nLab higher gauge field

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Contents

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Differential cohomology

differential cohomology

Ingredients

Connections on bundles

Higher abelian differential cohomology

Higher nonabelian differential cohomology

Fiber integration

Application to gauge theory

Contents

Idea

An ordinary gauge field (such as the electromagnetic field or the fields that induce the nuclear force) is a field (in the sense of physics) which is locally represented by a differential 1-form (the “gauge potential”) and whose field strength is locally a differential 2-form. For instance, in the case of the electromagnetic field this differential 2-form is the Faraday tensor.

Roughly speaking, a higher gauge field is similarly a field which is locally represented by differential forms of higher degree.

An explanation as to why an ordinary gauge field has a gauge potential given locally by a differential 1-form AA is that the trajectory of a charged particle is a 1-dimensional curve in spacetime XX, its worldline, hence a smooth function γ:Σ 1→X\gamma \colon \Sigma_1 \to X, and the canonical way to produce an action functional on the mapping space of such curves is the integration of 1-forms over curves:

exp(iℏS gauge):γ↦Pexp(∫ Σ 1γ *A). \exp(\tfrac{i}{\hbar} S_{gauge}) \;\colon\; \gamma \mapsto P \exp\left( \int_{\Sigma_1} \gamma^\ast A \right) \,.

This is the parallel transport map or the holonomy map, if Σ 1\Sigma_1 is a closed manifold. The contribution to the Euler-Lagrange equation of the particle obtained from the variation of this action functional is the Lorentz force which is exerted by the background gauge field on the particle.

When one generalizes in this picture from 0-dimensional particles with 1-dimensional worldlines to pp-dimensional particles (often called “p-branes”) with (p+1)(p+1)-dimensional worldvolumes γ p+1:Σ p+1→X\gamma_{p+1} \colon \Sigma_{p+1} \to X, then one needs, locally, a differential (p+1)-form A p+1A_{p+1} on spacetime XX

exp(iℏS highergauge):γ↦Pexp(∫ Σ p+1γ p+1 *A p+1). \exp(\tfrac{i}{\hbar} S_{higher\,gauge}) \;\colon\; \gamma \mapsto P \exp\left( \int_{\Sigma_{p+1}} \gamma_{p+1}^\ast A_{p+1} \right) \,.

The field strength or flux of such a higher gauge field is, accordingly, locally the (p+2)(p+2)-form F p+2F_{p+2}.

The archetypical example of such a higher gauge field is the (hypothetical) Kalb-Ramond field or B-field (a precursor of the axion field under KK-compactification) to which the charged 1-brane, the “string”, couples. This is locally a differential 2-form B 2B_2, and the gauge-coupling term in the action functional for the string is accordingly, locally, of the form

exp(iℏS stringygauge):γ↦Pexp(∫ Σ 2γ 2 *B 2). \exp(\tfrac{i}{\hbar} S_{stringy\,gauge}) \;\colon\; \gamma \mapsto P \exp\left( \int_{\Sigma_{2}} \gamma_2^\ast B_2 \right) \,.

This continues: next one may consider “2-branes”, i.e. membranes, and these will couple to a 3-form gauge field. For instance, the membrane which gives the name to M-theory (the M2-brane) couples to a 3-form field called the supergravity C-field.

But there is an important further aspect to higher gauge fields which makes this simple picture of higher degree differential forms drastically more rich:

Where an ordinary gauge field has gauge transformations A 1↦A′ 1A_1 \mapsto A'_1 given locally by smooth functions (0-forms) λ 0\lambda_0 via the de Rham differential d dRd_{dR}

A′ 1=A 1+d dRλ 0 A'_1 = A_1 + d_{dR} \lambda_0

so a higher gauge field has higher gauge transformations given locally by pp-forms λ p\lambda_p:

A′ p+1=A p+1+d dRλ p. A'_{p+1} = A_{p+1} + d_{dR} \lambda_{p} \,.

But for p>0p \gt 0 then a crucial new effect appears: these gauge transformations, being higher differential forms themselves, have “gauge-of-gauge transformations” between them, given by lower degree forms.

This phenomenon implies that higher gauge fields have a rich global (“topological”) structure, witnessed by the higher analog of their instanton sectors. Namely, while a higher gauge field to which a p-brane may couple is locally given by a (p+1)(p+1)-form A p+1A_{p+1}, as one moves across coordinate charts this form gauge transforms by a pp-form, which then itself, as one passes along two charts, transforms by a (p−1)(p-1)-form, and so on.

The global structure for higher gauge fields obtained by carrying out this globalization via higher gauge transformations is the higher analog of that of a fiber bundle with connection on a bundle in higher differential geometry. This is sometimes known as a gerbe or, more generally, a principal infinity-bundle.

In fact the situation that there is just one gauge potential of degree (p+1)(p+1) with field strength of degree (p+2)(p+2) is just the simplest case, the “ordinary” case. More abstractly one says that such higher gauge fields are cocycles in ordinary differential cohomology.

More generally it may happen in higher gauge theory that the gauge potential is a formal linear combination of differential forms in various degrees.

The canonical example of this phenomenon is the RR-field in string theory. This has, locally, a gauge potential which is a differential form in every even degree, or every odd degree. If one is careful about the higher gauge transformations in this situation to find the correct global structure (the “instanton sector”) of the higher gauge field, then one finds that this now is a cocycle in a differential generalized cohomology, namely, in what is called differential topological K-theory. This may be understood as a higher and generalized form of the famous Dirac charge quantization condition for the electromagnetic field, see Freed 00. A lot of the fine detail of the anomaly cancellation in type II string theory depends on being careful about the global nature of this K-theoretic higher gauge RR-field (Distler-Freed-Moore 09)

Examples

Among ordinary gauge theories there is

  1. “metric” gauge theory:

    1. Maxwell theory,

    2. Yang-Mills theory

  2. topological gauge theory:

(as well as mixed cases, such as the ABJM model).

While the fields of these theories are gauge fields of the same kind in all cases (modeled by connections on principal bundles), the nature of these theories is different, not the least because the first class depends on (couples to) a background (pseudo) Riemannian metric (gravity) while the second does not (is a topological field theory).

Moreover, even though Maxwell theory (electromagnetism) may be understood as the special case of Yang-Mills theory for gauge group specialized to the circle group U(1)U(1), this makes a crucial difference for the nature of these theories and their higher generalizations.

Some example of higher gauge fields…

  1. …of higher Maxwell type:

    the fields appearing in higher-dimensional supergravity theories:

  2. …of higher Chern-Simons type:

Higher gauge theories of Maxwell type

Consider

  • D=1+d∈ℕ ≥2D = 1 + d \in \mathbb{N}_{\geq 2} the spacetime dimension,

  • X DX^D a DD-dimensional spacetime manifold,

  • with Hodge star-operator on differential forms

    ⋆:Ω dR p(X D)→Ω dR D−p(X D) \star \,\colon\, \Omega^p_{dR}(X^D) \to \Omega^{D-p}_{dR}(X^D)

    satisfying (see there)

    ⋆⋆=−(−1) p(D−p), \star \, \star = - (-1)^{p(D-p)} \,,

  • I∈SetI \,\in\, Set an index set of flux species

    (subsuming both electric fluxes and magnetic fluxes, both now allowed of further different species),

  • (deg i∈ℕ ≥1) i∈I\big( deg_i \,\in\, \mathbb{N}_{\geq 1} \big)_{i \in I} an indexed set of degrees,

  • F→≡(F (i)∈Ω dR deg i(X D)) i∈I\vec F \,\equiv\,\big( F^{(i)} \,\in\, \Omega^{deg_i}_{dR}(X^D) \big)_{i \in I} an indexed set of flux densities

  • P→=(P (i)) i∈I\vec P = \big( P^{(i)} \big)_{i \in I} an indexed set of graded-symmetric polynomials in II variables,

  • μ→\vec \mu an invertible I×II \times I-matrix

then the corresponding higher Maxwell-type equations on these differential forms/flux densities are, in duality-symmetric form (cf. SS23):

  1. the system of Bianchi identity differential equations (cf. at Gauss law)

    (1)dF→=P→(F→) \mathrm{d}\,\vec F \,=\, \vec P\big( \vec F \big)

  2. the self-Hodge duality condition

    (2)⋆F→=μ→(F→). \star \, \vec F \,=\, \vec \mu\big(\vec F\big) \,.

The full field content of a higher gauge theory with these flux densities is to involve gauge potentials for these flux densities. A systematic way to get hold of these is to choose a flux quantization law for the given flux densities, defining a (nonabelian) generalized cohomology theory. Then a full higher gauge field is a cocycle in the corresponding (nonabelian) differential cohomology.

Higher gauge theories of Chern-Simons type

See at infinity-Chern-Simons theory

References

General

Introduction and exposition:

For technical introduction to the RR-field as a higher gauge field (for more see at Dirac charge quantization):

Deriving higher gauge symmetries through the variational bicomplex:

The aspect of flux quantization laws:

Foundations of higher prequantum field theory:

For foundations of higher gauge theory formalized in homotopy type theory see

On higher gauge theory in condensed matter physics:

  • Chenchang Zhu, Tian Lan, Xiao-Gang Wen, Topological non-linear σ-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems [arXiv:1808.09394]

  • J.P. Ang, Abhishodh Prakash, Higher categorical groups and the classification of topological defects and textures, (arXiv:1810.12965)

Specifically on topological phases of matter via higher lattice gauge theory:

Further on higher lattice gauge theory:

Higher gauge theory of the Green-Schwarz mechanism

Discussion of higher gauge theory modeling the Green-Schwarz mechanisms for anomaly cancellation in heterotic string theory, on M5-branes, and in related systems in terms of some kind of nonabelian differential cohomology (ordered by arXiv time-stamp):

Last revised on June 18, 2024 at 10:49:44. See the history of this page for a list of all contributions to it.

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