nLab Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories


under construction


Quantum field theory

Differential cohomology

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The article considers the generalization of a phenomenon in spontaneous symmetry breaking in abelian (higher) gauge theory, that was considered earlier in (Banks-Seiberg 08), to nonabelian higher gauge theory, hence involving not just differential 1-form gauge potentials but 2-form gauge potentials (“nonabelian B-fields”), hence to principal 2-connections.

Specifically, in (Banks-Seiberg 08, (2.4)-(2.9)) is an argument saying that some properties of the low energy limit of abelian Yang-Mills like gauge theories in four dimensions with (Higgs-like) spontaneous symmetry breaking to a discrete gauge cyclic group p\mathbb{Z}_p are described by a dual BF-theory, hence a higher gauge theory represented by a 2-connection instead of an ordinary connection. Here the duality is electric-magnetic duality, but applied not to the 1-form gauge field but to the Higgs boson-like scalar field, whose magentic dual in 4d is indeed a (4(0+1)1=2)(4 - (0+1) - 1 = 2)-form higher gauge field. See at connection on a 2-bundle for more on this.

In the nonabelian case the proposed mechanism is no longer electric-magnetic duality of a Higgs boson-like field, but instead to condensation of monopoles.

The abelian phenomenon is reviewed and expanded on in its relation to higher dimensional defect quantum operators in section III. Generalization to the nonabelian case is then in section IV. An outlook on the relation to structures such as in geometric Langlands duality is in section V.

Followups include

under construction


II. Surface operators in abelian gauge theories


III. TQFTs for abelian gapped phases

Here is an observation about how the idea in that section might be formalized. (Due to discussion among Domenico Fiorenza, Hisham Sati, and Urs Schreiber).

The moduli stack B n\mathbf{B}\mathbb{Z}_n of “discrete gauge fields” for gauge group n:=/n\mathbb{Z}_n := \mathbb{Z}/n\mathbb{Z} with nn \in \mathbb{N} may be realized as the following homotopy fiber of smooth moduli stacks of circle connections

B n BU(1) conn () n * 0 BU(1) conn \array{ \mathbf{B}\mathbb{Z}_n &\to& \mathbf{B}U(1)_{conn} \\ \downarrow &\swArrow& \downarrow^{\mathrlap{(-)^n}} \\ \ast &\stackrel{0}{\to}& \mathbf{B}U(1)_{conn} }

(in Smooth∞Grpd). By construction, an element XB nX \to \mathbf{B}\mathbb{Z}_n of this homotopy fiber is equivalently

  • a 1-form gauge field F˜ A:XBU(1) conn\tilde F_A \colon X \to \mathbf{B}U(1)_{conn} with local connection 1-form AA;

  • a 0-form gauge field ϕ:F˜ A0\phi \colon \tilde F_A \to 0 trivializing F˜ A\tilde F_A, with local component ϕ\phi locally satisfying dϕ=nAd \phi = n A.

This data is what motivates the discussion in the article.

The electric-magnetic duality may be formulate as a “differential higher Pontryagin duality

B 2^ n[B n,B 3U(1) conn], \mathbf{B}^2 \hat \mathbb{Z}_n \simeq [\mathbf{B}\mathbb{Z}_n, \mathbf{B}^3 U(1)_{conn}] \,,

where ^ n[ n,U(1)] n\hat \mathbb{Z}_n \coloneqq [\mathbb{Z}_n, U(1)] \simeq \mathbb{Z}_n is the Pontryagin dual group.

For this higher moduli stack of discrete higher gauge fields we have a local description by smooth data as before: it fits into the homotopy fiber sequence

B 2 n B 2U(1) conn () n * 0 B 2U(1) conn. \array{ \mathbf{B}^2\mathbb{Z}_n &\to& \mathbf{B}^2 U(1)_{conn} \\ \downarrow &\swArrow& \downarrow^{\mathrlap{(-)^n}} \\ \ast &\stackrel{0}{\to}& \mathbf{B}^2 U(1)_{conn} } \,.

In analogy to the above, this now describes the space of field configurations consisting of a circle 2-group-principal 2-connection BB and a trivialization of its nnth power by a 1-form connection AA, i.e. dAnB=0d A - n B = 0.

This one may also understand as the vanishing condition on the 2-form curvature of a 2-connection. This is the observation that drives the second part of the article.

A. Abelian Higgs phases

B. Abelian confining phases

C. Oblique confining phases


D. Mixed phases


IV. TQFTs for nonabelian gapped phases


A. Nonabelian Higgs phases


B. Nonabelian confining phases


1. Monopole condensation

A notce concerning the discussion of monopoles on the bottom right of p. 5.

Let GG be a connected Lie group. A subgroup

Γ 0π 1(G) \Gamma_0 \hookrightarrow \pi_1(G)

of its fundamental group defines a covering t:HGt \colon H \to G, exhibited by the following pasting diagram of homotopy pullbacks of smooth ∞-groupoids:

π 1(G)/Γ 0 H Π(H) Bπ 1(H)BΓ 0 t Π(t) * e G Π(G) τ 1 Bπ 1(G) \array{ \pi_1(G)/\Gamma_0 &\to& H &\to& \Pi(H) &\to& \mathbf{B}\pi_1(H) \simeq \mathbf{B}\Gamma_0 \\ \downarrow && \downarrow^{\mathrlap{t}} && \downarrow^{\mathrlap{\Pi(t)}} && \downarrow \\ \ast &\stackrel{e}{\to}& G &\stackrel{}{\to}& \Pi(G) &\stackrel{\tau_1}{\to}& \mathbf{B} \pi_1(G) }

Here ()Π()(-) \to \Pi(-) is the unit of the shape modality. In the given low degree case the above diagram can be checked explicitly by standard means, but it may be worthwhile to note that the whole situation follows fully generally as by the Galois theory in cohesive (Schreiber, section 3.8.6).

2. Nonabelian gerbes and nonabelian B-fields

3. A TQFT for a nonabelian confining phase

C. A TQFT for a nonabelian oblique confining phase



The argument relating aspects of the low energy limit of spontaneous symmetry breaking in ordinary gauge theory by electric-magnetic duality of the Higgs boson-like field to BF-type higher gauge theory is in section 2 of

following discussion of D-brane/NS5-brane phenomena in


The formulation of principal 2-connections via their higher parallel transport used in section IV is from

expanding on

  • John Baez, Urs Schreiber, Higher gauge theory, in A. Davydov et al. Categories in Algebra, Geometry and Mathematical Physics, Contemp. Math. 431, AMS, Providence, Rhode Island, 2007, pp. 7-30 (arXiv:math/0511710, web)

Some of the above comments on the formulation in smooth ∞-groupoids refer to

category: reference

Last revised on September 9, 2013 at 11:00:08. See the history of this page for a list of all contributions to it.