Gauge field ($A_{\mu }$)

A gauge field is a connection associated with a local symmetry in a gauge theory. Its gauge potential defines the internal-symmetry contribution to a covariant derivative. For an Abelian gauge field, the gauge field strength is $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$.

For a matrix-valued gauge field $A$, let $F=dA+A\wedge A$ be its gauge field strength; products below combine matrix multiplication with the wedge product of differential forms. The five-form can be written $Q_5=\mathrm{Tr}(A(dA{)}^2+\frac{3}{2}{A}^{3}dA+\frac{3}{5}{A}^5)$. Applying the exterior derivative, using its graded product rule and the graded cyclic property of the matrix trace, gives $d{Q}_5=\mathrm{Tr}(F^3)$. The six-form is an invariant polynomial in the gauge curvature, while its local potential depends on the connection.

For $F=dA$ on a five-dimensional oriented spacetime, the action $S=-\frac{1}{2}\int F\wedge \ast F+c\int A\wedge F\wedge F$ has bulk variation $\int \delta A\wedge (-d\ast F+3cF\wedge F)$. Two integrations by parts make the three variations of the Chern-Simons 5-form contribute equally. Under $A\mapsto A+d\lambda$, the extra action is $c\int d(\lambda F\wedge F)$, a boundary term by the Generalized Stokes theorem. Its bulk field equation is gauge invariant. The Chern-Simons term is metric-independent and has no bulk contribution to the stress-energy tensor when the differential-form field is held fixed in metric variation.

The boundary four-form is $\Theta =\mathrm{Tr}[\delta A(FA+AF-\frac{1}{2}{A}^3)]$. To derive the identity, put $B=dA$ and $a=\delta A$. The variations of the three polynomial terms in $Q_5$ are respectively $3\mathrm{Tr}(a{B}^2)+d\mathrm{Tr}[a(BA+AB)]$, $3\mathrm{Tr}[a(A^{2}B+B{A}^2)]+\frac{3}{2}d\mathrm{Tr}(a{A}^3)$, and $3\mathrm{Tr}(a{A}^4)$. Their sum gives the identity because $F=B+A^2$. For variations in a Lie algebra with basis $T_{\alpha }$, the bulk field equations are $\mathrm{Tr}(T_{\alpha }{F}^2)=0$, not necessarily the unprojected matrix equation $F^2=0$. Conjugation of both the Lie algebra and the gauge curvature proves covariance of these equations under Yang-Mills gauge transformations.

For vector coupling to a Dirac fermion, write the Hermitian color connection ${\mathcal{A}}_{\mu }=A_{\mu }^a{T}^a$. Charge conjugation reverses the vector current and transposes its color matrix, requiring ${\mathcal{A}}_{\mu }^C=-{\mathcal{A}}_{\mu }^T$. Combining with parity gives ${\mathcal{A}}_{\mu }^{CP}(x)=-P_{\mu }{}^{\nu }{\mathcal{A}}_{\nu }^T(x_P)$, where $P=\mathrm{diag}(1,-1,-1,-1)$. The transpose acts only on color indices; intrinsic fermion phases cancel in the bilinear.

An Abelian $p$-form gauge potential $A_p$ is an antisymmetric tensor field with gauge equivalence $A_p\mapsto A_p+d{\Lambda }_{p-1}$ and field strength $F_{p+1}=d{A}_p$. Its field strength obeys the Bianchi identity for an Abelian p-form. The potential can have global gauge data even when its local field strength vanishes.

A free massive $p$-form obeys the displayed equation, with the divergence constraint following by applying the codifferential. For $m>0$ and $0\le p\le D-1$, a rest frame leaves an antisymmetric tensor on $D-1$ spatial directions, giving $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$ physical polarizations. The mass term removes the ordinary gauge invariance, unless a Stueckelberg mechanism restores a redundant description.

// Target: quantum-field-theory.bigb

For $p\ge 1$, introduce a $(p-1)$-form $C$ and replace the mass term of a massive p-form field by one built from $A-m^{-1}dC$. The transformations $A\mapsto A+d\Lambda$, $C\mapsto C+m\Lambda$ leave this combination and $dA$ invariant. Gauge fixing $C=0$ recovers the original massive theory. Its massless-component counts add as $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)+\left(\genfrac{}{}{0ex}{}{D-2}{p-1}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$, explaining the longitudinal degrees of freedom and the corresponding assembly of nonzero Kaluza-Klein modes.

For a free massive p-form field, the displayed dual has degree $D-p-1$. The massive field equation reconstructs $A$ locally from $m^{-1}\ast dB$ with the signature-dependent sign. It also gives $\delta B=0$ and $\delta dB+m^{2}B=0$. The polarization counts agree because $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{D-p-1}\right)$. The construction depends on nonzero mass and therefore differs from massless p-form duality by one potential rank.

// Target: physics.bigb

A massless Abelian $p$-form in $D$-dimensional flat spacetime transforms on shell as an antisymmetric rank-$p$ tensor of the rotational little group $SO(D-2)$. Its physical state count is $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$. A compatible real self-duality constraint halves this count. A four-dimensional three-form has no local propagating polarization.

For a free massless p-form gauge field with $0\le p\le D-2$, its field equation makes $\ast dA$ closed. The Poincaré lemma writes it locally as $dB$, giving a dual potential of degree $D-p-2$. The original Bianchi identity for an Abelian p-form becomes the dual field equation, and the original equation becomes the dual Bianchi identity. The two descriptions have equal little group counts $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$. Global fluxes and top forms require separate treatment.

// Target: quantum-field-theory.bigb

A three-form potential in four dimensions has no local massless polarizations. Its four-form field strength can be a constant multiple of the spacetime volume form. The equation $d(g_3^{-2}\ast F_4)=0$ leaves a constant flux parameter, which can affect vacuum energy despite supplying no local wave.

Locally in four dimensions, a massless two-form is dual to one real scalar field. In signature $(-+++)$, the first-order density $-H^2/(12g_B^2)+{ϵ}^{\mu \nu \rho \sigma }{H}_{\mu \nu \rho }{\partial }_{\sigma }a/6$ gives $H^{\mu \nu \rho }=g_B^2{ϵ}^{\mu \nu \rho \sigma }{\partial }_{\sigma }a$. Eliminating $H$ in both terms produces $-g_B^2(\partial a{)}^2/2$, so the fixed-normalization kinetic coefficient is inverted. Global flux sectors and scalar periodicities need separate treatment.

For an Abelian p-form gauge field with local field strength $F=dA$, the exterior derivative identity $d^2=0$ gives $dF=0$. In a first-order duality formulation, this identity can be enforced with a Lagrange multiplier, whose elimination restores the local gauge potential.

An antisymmetric tensor potential $B$ has gauge transformation $B\mapsto B+d\alpha$, where $\alpha$ is a one-form. Its field strength is the three-form $H=dB$, invariant because $d^2=0$. The Kalb–Ramond field is the string-theory example.

The coupling $q{\int }_{\Sigma }{X}^{\ast }B$ integrates the pullback of the two-form gauge field over the string worldsheet. Its gauge variation is $q{\int }_{\partial \Sigma }{X}^{\ast }\alpha$, by Stokes theorem. It therefore vanishes for a closed worldsheet, while an open surface requires suitable boundary fields or boundary-state transformations.

The time component of a gauge field's equation of motion constrains the spatial fields and their momenta. In a Yang-Mills theory it has the covariant form $D_i{E}_i=\rho$; its Abelian form is Gauss's law. Time-dependent collective coordinates of a gauge-theory soliton require a temporal gauge field, or an equivalent horizontal projection, that satisfies this constraint before the kinetic metric is read from the action.

A gauge boson is a spin-one quantum excitation of a gauge field. Each generator of the gauge group supplies one gauge-boson species before symmetry breaking.

With canonically normalized gauge field kinetic terms, a gauge-boson mass matrix is the real symmetric matrix of coefficients in the displayed quadratic mass term. Its eigenvalues are squared physical gauge boson masses. For a complex scalar field vacuum ${\Phi }_0$ in the Higgs mechanism, $(M^2{)}_{ab}=2\mathrm{Re}[(g_a{T}_a{\Phi }_0{)}^{†}(g_b{T}_b{\Phi }_0)]$. Thus for real $x_a$, $x^T{M}^{2}x=2\Vert {\sum }_a{x}_a{g}_a{T}_a{\Phi }_0{\Vert }^2\ge 0$: it is a Gram matrix and a positive semidefinite matrix. An unbroken generator annihilates the vacuum expectation value and lies in its null space. Diagonalizing the matrix identifies the massive and massless field combinations.

With positive canonical kinetic terms and nonzero coupling $g$, expansion of the covariant scalar kinetic term gives the displayed gauge-boson mass matrix. It is a Gram matrix: for real $c_a$, $c^T{M}_A^{2}c=g^2|{\sum }_a{c}_a{T}_a{\varphi }_0{|}^2$. Its kernel is the Lie algebra of the stabilizer subgroup of the vacuum. Thus its rank is $\dim G-\dim H$, and exactly that many gauge-field combinations acquire nonzero squared masses. Removal of these gauge-orbit directions does not remove accidental zero eigenvalues of the scalar mass matrix on the transverse complement.

The gauge group is the group of local internal symmetries. Its Lie algebra labels the gauge-boson species, while a matter field carries a representation of the group.

A gauge generator is the representation matrix of a basis element of the Lie algebra of a continuous gauge group on a specified set of fields. For Hermitian generators $T^a$, an infinitesimal transformation is $\delta \varphi =i{\alpha }^a{T}^a\varphi$, and $[T^a,T^b]=i{f}^{ab}{}_c{T}^c$. Gauge invariance of a superpotential consequently requires $W_i(T^a{)}^i{}_j{\varphi }^j=0$ for each generator.

A gauge group representation specifies how a matter field transforms under a gauge group. It fixes the gauge charges and whether gauge-invariant mass terms can pair a field with a field in the conjugate group representation.

A gauge coupling is the constant multiplying the gauge potential in the gauge covariant derivative. Its normalization depends on that of the Lie-algebra generators; each simple or Abelian factor of a gauge algebra may have an independent coupling.

For $\mathcal{L}=-\mathcal{N}{F}_{\mu \nu }^a{F}^{a\mu \nu }/4$ with $\mathcal{N}>0$, rescale ${\mathcal{A}}^a=\sqrt{\mathcal{N}}{A}^a$ and $g_c=g/\sqrt{\mathcal{N}}$. The gauge field strength then has canonical component normalization. Masses inferred from matter couplings must use $g_c$, not an unnormalized coefficient. A matrix-trace convention $\mathrm{Tr}(t^a{t}^b)=\mathcal{N}{\delta }^{ab}$ fixes this factor.

A Wilson line is the path-ordered parallel transporterUnder a gauge transformation $V$, it transforms as $U(y,x)\mapsto V(y)U(y,x)V(x{)}^{-1}$.

With gauge covariant derivative $D=d+A$, a Wilson loop is the matrix trace in a specified group representation of parallel transport around a closed curve. The holonomy of a connection changes by conjugation under gauge equivalence of principal connections, so its trace is gauge-invariant. It detects global connection data even when the local gauge curvature is zero.

Expanding a short Wilson line determines the inhomogeneous gauge-potential transformation. For $V=1+i\alpha$ and Hermitian generators,

For $D_{\mu }={\partial }_{\mu }+A_{\mu }$ and a matter field transforming as $\varphi \mapsto g\varphi$, covariance of $D_{\mu }\varphi$ requires

The curvature of a non-Abelian gauge field is $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+[A_{\mu },A_{\nu }]$. It satisfies $[D_{\mu },D_{\nu }]=F_{\mu \nu }$ and transforms as $F_{\mu \nu }\mapsto g{F}_{\mu \nu }{g}^{-1}$.

In complexified four-dimensional space-time, antisymmetry giveswhere both gauge curvature two-component spinors are symmetric. The two summands are the two duality components. In the convention where the primed symmetric part is self-dual, the Anti-self-dual Yang-Mills equations are ${\phi }_{A^{\prime }{B}^{\prime }}=0$. Equivalently ${\pi }^{A^{\prime }}{\pi }^{B^{\prime }}{F}_{A{A}^{\prime }B{B}^{\prime }}=0$ for every primed two-component spinor $\pi$, since a quadratic polynomial vanishing for every two-component spinor has all symmetric coefficients zero.

For the ordinary three-dimensional cross product, $D_{\mu }\Phi ={\partial }_{\mu }\Phi +e{A}_{\mu }\times \Phi$ requires $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+e{A}_{\mu }\times A_{\nu }$. This follows from $[A\times ,B\times ]=(A\times B)\times$. The same curvature then transforms covariantly under local rotations. Using a minus sign in its quadratic term while retaining the plus sign in $D$ breaks local gauge invariance: a pure gauge for $D$ has $F_{+}=0$ but generally $F_{-}=-2e{A}_{\mu }\times A_{\nu }\ne 0$. Reversing both signs is an alternative consistent convention.

For ${\mathcal{F}}_{\mu \nu }={\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }+ig[{\mathcal{A}}_{\mu },{\mathcal{A}}_{\nu }]$, the CP transformation of a non-Abelian gauge connection gives ${\mathcal{F}}_{\mu \nu }^{CP}(x)=-P_{\mu }{}^{\alpha }{P}_{\nu }{}^{\beta }{\mathcal{F}}_{\alpha \beta }^T(x_P)$. The commutator term transforms with the same sign as the derivative terms because $[X^T,Y^T]=-[X,Y{]}^T$. A simple component check is $F_{0i}\mapsto +F_{0i}^T$ and $F_{ij}\mapsto -F_{ij}^T$, evaluated at the reflected point.

Write $B_i=({\partial }_{i}g)g^{-1}$ and use $D=d+A$. The right Maurer-Cartan equation gives ${\partial }_x{B}_y-{\partial }_y{B}_x=[B_x,B_y]$. Thus $A_i=\alpha B_i$ has the displayed gauge curvature. Both $\alpha =0$ and $\alpha =-1$ are universally flat; the latter is a pure gauge potential. Commuting $B_x,B_y$ give additional flat cases. For the opposite convention $D=d-A$, the corresponding component formula instead has $\alpha (1-\alpha )$.

For $g=\exp (-i(x{\sigma }_1+y{\sigma }_2)/2)$, write $r=\sqrt{x^2+y^2}$ and $n=(x/r,y/r,0)$. The Pauli matrix multiplication law gives $g=\cos (r/2)I-i\sin (r/2)n\cdot \sigma$. On the circles $r=2\pi k$, the angular derivative is zero, so the two Cartesian right logarithmic derivatives are proportional to the same radial generator. They commute, giving zero gauge curvature for every scaling constant. At the origin their commutator is $-i{\sigma }_3/2$, which need not vanish.

With gauge covariant derivative $D=d+A$, transforming the zero connection by a smooth group-valued map $g$ gives $A=-dg{g}^{-1}$. This potential has zero gauge curvature. A flat potential is locally of this form; global reconstruction can also depend on topology and holonomy. The sign changes when the definition of the covariant derivative changes.

The Jacobi identity for gauge covariant derivatives gives $D_{[\mu }{F}_{\nu \rho ]}=0$. In four dimensions this is $D_{\mu }{\tilde{F}}^{\mu \nu }=0$, with ${\tilde{F}}^{\mu \nu }=\frac{1}{2}{\epsilon }^{\mu \nu \rho \sigma }{F}_{\rho \sigma }$. It holds independently of the Yang-Mills equations.

A flat connection has vanishing curvature. On a simply connected patch it is gauge equivalent to the zero connection, because the parallel-transport equation has path-independent solutions.

For $D_i={\partial }_i+A_i$, compatibility for arbitrary initial vectors is equivalent to zero gauge curvature. An invertible parallel bundle frame $S$ obeys $D_{i}S=0$, so $[D_i,D_j]S=F_{ij}S=0$ forces $F_{ij}=0$. Conversely, solve the two ordinary transport equations along one coordinate axis and then along parallel lines. The unused transport residual solves a homogeneous transport equation with zero initial value when the gauge curvature is zero; it therefore vanishes. Thus $A_i=-({\partial }_{i}S)S^{-1}=g^{-1}{\partial }_{i}g$ with $g=S^{-1}$. One particular parallel vector requires only $F_{ij}v=0$, a weaker condition.

The holonomy of a flat connection is invariant under based homotopy of loops, giving a group homomorphism from the fundamental group into the structure group after choosing a frame at the base point. Changing that frame conjugates the representation. Its associated bundle is constructed from the universal cover and this monodromy. It differs from the natural representation of a Riemannian holonomy group on a tangent fiber.

For a connected base and a chosen frame at one point, a flat connection gives a representation of the fundamental group by parallel transport. Conversely, $(\tilde{X}\times G)/{\pi }_1(X)$ with monodromy representation has a flat product connection. Based gauge classes correspond to representations, and unbased classes to conjugacy classes. On a fixed underlying bundle only representations whose associated flat bundle has that bundle type are allowed.

// Destination: geometry-and-topology.bigb

A gauge covariant derivative combines an ordinary derivative with a gauge field so that differentiating a field preserves its gauge-transformation law. In a matrix representation one common convention is $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }$.

For the Killing form or another invariant bilinear form on a Lie algebra,The connection terms cancel by invariance. Thus, for compactly supported variations, a gauge covariant derivative can be moved to the other factor in an integral with a minus sign, just as an ordinary derivative.

// Target: quantum-field-theory.bigb

If a complex scalar field uses $D_{\mu }\varphi =({\partial }_{\mu }+ie{A}_{\mu })\varphi$, its conjugate uses $(D_{\mu }\varphi {)}^{\ast }=({\partial }_{\mu }-ie{A}_{\mu }){\varphi }^{\ast }$. Under $\varphi \mapsto e^{-ie\alpha }\varphi$ and $A\mapsto A+d\alpha$, the two derivatives transform with opposite phases. Their product is gauge invariant. Applying the same plus-charge operator to both factors leaves an unwanted term proportional to ${\partial }_{\mu }\alpha$ and fails gauge invariance.

With $D_{\mu }={\partial }_{\mu }+R(A_{\mu })$ and ${\delta }_X{A}_{\mu }=-ϵ{\partial }_{\mu }X+ϵ[X,A_{\mu }]$, the derivative terms in ${\delta }_X(D_{\mu }\varphi )$ cancel. The remaining terms combine using $R([X,A_{\mu }])=[R(X),R(A_{\mu })]$, so the covariant derivative transforms in the same Lie algebra representation as the scalar. This convention absorbs the gauge coupling into the connection.

On an adjoint-valued field $X$, the covariant derivative is $D_{\mu }X={\partial }_{\mu }X+[A_{\mu },X]$. If $X\mapsto gX{g}^{-1}$, then $D_{\mu }X\mapsto g(D_{\mu }X)g^{-1}$.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and matter transformation ${\varphi }^{\prime }=h\varphi$. Covariance $D_{\mu }^{\prime }{\varphi }^{\prime }=h{D}_{\mu }\varphi$ forces $A_{\mu }^{\prime }=h{A}_{\mu }{h}^{-1}+(i/g)({\partial }_{\mu }h)h^{-1}$. For $h=1-ig\omega +O({\omega }^2)$, expansion gives the displayed adjoint covariant derivative variation. With an orthonormal Hermitian Lie-algebra basis, $\delta A_{\mu }^a={\partial }_{\mu }{\omega }^a-g{f}^{abc}{A}_{\mu }^b{\omega }^c$. Thus Lorenz-gauge variation is $\delta (\partial \cdot A)=\partial \cdot \mathcal{D}\omega$. Choosing the Faddeev-Popov operator as $\mathcal{M}=-\partial \cdot \mathcal{D}$, with its field-independent sign absorbed in normalization, gives ghost action $-\bar{c}\partial \cdot \mathcal{D}c$; consistent propagator and vertex signs follow from this choice.

For Lie-algebra-valued fields and a constant invariant bilinear form on a Lie algebra, the connection terms in the displayed identity cancel. Indeed, invariance gives $\kappa ([A,Y],Z)+\kappa (Y,[A,Z])=0$. Ordinary integration by parts then yields $\int \kappa (D_{\mu }Y,Z)=-\int \kappa (Y,D_{\mu }Z)$ when boundary terms vanish, for example for compactly supported variations. This preserves covariance in variational calculations of Yang-Mills equations.

Gauge invariance is invariance under local changes of gauge representative. Gauge-invariant observables depend only on the physical gauge orbit.

A gauge theory uses this redundancy to describe the same physical state by different field representatives.

A spatial transformation is a symmetry of a gauge-field configuration when its pullback lies in the gauge orbit of the original configuration. It need not fix a particular representative. For a planar vortex, rotation changes its winding phase and a constant $U(1)$ transformation compensates. For a hedgehog ansatz for a monopole, a spatial rotation is compensated by the corresponding internal adjoint $SU(2)$ rotation. Gauge-invariant densities remain invariant without choosing a special gauge.

A local field combination is a gauge-invariant operator when its gauge representations and charges combine to a singlet. In an effective field theory, such operators may be multiplied by Wilson coefficients and inverse powers of a heavy scale. Multiplication by ${\varphi }^{†}\varphi$ preserves the invariance of an existing Higgs Yukawa operator.

Gauge redundancy means that multiple field configurations or polarization representatives describe the same physical state. Physical amplitudes must be unchanged when an external polarization is shifted by a pure-gauge component.

A gauge orbit is the set of all field configurations related by gauge transformations. Every point on one orbit represents the same physical gauge configuration.

An Abelian gauge theory has a commutative gauge group. Its gauge field strength has no commutator term, so the pure gauge Lagrangian has no cubic or quartic gauge-boson self-interactions.

For $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, the local U(1) transformation is

A charged scalar field coupled to an Abelian gauge field through $D_{\mu }={\partial }_{\mu }-ie{a}_{\mu }$. Under $\varphi \mapsto e^{i\alpha }\varphi$ and $a_{\mu }\mapsto a_{\mu }+e^{-1}{\partial }_{\mu }\alpha$, its gauge covariant derivative transforms with the same phase. The potential determines whether the perturbative spectrum has a massless photon or exhibits the Higgs mechanism.

With scalar charges $e$ and $2e$, both $\psi$ and ${\varphi }^2$ transform with phase $e^{2i\alpha }$. Therefore $M^2|\psi -b{\varphi }^2{|}^2$ is a gauge-invariant potential term. It contains a nonzero direct cubic interaction $-M^2(b{\psi }^{\ast }{\varphi }^2+h.c.)$. Adding positive quadratic and quartic modulus terms gives a real bounded-below potential. All expanded interactions have field degree at most four.

A large gauge transformation does not approach the identity at the relevant spatial or asymptotic boundary. Although it leaves the local field strength unchanged, its boundary charge can act nontrivially on physical states and constrain soft limits through a Ward identity.

Yang-Mills theory is the non-Abelian gauge theory with field strengthUnder an infinitesimal gauge transformation it transforms covariantly as $\delta F_{\mu \nu }=-ig[F_{\mu \nu },\alpha ]$.

If a Lie algebra has an invariant symmetric nondegenerate form $h$, its curvature $F$ transforms in the Adjoint representation and $\mathcal{L}=-h(F_{\mu \nu },F^{\mu \nu })/4$ is gauge invariant. Infinitesimal invariance follows from $h([\theta ,X],Y)+h(X,[\theta ,Y])=0$. For compact generators with $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, this gives $-\mathrm{tr}(F_{\mu \nu }{F}^{\mu \nu })/2=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$. An arbitrary index contraction is not gauge invariant for an arbitrary Lie algebra.

A theta vacuum is a coherent state $|\theta ⟩={\sum }_n{e}^{in\theta }|n⟩$ formed from gauge vacua of different winding numbers. Large gauge transformations shift $n$ and act on this state by a phase. The Yang-Mills theta term gives the corresponding weighting of spacetime topological sectors.

In four dimensions the Yang-Mills theta term is proportional to $\theta \mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. Its density is a total derivative, but nontrivial gauge-field topology can make its spacetime integral physically relevant in the quantum theory.

The term weights topological sectors and supplies the angle labeling a theta vacuum.

For $F=dA+A\wedge A$, the gauge-theory Bianchi identity is $DF=0$ and the variation is $\delta F=D\delta A$. The graded product rule for the covariant exterior derivative gives the displayed boundary integral. Thus a constant-coefficient Yang-Mills theta term leaves the bulk Yang-Mills equations unchanged, but can change the boundary variational condition. Its four-form density is exactly gauge-invariant by conjugation of $F$ and cyclicity of the matrix trace, even if the gauge parameter is nonzero on the boundary. Gauge changes of a local Chern-Simons 3-form must not be confused with gauge changes of this globally defined curvature density.

The color-contracted density ${ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }^a{F}_{\rho \sigma }^a$ is even under charge conjugation and odd under parity. In matrix form the two charge-conjugation signs cancel and transposition leaves the trace of the product unchanged. Parity contributes its orientation-reversing determinant to the four-index Levi-Civita symbol. Thus the density is CP odd. A generic fixed coefficient violates CP symmetry; being a local total derivative does not eliminate effects from nontrivial gauge topology.

The physical QCD theta angle is $\bar{\theta }={\theta }_{\mathrm{QCD}}+\arg \det (M_u{M}_d)$. An anomalous chiral quark rotation shifts the two terms oppositely, so only their sum is invariant.

The strong CP problem asks why the physical QCD theta angle is smaller than roughly $10^{-10}$ even though no Standard Model symmetry requires this small value.

Peccei–Quinn theory promotes the physical QCD theta angle to a dynamical field through an anomalous global $U(1)$ symmetry. Nonperturbative QCD drives its vacuum expectation value to a CP-conserving minimum.

The axion is the pseudo-Goldstone boson of spontaneously broken Peccei–Quinn symmetry. Its QCD-generated potential dynamically relaxes the physical theta angle toward zero.

The electroweak theta angle multiplies the $SU(2{)}_L$ topological density. In the renormalizable Standard Model it is unphysical because an anomalous baryon-plus-lepton rephasing shifts it, but explicit baryon-plus-lepton violation can make an invariant combination observable.

The Chern-Simons current satisfies ${\partial }_{\mu }{K}^{\mu }=\mathrm{Tr}(F_{\mu \nu }{}^{\star }{F}^{\mu \nu })$. It is not gauge invariant even though its divergence is.

In four dimensions this current is dual to the degree-three Chern-Simons form.

With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }$ and $[T^a,T^b]=i{f}_{abc}{T}^c$, the Yang-Mills field strength has the sign $F_{\mu \nu }^a=G_{\mu \nu }^a-g{f}_{abc}{A}_{\mu }^b{A}_{\nu }^c$. Expanding $-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a cross term $(g/2)G_{\mu \nu }^a{f}_{abc}{A}^{b\mu }{A}^{c\nu }$; antisymmetry makes its two derivative terms equal, proving the cubic expression. Squaring the nonlinear term gives the quartic expression. Both vertices arise from the same gauge coupling, rather than independent free parameters.