wang: Strings and membranes from $\mathcal{A}-theory five brane [arXiv:2410.11197]