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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.

An asymptotically free theory has a running coupling that tends to zero at arbitrarily high energy. Its negative weak-coupling beta function makes short-distance perturbation theory increasingly accurate.

In Quantum chromodynamics, this means that a short-distance probe can resolve quarks with a small QCD coupling, even though isolated quarks are absent from the observed hadron spectrum. This made asymptotic freedom a bridge between the approximately free constituents of the parton model and an interacting quantum field theory. Asymptotic freedom supplies control at large momentum transfer; it does not itself prove confinement.^[ref]

Use the four-dimensional QCD coupling convention ${\alpha }_s=g_s^2/(4\pi )$. For the SU(3) gauge group and $n_f$ active quark flavors in the fundamental representation, the one-loop order beta function isHere $n_f$ counts quark flavors, not the three values of color charge. The chain rule converts the first equation to the last; changing from $\ln \mu$ to $\ln {\mu }^2$ supplies a factor of two. For ${\beta }_0>0$, the negative sign makes $g_s=0$ an attractive high-energy renormalization-group fixed point.^[ref]

For ${\beta }_0>0$, set the renormalization scale $\mu =Q$, where $Q$ is a positive energy scale. The one-loop running of the strong coupling then givesThe strong-coupling scale $\Lambda$ has the same units of measurement as $Q$, so the logarithm has a dimensionless argument. This is a fixed-$n_f$ approximation on the positive branch $Q>\Lambda$; controlled perturbative quantum field theory requires a small QCD coupling and suitably small omitted corrections, not merely a positive denominator. Heavy quark masses require matching the QCD scale across a quark threshold, changing $n_f$ and the associated $\Lambda$. Both the renormalization scheme and the loop order matter when quoting a numerical $\Lambda$.

The bino is the spin-one-half gaugino of the hypercharge $U(1{)}_Y$ gauge group. Its adjoint representation is the singlet $(1,1,0)$, so it has no hypercharge of its own. After electroweak symmetry breaking, it can mix with the neutral wino and higgsinos into neutralinos.

A gauge theory whose left- and right-chiral fermions transform differently under its gauge group. The weak charged current gives a physical example. A chiral representation assignment must satisfy the relevant gauge anomaly cancellation conditions to define a consistent quantum theory.

Color charge is the gauge group charge of Quantum chromodynamics, associated with its $SU(3)$ symmetry. A quark transforms in the fundamental three-dimensional representation, an antiquark in its conjugate, and a lepton is a singlet. The word color is a label for this charge, unrelated to visible light.

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 field is a gauge singlet if it transforms in the trivial representation of the entire gauge group: every gauge transformation leaves it unchanged. A singlet of only one factor need not be a gauge singlet of the product group. For example a right-handed neutrino with representation $(1,1{)}_0$ is a Standard Model gauge singlet, whereas a color-singlet charged lepton is still acted on by the electroweak group. A singlet can have ordinary derivatives in its kinetic term and gauge-invariant masses, subject to Lorentz invariance and any additional global symmetries.

Set $\epsilon =i{\tau }_2$, whose Pauli matrix identity is $\epsilon {\tau }_i^{\ast }=-{\tau }_i\epsilon$. Exponentiation gives $\epsilon U^{\ast }=U\epsilon$ for $U\in SU(2)$. Hence complex conjugating a Higgs doublet and multiplying by $\epsilon$ restores the fundamental representation of $SU(2)$, while reversing hypercharge. A Higgs field of hypercharge $1/2$ yields a conjugate doublet of hypercharge $-1/2$. Its product ${\tilde{\varphi }}^{†}L$ with a lepton doublet $L$ of hypercharge $-1/2$ transforms in the trivial representation of the whole electroweak gauge group. At vacuum $(0,v{)}^T/\sqrt{2}$, this composite is $v{\nu }_L/\sqrt{2}$.

For the convention $D_{\mu }={\partial }_{\mu }+g{A}_{\mu }$, a gauge group transformation $\Phi \mapsto R\Phi$ acts on the gauge potential by $A_{\mu }\mapsto R{A}_{\mu }{R}^{-1}-g^{-1}({\partial }_{\mu }R)R^{-1}$. The gauge field strength transforms by conjugation. This makes $D_{\mu }\Phi$ transform in the same gauge group representation as $\Phi$.

The Standard Model uses the local gauge group $SU(3{)}_c\times SU(2{)}_L\times U(1{)}_Y$. Its Higgs field is a complex Higgs doublet of hypercharge $Y=1/2$. A bare mass term for a weak gauge boson, or for a charged fermion joining inequivalent left- and right-handed gauge representations, would not respect the unbroken electroweak construction. Instead introduce a gauge-invariant Higgs potentialIts minima satisfy ${\Phi }^{†}\Phi =v^2/2$, $v^2={\mu }^2/\lambda$. Choosing a representative $⟨\Phi ⟩=(0,v{)}^T/\sqrt{2}$ leaves $Q=T_3+Y$ unbroken. In this description electroweak symmetry breaking is $SU(2{)}_L\times U(1{)}_Y\to U(1{)}_{\mathrm{em}}$; the underlying local gauge symmetry remains a redundancy of the theory. A vacuum representative is not a physical explicit violation of gauge invariance.

In unitary gauge, write $\Phi =(0,v+H{)}^T/\sqrt{2}$. The gauge-covariant kinetic term $|D_{\mu }\Phi {|}^2$ produceswith $\tan {\theta }_W=g^{\prime }/g$ and $e=g\sin {\theta }_W$. Three angular scalar directions supply the longitudinal gauge-boson polarizations of $W^{+},W^{-},Z$, while one real radial Higgs boson survives with $m_H^2=2\lambda v^2$. Counting physical degrees of freedom checks the Higgs mechanism: the electroweak gauge sector and scalar doublet have $4\times 2+4=12$ before the rearrangement and $3\times 3+2+1=12$ afterwards.

The Higgs potential also fixes the scalar self-interactions:The color gauge group is unbroken, so gluons remain massless. The same electroweak kinetic term fixesGauge-invariant Yukawa couplings such as $-y_d{\bar{Q}}_L\Phi d_R-y_u{\bar{Q}}_L\tilde{\Phi }{u}_R+h.c.$ give $m_f=y_{f}v/\sqrt{2}$ and the physical coupling $-m_{f}H\bar{f}f/v$. These relations predict stronger Higgs couplings to heavier massive particles; they do not determine the Yukawa parameters themselves. The minimal Standard Model does not supply observed neutrino masses. The surviving Higgs boson is neutral and has spin zero; its mass and self-coupling are tied to the curvature of the Higgs potential, not fixed by the gauge couplings alone. Higgs exchange also provides the Higgs cancellation in longitudinal vector scattering, maintaining the required high-energy behavior of weak amplitudes.

At a proton collider, important production mechanisms are gluon-fusion Higgs production through a heavy-quark loop, vector-boson fusion with two outgoing quark jets, associated $WH$ or $ZH$ production, and production with a top quark pair. At an electron–positron collider, Higgsstrahlung $e^{+}{e}^{-}\to ZH$ is a clean example. Tree-level decays include Higgs decay to a fermion pair and, when allowed, Higgs decay to two W bosons or Higgs decay to two Z bosons. Below an on-shell vector-pair threshold, one or both vectors can be virtual. The $gg$ and $\gamma \gamma$ channels arise through loops; neutral colorless $H$ has no corresponding direct tree-level gauge coupling. Rates and branching fractions depend on both couplings and kinematics, so “heaviest available particle” is a guide rather than a complete rule.

The Higgs boson discovery in 2012 supplied an experimental test. The ATLAS experiment and CMS experiment reconstructed two-photon and four-lepton invariant masses in Large Hadron Collider proton collisions. Electromagnetic calorimeters measure photon energies, while tracking and muon systems identify charged leptons; narrow reconstructed peaks can be distinguished from smooth backgrounds with control samples and a likelihood analysis. The ATLAS discovery paper reported a new resonance near $126.5\mathrm{GeV}$. An observed mass peak supports the existence of a new state; production channels, decay rates and angular distributions test whether it has the particular Higgs couplings and quantum numbers.^[ref]

Finally, the hierarchy problem asks why the electroweak scalar mass scale remains much smaller than a putative heavy physical scale without delicate cancellations. A cutoff estimate suggests corrections $\delta m_H^2\sim {\Lambda }^2/(16{\pi }^2)$ times couplings, but the quadratic cutoff term itself depends on the regularization convention. The physical concern persists when a heavy state of mass $M$ couples to the Higgs field and produces threshold corrections of order its coupling times $M^2/(16{\pi }^2)$. Matching a much smaller Higgs mass may then require cancellation between independent parameters. Supersymmetry, compositeness and other protective structures propose different resolutions; renormalizability of the Standard Model does not by itself resolve this sensitivity or establish that new particles must occur at one particular mass.

The left- and right-chiral components of a charged Standard Model fermion transform in different electroweak group representations. A bare Dirac mass term $-m{\bar{f}}_L{f}_R+h.c.$ is therefore not invariant under the full electroweak gauge group. The Higgs doublet supplies the missing transformation properties. With $\tilde{H}=i{\sigma }_2{H}^{\ast }$, the gauge-invariant Yukawa interactions arewhere each $Y_f$ is a flavor-space Yukawa matrix. For example, hypercharges in the down-quark term add to $-1/6+1/2-1/3=0$, while those in the up-quark term add to $-1/6-1/2+2/3=0$. Thus the Higgs and conjugate Higgs are required in different terms.

After electroweak symmetry breaking, choose unitary gauge with $H=(0,(v+h)/\sqrt{2}{)}^T$. The fermion mass matrices becomeA singular value decomposition $U_{fL}^{†}{M}_f{U}_{fR}=\mathrm{diag}(m_{f,1},m_{f,2},m_{f,3})$ defines nonnegative physical tree-level masses. The mismatch $U_{uL}^{†}{U}_{dL}$ is the Cabibbo-Kobayashi-Maskawa matrix in the weak charged current. The corresponding physical Higgs boson couples diagonally as $-(m_f/v)h\bar{f}f$ in the minimal one-doublet theory. This relates its coupling strength to each mass, but the measured Yukawa couplings remain independent input parameters: the minimal theory does not predict their hierarchy.

The minimal renormalizable Standard Model contains no right-handed neutrino, so it has neither a neutrino Dirac Yukawa term nor a gauge-invariant renormalizable left-handed Majorana mass term. Its neutrinos are massless. Observed neutrino oscillations require going beyond that minimal field content or operator set: adding right-handed singlets permits Dirac masses, while the dimension-five Weinberg operator gives a Majorana neutrino mass of order $v^2/\Lambda$ after symmetry breaking. These alternatives should be distinguished from the charged-fermion Yukawa mechanism.

Gauge boson self-interactions follow from noncommuting generators, rather than a separate assumption that the force carriers should attract one another. With $D_{\mu }={\partial }_{\mu }+ig{A}_{\mu }^a{T}^a$ and $[T^a,T^b]=i{f}^{abc}{T}^c$, the Yang-Mills field strength isThe last term is required so that $F_{\mu \nu }$ transforms covariantly under the gauge group. Let $G_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a$. Expanding $\mathcal{L}=-F_{\mu \nu }^a{F}^{a\mu \nu }/4$ gives a quadratic kinetic term and the cubic and quartic Yang-Mills self-interactionsThe structure constants and one gauge coupling fix both types of vertex. For an Abelian group the structure constants vanish: the renormalizable pure electromagnetic field action has no elementary three- or four-photon vertex. Photon scattering through charged-particle loops is a different statement, not a contradiction.

In Quantum chromodynamics, the non-Abelian $SU(3{)}_c$ structure produces three- and four-gluon vertices; the gluons themselves carry color charge. These vertices influence multi-jet production and the renormalization group running of the strong coupling. In the electroweak sector the $SU(2{)}_L$ fields mix with the Abelian field after electroweak symmetry breaking. The resulting charged-vector vertices include $W^{+}{W}^{-}\gamma$ and $W^{+}{W}^{-}Z$, with couplings $e$ and $g\cos {\theta }_W$, together with the related four-boson vertices. Neutral-only triple vertices such as $ZZ\gamma$ are absent at tree level in the Standard Model.

The electroweak triple gauge coupling can be measured using $e^{+}{e}^{-}\to W^{+}{W}^{-}$. The neutrino exchange and neutral-vector exchange diagrams interfere: the gauge-related coefficients cancel the most rapidly growing longitudinal-polarization terms. Their relative strengths and angular dependence therefore test the theory more sharply than merely observing a pair of massive vector bosons. The electron-mass-suppressed Higgs contribution completes the corresponding mass-dependent cancellation. Adding arbitrary independent three- and four-vector couplings would generally destroy these relations.

At the Large Electron–Positron Collider, the L3 experiment fitted charged triple-gauge parameters using hadronic and semileptonic W-pair events, complemented by single-W and single-photon channels. These measurements agreed with the predicted self-couplings. Their interpretation uses reconstructed decay products and differential distributions, rather than direct pictures of a three-boson vertex.^[ref]

The unbroken group is the stabilizer subgroup of the chosen vacuum expectation value:Its Lie algebra consists of generators $T$ with $T{\varphi }_0=0$. The vacuum manifold is the orbit $G{\varphi }_0\simeq G/H$ under the stated transitivity assumption. Its tangent space is the image of $T\mapsto T{\varphi }_0$, so its dimension is $r=\dim G-\dim H$.

Let $K_{ij}={\partial }_i{\partial }_{j}V({\varphi }_0)$ be the scalar mass matrix, with canonical scalar kinetic terms. Invariance of the scalar potential gives $({\partial }_{i}V)(T_a\varphi {)}_i=0$ for every $\varphi$. Differentiate with respect to ${\varphi }_j$ and evaluate at the minimum, where ${\partial }_{i}V=0$:Thus every independent tangent direction is a zero eigenvector of $K$. These are the Goldstone directions in the scalar mass matrix; for a global symmetry they are $r$ physical massless Goldstone bosons. For a generic nondegenerate minimum transverse to the orbit, they are precisely all the zero eigenvalues. Without that nondegeneracy assumption, additional massless scalars are possible.

For a local gauge group, use the orthogonal unitary-gauge slice near a scalar vacuum. Since ${\varphi }_0^T{T}_a{\varphi }_0=0$ by antisymmetry, the proposed condition is equivalent toIt sets the gauge-orbit components of the scalar fluctuation to zero. This is an admissible local unitary gauge: on a basis of independent broken tangent vectors $v_a=T_a{\varphi }_0$, the variation of these conditions under the broken gauge parameters is $v_a^T{v}_b$. This Gram matrix is invertible on that basis, so the implicit function theorem selects the required gauge transformation near the vacuum. Unbroken transformations remain as a residual $H$ gauge symmetry; no global gauge slice through zeros of the scalar field is being asserted.

Write $\varphi ={\varphi }_0+h$. To quadratic order, the covariant kinetic term containsThe middle term vanishes in this unitary gauge. The gauge-boson mass rank from a real scalar vacuum is therefore determined byThese are squared masses: the physical gauge boson masses are square roots of the matrix's nonzero eigenvalues after diagonalization. For real $c_a$,Its kernel consists exactly of the unbroken generators. With $g\ne 0$, the rank is $r$, so exactly $\dim G-\dim H$ gauge-field combinations become massive. Each removed scalar Goldstone mode supplies the extra longitudinal polarization of a massive gauge boson.

The remaining physical scalar fluctuations lie in the orthogonal complement of the orbit tangent space. They have no massless modes if $K$ is positive definite on that complement. The unconditional absence of massless scalars does not follow from the printed assumptions. For a concrete counterexample, take a real doublet, $G=SO(2)$ andIts entire minimum set is one circle orbit, as required, but its Hessian matrix vanishes at the minimum. Gauging the rotation gives one vector of squared mass $g^2{v}^2$ and removes the angular scalar. The radial scalar remains massless at quadratic order, since its potential starts at fourth order in the radial fluctuation. This is an accidental massless radial mode beyond Goldstone modes. Thus the intended no-massless-scalar conclusion needs nonzero transverse curvature.

In the Standard Model, the Higgs doublet contains four real components and breaks $SU(2{)}_L\times U(1{)}_Y$ to $U(1{)}_{\mathrm{em}}$. Three Goldstone bosons become the longitudinal components of $W^{\pm }$ and $Z$, while the photon remains massless. The remaining radial Higgs boson is massive for the ordinary quartic potential with positive curvature. With $⟨H⟩=(0,v_{\mathrm{EW}}/\sqrt{2}{)}^T$,The colour gauge group is unbroken. This is the Higgs mechanism, with the generic curvature condition realized by the standard Higgs potential.

Specify the gauge group, its gauge group representations and gaugings, a real Kähler potential $K$, a holomorphic superpotential $W$, and a symmetric holomorphic gauge kinetic function $f_{ab}$, with positive kinetic matrices. The latter includes the gauge couplings and theta angles; allowed Abelian Fayet–Iliopoulos terms require constants ${\xi }_a$. For local supersymmetry, also specify the Planck mass; constant Fayet–Iliopoulos terms in conventional N=1 supergravity require a compatible gauged R-symmetry and transformation of $W$. $W$ and $f$ are holomorphic; $K$ generally is not. Radiative corrections generally change $K$. The Wilsonian effective action obeys superpotential non-renormalization perturbatively, although physical couplings run through field normalization and nonperturbative corrections can change $W$. The gauge kinetic function does receive corrections while remaining holomorphic; its perturbative Wilsonian running has one-loop exactness of the holomorphic gauge kinetic function, not all-order constancy of the physical gauge coupling.

For a massless supermultiplet, put the momentum along the third axis. The matrix $2\sigma \cdot p$ in the supersymmetry algebra has rank one, so only one fermionic oscillator acts nontrivially. It relates states whose helicities differ by $\frac{1}{2}$. Including the antiparticles required by CPT completion of a supermultiplet gives the following complete spectra.

A chiral multiplet contains a complex scalar $A$ and a Weyl spinor $\psi$. Its scalar particle and antiparticle have helicity $0$; its fermion particle and antiparticle have helicities $-\frac{1}{2}$ and $+\frac{1}{2}$. Therefore $n_B=2=n_F$, with maximum absolute helicity $\frac{1}{2}$.

A supersymmetric vector multiplet contains a massless gauge vector $A_{\mu }$, with helicities $+1,-1$, and a Majorana spinor gaugino $\lambda$, with helicities $+\frac{1}{2},-\frac{1}{2}$. Again $n_B=2=n_F$, now with maximum absolute helicity $1$. The chiral $F$ and vector $D$ auxiliary fields contribute no on-shell states.

The Higgs mechanism is compatible with unbroken supersymmetry. A supersymmetric vacuum may have nonzero scalar vacuum expectation values that break the internal gauge symmetry while obeying both F-flatness and D-flatness. For example, oppositely charged chiral superfields with $W=0$, no Fayet–Iliopoulos term and equal nonzero scalar magnitudes have $F=D=0$ but break the Abelian gauge group. In the supersymmetric Higgs mechanism, a massless vector multiplet combines with the chiral multiplet along the broken gauge direction. One real scalar becomes the longitudinal vector polarization; the other remains a physical real scalar, and the gaugino combines with the Higgsino into a Dirac spinor. The resulting massive vector multiplet has $3+1=4$ bosonic and $4$ fermionic states with a common mass, as in part (i). Any other uneaten chiral multiplets remain separately in the spectrum.

Let $Y$ be the compact coordinate, identified modulo $2\pi R$. The closed string may wrap the circle:Single-valued momentum eigenfunctions $e^{ipY}$ require $p=n/R$, $n\in \mathbb{Z}$. These are the momentum and winding modes. Write the chiral zero-mode momenta asThe left-moving string oscillators will be tilded: $N_L=\tilde{N}$, $N_R=N$. This fixes the sign in compact-circle closed-string level matching.

Define the lower-dimensional mass by $M^2=-p_{\mathrm{noncompact}}^2$. The two closed-string physical-state Virasoro conditions, with bosonic string intercept one, areTheir average and difference derive the complete free bosonic string mass spectrum on the circle:The allowed states have nonnegative integer oscillator levels and obey the Virasoro constraints within each chiral sector. Compactification has not removed oscillator polarizations; it has added quantized compact momentum and allowed winding while changing the lower-dimensional interpretation of those polarizations.

At large $R$, the momentum and winding modes with $w=0$ have closely spaced momentum energies $|n|/R$, whereas nonzero winding costs $|w|R/{\alpha }^{\prime }$. At small $R$ their roles reverse. The precise equality is T-duality:It leaves $p_L$ unchanged and reverses $p_R$, and leaves the mass formula and closed-string level matching unchanged. On the string embedding map it reverses the right-moving part of $Y$. Therefore the small-radius spectrum is the large-radius spectrum of the dual theory, rather than a spectrum in which all charged states simply become heavy.

At a generic radius, the neutral level $(N_L,N_R)=(1,1)$ contains the lower-dimensional graviton, Kalb–Ramond field and dilaton, two gauge bosons from the mixed components of the metric and two-form, and the radius modulus of a string compactification. The two vector charges are the Cartan charges of $U(1{)}_L\times U(1{)}_R$.

Consider instead the following charged states:Every state satisfies compact-circle closed-string level matching. The single noncompact string oscillator supplies a vector polarization. At the massless point, the level-one Virasoro constraint imposes transverse polarization and its physical null string state identifies polarizations differing by the noncompact momentum. Thus these are genuine lower-dimensional vector states. Their squared mass isThus four additional charged massless spin-one particles appear atFor $(N_L,N_R)=(0,1)$ the charges have $p_R=0$, $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$; for $(1,0)$ they have $p_L=0$, $p_R=\pm 2/\sqrt{{\alpha }^{\prime }}$. This identifies the two pairs of charged roots.

The enhanced gauge group can be derived rather than just inferred from counting. At the self-dual circle, the chiral compact boson satisfies $Y_L(z)Y_L(u)\sim -({\alpha }^{\prime }/2)\log (z-u)$. The self-dual circle current algebra has currentsThe exponentials have conformal weight ${\alpha }^{\prime }{p}_L^2/4=1$. Free-boson contractions give the operator product expansionsThey form the level-one SU(2) Lie algebra, and there is an independent right-moving copy. The string vertex operators $J_L^a\bar{\partial }{X}^{\mu }$ and $\partial X^{\mu }{J}_R^a$ give the six massless vectors. ConsequentlyChanging $R$ gives the four charged vectors the mass found above, consistent with the radius deformation as a Higgs mechanism.

For completeness, the enhanced spectrum also has scalar fields. Using a compact rather than noncompact oscillator in the four singly excited charged families gives four scalars. At $(N_L,N_R)=(0,0)$, closed-string level matching requires $nw=0$; the mass equation then gives the four charges $(\pm 2,0),(0,\pm 2)$. These exceptional massless ground states of a bosonic circle give another four scalars. Together with the radius scalar they form nine scalars in the $(3,3)$ representation, in addition to the neutral dilaton. The ever-present neutral bosonic tachyon still has $M^2=-4/{\alpha }^{\prime }$. The enhancement establishes the additional massless vectors, not stability of the entire bosonic vacuum.

At two derivatives, specify the gauge group and matter representations, a real Kähler potential $K(\Phi ,\bar{\Phi })$, a holomorphic superpotential $W(\Phi )$ and a symmetric holomorphic gauge kinetic function $f_{ab}(\Phi )$. Gauge couplings and topological angles are contained in $f$; global theories can also have allowed Fayet–Iliopoulos terms for Abelian factors. Local theories require the gravitational coupling and consistent gauging: a constant Fayet–Iliopoulos term is not an independent addition to arbitrary supergravity data. A constant in $W$ affects supergravity although it is irrelevant in the global theory. $K$ is not holomorphic or protected; $W$ has no perturbative Wilsonian renormalization; holomorphic $f$ is perturbatively one-loop exact. Nonperturbative corrections can modify $W$ and $f$, and wave-function renormalization makes physical couplings run despite superpotential non-renormalization. Higher-derivative actions require further data.

Take $0\le \sigma <2\pi$. Variation of the conformal-gauge action, with periodic variations, gives $({\partial }_{\tau }^2-{\partial }_{\sigma }^2)X^{\mu }=0$. Hence $X^{\mu }=X_R^{\mu }(\tau -\sigma )+X_L^{\mu }(\tau +\sigma )$. In an uncompactified direction, periodicity allows equal linear slopes and independent periodic parts. Expanding the periodic parts in a Fourier series yields the closed-string mode expansionHere ${\alpha }_m$ labels the right-moving sector and ${\tilde{\alpha }}_m$ the left-moving sector; exchanging the names is harmless if done consistently. Integration of the canonical momentum density $(2\pi {\alpha }^{\prime }{)}^{-1}{\dot{X}}^{\mu }$ over the string gives $p^{\mu }$, fixing the zero-mode coefficient. Reality gives ${\alpha }_m^{†}={\alpha }_{-m}$ and the analogous relation for tildes, with ${\alpha }_0={\tilde{\alpha }}_0=\sqrt{{\alpha }^{\prime }/2}p$.

The closed-string physical-state Virasoro conditions arewhere $L_0={\alpha }^{\prime }{p}^2/4+N$ and ${\tilde{L}}_0={\alpha }^{\prime }{p}^2/4+\tilde{N}$. Thus closed-string level matching gives $N=\tilde{N}$, and $M^2=-p^2=4(N-1)/{\alpha }^{\prime }$. The massless level is $N=\tilde{N}=1$. Its general state isThe string oscillator commutators give $[L_m,{\alpha }_n^{\mu }]=-n{\alpha }_{m+n}^{\mu }$: the two terms obtained by commuting through the quadratic generator coincide and cancel its factor $1/2$. Therefore $L_1$ replaces ${\alpha }_{-1}^{\mu }$ by ${\alpha }_0^{\mu }$, while $L_{m\ge 2}$ produces positive string oscillators that annihilate this level. The same holds independently for tildes. The complete polarization conditions areNull string states identify ${\zeta }_{\mu \nu }$ with ${\zeta }_{\mu \nu }+p_{\mu }{\lambda }_{\nu }+{\tilde{\lambda }}_{\mu }{p}_{\nu }$, with transverse $\lambda ,\tilde{\lambda }$. At nonzero momentum this leaves $24\cdot 24=576$ physical polarizations. Their symmetric traceless, antisymmetric and transverse-trace parts are the graviton, Kalb–Ramond field and dilaton.

The original PDF's condition ${\alpha }^{\prime }{p}^2=4$ is inconsistent with calling the displayed excited states massless. It is the on-shell condition of the string oscillator vacuum as a physical tachyon. For the level-one state, that value would give $L_0={\tilde{L}}_0=2$, violating the physical conditions. The ket $|0;p⟩$ used as the basis for a massless excitation must instead carry the excitation's momentum $p^2=0$; by itself it is then not an on-shell physical vacuum state.

For the compact coordinate, $X^1(\tau ,\sigma +2\pi )=X^1(\tau ,\sigma )+2\pi wR$, $w\in \mathbb{Z}$. Its momentum is $n/R$, $n\in \mathbb{Z}$, since the centre-of-mass wavefunction is single-valued on the circle. Thus its zero modes becomewithThese are momentum and winding modes. Let $k$ be the 25-dimensional noncompact momentum and $M^2=-k^2$. The two zero-mode constraints now giveTheir difference and average give, respectively,This also displays T-duality $R\leftrightarrow {\alpha }^{\prime }/R$, $n\leftrightarrow w$, with one chiral momentum reversed.

At a generic radius, the massless solutions are only $n=w=0$, $N=\tilde{N}=1$. Their fields in 25 dimensions are a graviton (275 polarizations), antisymmetric two-form (253), dilaton (one), two abelian vectors from $G_{a1}$ and $B_{a1}$ (23 each), and the radius scalar (one). Hence the generic massless spectrum has $275+253+1+46+1=576$ polarizations, with gauge group $U(1{)}_L\times U(1{)}_R$. Here generic excludes isolated radii where exceptional massless ground states of a bosonic circle occur. Indeed $N=\tilde{N}=0$ forces $nw=0$, and becomes massless at $R=|n|\sqrt{{\alpha }^{\prime }}/2$ or $R=2\sqrt{{\alpha }^{\prime }}/|w|$. These additional thresholds matter in the bosonic theory.

At the self-dual circle $R=\sqrt{{\alpha }^{\prime }}$, masslessness and level matching readSince $N,\tilde{N}\ge 0$, their sum is at most two. For sum two, $n=w=0$ and level matching gives the ordinary $(1,1)$ family. For sum one the four possibilities areThey have one right-moving ${\alpha }_{-1}$ or one left-moving ${\tilde{\alpha }}_{-1}$ excitation, as their levels specify. Each family has 24 physical polarizations: a 25-dimensional vector with 23 polarizations and one compact-oscillator scalar. Thus these families add 96 states.

For sum zero, level matching forces $nw=0$, while the mass equation requires $n^2+w^2=4$. It gives four additional scalar states with charges $(\pm 2,0)$ and $(0,\pm 2)$, without any string oscillators. They must not be omitted. The massless spectrum at the bosonic self-dual circle therefore has 100 additional polarizations, 676 in total. The four new vector fields enhance the gauge group to $SU(2{)}_L\times SU(2{)}_R$. Including the original radius scalar, the nine non-dilaton scalars form its $(3,3)$ representation. Equivalently, the full count is $275+253+1+6\cdot 23+9=676$.

Ordinary electromagnetism uses a one-form gauge potential, its gauge transformation and its curvature. On superspace the corresponding object is a superspace gauge connection, with both vector and spinor components. Write ${\nabla }_A=D_A+i{\mathcal{A}}_A$. Its graded commutator contains the flat-frame torsion plus supercurvature. The conventional constraints set the two undotted-spinor curvatures, the two dotted-spinor curvatures and the conventional mixed spinor curvature to zero; the mixed derivative algebra then determines the vector connection. The remaining independent gauge curvature is packaged into a chiral field-strength superfield. A real vector superfield $V=V^{†}$ is a prepotential solving these constraints, rather than four unrelated ordinary gauge potentials.

Use the left-derivative conventions of Question 5, with $D^2=D^{\alpha }{D}_{\alpha }$ and ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$. An Abelian supergauge transformation has a chiral parameter $\Lambda$, and one consistent normalization isThe parameter is chiral so that the transformed matter is still a chiral superfield; its complex components remove more of the prepotential than an ordinary real gauge parameter. In Wess-Zumino gauge, one convenient expansion isIts components are the gauge potential $A_m$, the gaugino $\lambda$ and the real auxiliary field $D$. Wess-Zumino gauge removes the extra scalar and spinor components of a general real superfield. Supersymmetry transformations usually leave that gauge slice and require a compensating supergauge transformation. For a residual chiral parameter with real lowest component $a(x)$, the expansion gives $A_m^{\prime }=A_m+2{\partial }_{m}a$ and ${\varphi }^{\prime }=e^{-2iqa}\varphi$; thus $\alpha =2a$ is the ordinary Maxwell gauge parameter.

The Abelian field strength is the Abelian field-strength chiral projectionThree barred derivatives vanish because there are only two anticommuting dotted components, so ${\bar{D}}_{\dot{\beta }}{W}_{\alpha }=0$. Gauge invariance follows directly: $D_{\alpha }\bar{\Lambda }=0$ by antichirality, while the mixed derivative algebra gives ${\bar{D}}^2{D}_{\alpha }\Lambda =0$ for chiral $\Lambda$. In chiral coordinates its component structure iswhere ${\sigma }^{mn}=({\sigma }^m{\bar{\sigma }}^n-{\sigma }^n{\bar{\sigma }}^m)/4$ fixes the tensor normalization. Overall gaugino phase choices do not change the gauge-invariant field strength or its Bianchi constraint.

Reality of $V$ and the derivative algebra imply the superspace Yang-Mills Bianchi identityIndeed, the two sides are the same differential operator on $V$, because $D^{\alpha }{\bar{D}}^2{D}_{\alpha }={\bar{D}}_{\dot{\alpha }}{D}^2{\bar{D}}^{\dot{\alpha }}$. This is the superspace counterpart of $dF=0$, and its components contain ${\partial }_{[m}{F}_{np]}=0$. It is a defining curvature constraint, not the Maxwell equations. Varying the pure gauge action additionally sets the common real superfield $D^{\alpha }{W}_{\alpha }$ to zero; matter supplies a current instead.

The Supersymmetric Yang-Mills action and minimal matter coupling areThe chiral phase factors cancel in $\bar{\Phi }{e}^{2qV}\Phi$. The component matter derivative is $({\partial }_m+iq{A}_m)\varphi$, which transforms covariantly under the residual gauge transformation just derived. The same superspace term generates the associated matter–gaugino Yukawa coupling and $qD|\varphi {|}^2$ auxiliary coupling, so minimal coupling is not only replacing an ordinary derivative in the scalar action. The gauge action contains $-F_{mn}{F}^{mn}/(4g^2)$, the gaugino kinetic term and $D^2/(2g^2)$. Rescaling the vector multiplet to canonical kinetic normalization gives the physical coupling $qg$. The holomorphic superpotential must be gauge invariant; a single field of nonzero continuous Abelian charge admits no nonconstant polynomial superpotential without additional fields. For several fields, invariance requires ${\sum }_i{q}_i{\Phi }_i{\mathcal{W}}_i=0$. A Fayet–Iliopoulos term is additionally possible for an Abelian factor, shifting the algebraic $D$ equation.

For a non-Abelian gauge group, $V$ and $\Lambda$ are Lie-algebra-valued, and multiplication cannot be reordered. For matter in a specified representation, setThe same matter quadratic form remains invariant. Its curvature isAt first order in an Abelian $V$, this reduces to $-{\bar{D}}^2{D}_{\alpha }V/4$, fixing the factor $1/8$. In the chiral gauge frame, $\bar{\nabla }=\bar{D}$ and ${\nabla }_{\alpha }=e^{-2V}{D}_{\alpha }{e}^{2V}$ as an operator. The gauge-covariant Bianchi identity is ${\nabla }^{\alpha }{W}_{\alpha }={\bar{\nabla }}_{\dot{\alpha }}{\tilde{\bar{W}}}^{\dot{\alpha }}$, where $\tilde{\bar{W}}=e^{-2V}\bar{W}{e}^{2V}$ is the conjugate curvature expressed in that same frame. It follows from the graded Jacobi identity of the covariant derivatives. Taking a trace of $W^{\alpha }{W}_{\alpha }$ makes the chiral gauge action invariant. Its bosonic curvature becomes the ordinary Yang-Mills curvature, and the gaugino transforms in the Adjoint representation. A constant Fayet–Iliopoulos coefficient cannot be inserted for a simple non-Abelian group, since it would need an invariant linear functional annihilating every Lie bracket. These constructions connect the constrained superspace connection, its prepotential and its component gauge multiplet.