Four-Dimensional SCFT and Protected Observable Data

Use natural units $c=\mathrm{\hslash }=1$. The selected classical model is the rigid four-dimensional $\mathcal{N}=4$ Yang--Mills model with spacetime $M={\mathbb{R}}^{1,3}$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, and global gauge group $G=SU(2)$ rather than $SO(3)$. Its field data are one $SU(2)$ connection, six real adjoint scalars, and four adjoint Weyl fermions, with the coupling and action normalization fixed by the selected model data. The gauge coupling $g>0$ is dimensionless in four dimensions. The model's sixteen-supercharge invariance is adopted as part of that classical model; this definition does not add a proof of invariance, a quantum measure, or a continuum construction.

A four-dimensional $\mathcal{N}=4$ SCFT state-space datum used here consists of a ${\mathbb{Z}}_2$-graded complex Hilbert space $\mathcal{H}={\mathcal{H}}^{\bar{0}}\oplus {\mathcal{H}}^{\bar{1}}$ with positive inner product, a common invariant dense domain $\mathcal{D}$, and a complex-linear representation $\rho :{\mathfrak{g}}_{\mathbb{C}}\to \mathrm{End}(\mathcal{D})$ of the complexification ${\mathfrak{g}}_{\mathbb{C}}=\mathfrak{g}{\otimes }_{\mathbb{R}}\mathbb{C}$ of the real Lie superalgebra $\mathfrak{g}=\mathfrak{p}\mathfrak{s}\mathfrak{u}(2,2\mathrm{\mid }4)$, preserving parity and satisfying $\rho ([X,Y{]}_{\mathrm{s}})=[\rho (X),\rho (Y){]}_{\mathrm{s}}$ on $\mathcal{D}$. For homogeneous matrices its bracket is the graded commutator $[X,Y{]}_{\mathrm{s}}=XY-(-1{)}^{\mathrm{\mid }X\mathrm{\mid }\mathrm{\mid }Y\mathrm{\mid }}YX$. This algebra is the quotient by its central real line $\mathbb{R}i{I}_{4\mathrm{\mid }4}$ of the real form of the supertrace-zero $4\mathrm{\mid }4$ matrix Lie superalgebra preserving a Hermitian form of signature $(2,2)$ on the even block and positive signature on the odd block; for a block-diagonal matrix $A\oplus D$, the supertrace is $\mathrm{tr}A-\mathrm{tr}D$. Its even part is $\mathfrak{s}\mathfrak{u}(2,2)\oplus \mathfrak{s}\mathfrak{u}(4{)}_R$, with $\mathfrak{s}\mathfrak{u}(2,2)\cong \mathfrak{s}\mathfrak{o}(4,2)$; its odd generators are the Poincar'e supercharges $Q,\bar{Q}$ and the special superconformal charges $S,\bar{S}$. The conformal generators include translations $P_{\mu }$, Lorentz transformations $M_{\mu \nu }$, dilations $D$, and special conformal transformations $K_{\mu }$.

Use radial quantization throughout the representation-theoretic data below. Distinguish the raw anti-Hermitian compact matrix basis from physical Hermitian charges: for every real compact even generator $X$, require $\rho (X)$ to be essentially skew-adjoint and define $H_X=i\overline{\rho (X)}$. On the common invariant domain this gives $[H_X,H_Y]=i{H}_{[X,Y]}$, while the raw representation retains $[\rho (X),\rho (Y)]=\rho ([X,Y])$. The named compact Cartans $R,J_1,J_2$ below are charges $H_X$ in the selected normalizations, not raw images $\rho (X)$. The named radial operators $P_{\mu },K_{\mu },M_{\mu \nu },D,Q,\bar{Q},S,\bar{S}$ likewise denote images of a selected complex basis, with its phases and rescalings fixed by the displayed adjoint and supercharge anticommutator relations; they are not identified with anti-Hermitian real matrix generators. The closures of the generators have Hilbert adjoints $P_{\mu }^{†}=K_{\mu }$, $M_{\mu \nu }^{†}=M_{\mu \nu }$, and $D^{†}=D$; the chosen Hermitian generators of the compact $SU(4{)}_R$ and Lorentz $SU(2)$ subgroups are self-adjoint. The radial Hamiltonian is $E=D$, self-adjoint with spectrum bounded below. Thus translations $P_{\mu }$ are not asserted to be self-adjoint on the radial Hilbert space. A unitary representation here means that the inner product is positive, the generators have the common invariant dense domain just specified, and the displayed adjoint relations hold as operator equalities. A superconformal primary is a vector annihilated by all $K_{\mu }$ and all special superconformal charges; its descendants are obtained by applying $P_{\mu }$ and Poincar'e supercharges. A scalar superconformal primary is such a nonzero vector that is a simultaneous eigenvector of $D$ and $R$ and transforms trivially under both Lorentz $SU(2)$ factors, so $J_1=J_2=0$. This is only the representation-theoretic data used here; it does not define a local operator algebra or assert a continuum quantum field theory construction.

For the scalar-primary sector, use the selected $\mathcal{N}=1$ subalgebra and normalize its $U(1{)}_R$ generator $R$ so that the three adjoint chiral superfields have scalar charge $2\mathrm{/}3$. Write $d_{\mathcal{O}}$ for the eigenvalue of the dilation generator on a scalar superconformal primary $\mathrm{\mid }\mathcal{O}⟩$ and $r_{\mathcal{O}}$ for its $R$ charge. The scalar unitarity bound in this convention is $d_{\mathcal{O}}\ge 3\mathrm{\mid }{r}_{\mathcal{O}}\mathrm{\mid }\mathrm{/}2$; equality in the positive-charge branch is the chiral-primary shortening condition. The local conditional norm argument is given by the chiral-primary unitarity item on this page; no classification of all $\mathfrak{p}\mathfrak{s}\mathfrak{u}(2,2\mathrm{\mid }4)$ representations is asserted here.

For the superconformal index, use Euclidean radial quantization on a unit three-sphere, defined by writing the punctured Euclidean space as $(t,\mathrm{\Omega })=(\log \mathrm{\mid }x\mathrm{\mid },x\mathrm{/}\mathrm{\mid }x\mathrm{\mid })\in \mathbb{R}\times S^3$. Let $\mathcal{H}(S^3)$ be the resulting graded Hilbert space, $E$ its dimensionless radial Hamiltonian, and $F$ fermion parity. In this item's local index notation, $J_1$ is the Cartan of the dotted Lorentz $SU(2)$ factor and $J_2$ is the Cartan of the undotted factor; these local subscripts are opposite to the standard factor labels used on the separate right-handed-index page. Write ${\bar{Q}}_{\dot{\alpha }}^{\mathrm{s}\mathrm{t}\mathrm{d}}$ (also denoted ${\tilde{Q}}_{\dot{\alpha }}$) for standard dotted Poincaré supercharges and $Q_{\alpha }^{\mathrm{s}\mathrm{t}\mathrm{d}}$ for standard undotted ones, with $[R,{\bar{Q}}_{\dot{\alpha }}^{\mathrm{s}\mathrm{t}\mathrm{d}}]=+{\bar{Q}}_{\dot{\alpha }}^{\mathrm{s}\mathrm{t}\mathrm{d}}$ and $[R,Q_{\alpha }^{\mathrm{s}\mathrm{t}\mathrm{d}}]=-Q_{\alpha }^{\mathrm{s}\mathrm{t}\mathrm{d}}$. Select $Q_{\mathrm{c}\mathrm{h}}={\bar{Q}}_{\dot{-}}^{\mathrm{s}\mathrm{t}\mathrm{d}}\mathrm{/}\sqrt{2}$ and $Q_{\mathrm{a}\mathrm{c}}=Q_{-}^{\mathrm{s}\mathrm{t}\mathrm{d}}\mathrm{/}\sqrt{2}$, with radial Hilbert adjoints $S_{\mathrm{c}\mathrm{h}}=Q_{\mathrm{c}\mathrm{h}}^{†}$ and $S_{\mathrm{a}\mathrm{c}}=Q_{\mathrm{a}\mathrm{c}}^{†}$. The positive real rescaling fixes the factor $2$ in the anticommutators below relative to the standard component normalization. For the index-sector operator hypotheses and trace below, retain the shorthand $Q=Q_{\mathrm{c}\mathrm{h}}$ and $S=S_{\mathrm{c}\mathrm{h}}$; in particular this $Q$ is a dotted, physically barred supercharge, not an undotted $Q_{\alpha }^{\mathrm{s}\mathrm{t}\mathrm{d}}$. In this normalization the superconformal algebra gives the nonnegative operator

$\delta =2\{S,Q\}=E-2J_1-\frac{3}{2}R\mathrm{.}$

The chiral-branch component $Q_{\mathrm{c}\mathrm{h}}$ has R charge $+1$ and $2\{S_{\mathrm{c}\mathrm{h}},Q_{\mathrm{c}\mathrm{h}}\}=E-2J_1-3R\mathrm{/}2$; the opposite-chirality component $Q_{\mathrm{a}\mathrm{c}}$ has R charge $-1$ and $2\{S_{\mathrm{a}\mathrm{c}},Q_{\mathrm{a}\mathrm{c}}\}=E-2J_2+3R\mathrm{/}2$. On a scalar superconformal primary both Cartan eigenvalues are zero. Consequently positive-charge chiral shortening is $Q_{\mathrm{c}\mathrm{h}}\mathrm{\mid }\mathcal{O}⟩=0$ with $d_{\mathcal{O}}=3r_{\mathcal{O}}\mathrm{/}2$, while negative-charge anti-chiral shortening is $Q_{\mathrm{a}\mathrm{c}}\mathrm{\mid }\mathcal{O}⟩=0$ with $d_{\mathcal{O}}=-3r_{\mathcal{O}}\mathrm{/}2$; these are the two scalar-primary positivity bounds used on this page. The local index shorthand does not interchange the physical chiralities or reverse either anticommutator's R sign.

For the index sector require the closed odd operator $D_Q=\sqrt{2}(Q+S)$ to be self-adjoint with $D_Q^2=\delta$ as a closed composition equality. Require also $\delta =2(SQ+QS)$ as an equality on the full intersection of the two composition domains, not merely as an extension of an identity on test vectors. Supply an orthonormal basis of simultaneous eigenvectors of $\delta ,J_1^{\mathrm{\prime }},\dots ,J_k^{\mathrm{\prime }}$ and $(-1{)}^F$, with real energy/charge labels $(\lambda ,\mathbf{j})$, parity label $\epsilon \in \{+1,-1\}$, and maximal diagonal self-adjoint domains. The joint $(\lambda ,\mathbf{j})$ subspace includes the basis vectors of both parity labels; write $P_{\lambda ,\mathbf{j}}$ for its orthogonal projection. Here $\lambda \ge 0$, the joint energy/charge labels form a supplied countable set, and entire gap eigenspaces, including the protected kernel, may be infinite-dimensional. Require $P_{\lambda ,\mathbf{j}}D(T)\subset D(T)$ and $T{P}_{\lambda ,\mathbf{j}}\psi =P_{\lambda ,\mathbf{j}}T\psi$ for $T=Q,S,D_Q$ on $D(T)$. These combined-parity projections are the reducing projections; parity-specific projections are not required to commute with odd operators. The grading preserves each joint $(\lambda ,\mathbf{j})$ subspace and anticommutes with the odd operators on their domains. These reducing-domain conditions specify the actual spectral compatibility; common-domain commutators alone are insufficient.

For $\beta >0$ and finite real or complex chemical potentials $\mathit{\mu }$, define the weighted exponential first on the finite joint-eigenvector span by the eigenvalues $\exp (-\beta \lambda +{\sum }_a{\mu }_a{j}_a)$. Require its closure to be bounded trace class for every $\beta >0$ at the chosen $\mathit{\mu }$. This is the meaning of the exponential in the index observable, not an unspecified product of unbounded operators. Every joint block then has finite dimension: its weight is nonzero and infinitely many equal singular values would violate trace class. Infinite total gap multiplicities are permitted through infinitely many joint blocks. Define

$\mathcal{I}(\beta ,\mathit{\mu })={\mathrm{Tr}}_{\mathcal{H}(S^3)}((-1{)}^F\exp [-\beta \delta +{\sum }_{a=1}^k{\mu }_a{J}_a^{\mathrm{\prime }}])\mathrm{.}$

If the trace is not trace class, the same symbol may be used only for a specified formal series whose coefficients are defined; it does not denote a convergent partition function. The index is called protected against $\beta$-variation when the stated operator and trace hypotheses imply that positive-$\delta$ states cancel in the graded trace, as proved by the index-pair theorem on this page. A source's further claim of invariance under changing a coupling requires its own stated deformation hypotheses. A protected primary here means a named primary whose dimension is fixed by a specified shortening relation; the word alone asserts no deformation invariance.

For the quantum coupling family, use the source normalization

$\tau =\frac{\theta }{2\pi }+\frac{4\pi i}{g^2},\mathrm{Im}\tau >0,$

This is the physical component coupling family, not the formal Wilsonian coordinate ${\tau }_{\mathrm{h}}$ of the earlier zero-theta density bridge; no equality of the two parameter symbols is inferred. For the instanton-number statement, let $X$ be a nonempty compact, connected, oriented, smooth, Hausdorff, second-countable Riemannian four-manifold without boundary, with orientation fundamental class $[X]\in H_4(X;\mathbb{Z})$. Let $\pi :P\to X$ be a smooth right principal $SU(2)$ bundle in a fixed bundle sector, meaning an isomorphism class of such bundles under smooth isomorphisms over $X$. Let $\omega$ be a smooth principal connection, and choose local sections $s_i:U_i\to P$ on an open cover $\{U_i\}$ of $X$. Set $A_i=s_i^{\ast }\omega \in {\mathrm{\Omega }}^1(U_i;\mathfrak{s}\mathfrak{u}(2))$ and $F_i=d{A}_i+\frac{1}{2}[A_i\wedge A_i]\in {\mathrm{\Omega }}^2(U_i;\mathfrak{s}\mathfrak{u}(2))$. The local four-forms ${\mathrm{tr}}_2(F_i\wedge F_i)$ agree on overlaps by invariance of the defining-representation trace; denote the resulting global four-form by ${\mathrm{\Omega }}_{\mathrm{tr}}$. On $\mathfrak{s}\mathfrak{u}(2)$ use $B(U,V)=-{\mathrm{tr}}_2(UV)$ for $U,V\in \mathfrak{s}\mathfrak{u}(2)$, and let ${\mathrm{\Omega }}_B$ be the global four-form whose local expression is $B(F_i\wedge F_i)$, so ${\mathrm{\Omega }}_B=-{\mathrm{\Omega }}_{\mathrm{tr}}$. Assume full Axiom of Choice for the integral Chern-class comparison used here. For the associated Hermitian bundle $E=P{\times }_{SU(2)}{\mathbb{C}}^2$ with its standard Hermitian metric and integral second Chern class $c_2(E)\in H^4(X;\mathbb{Z})$, the supplied trace-normalized instanton-number definition gives

$k:=q_B(P,\omega )=\frac{1}{8{\pi }^2}{\int }_X{\mathrm{\Omega }}_B=-\frac{1}{8{\pi }^2}{\int }_X{\mathrm{\Omega }}_{\mathrm{tr}}=-⟨c_2(E),[X]⟩\in \mathbb{Z}\mathrm{.}$

Thus this integrality statement is for the specified $SU(2)$ bundle and defining trace normalization. The physical $\theta$ angle weights its sectors by $e^{i\theta k}$. For the opposite, positive-ASD counting convention $k_{\mathrm{A}}=+{\int }_X{\mathrm{\Omega }}_{\mathrm{tr}}\mathrm{/}(8{\pi }^2)$ on the same oriented manifold, one has $k_{\mathrm{A}}=-k$ by definition. For the Euclidean continuation on $X$, let $F_{\omega }\in {\mathrm{\Omega }}^2(X,\mathrm{ad}P)$ be the curvature represented locally by the $F_i$ above; contraction with the invariant form $B$ makes the following integral global. With real $\theta$, adopt the gauge-sector action and classical field weight

$S_E^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}[\omega ]=\frac{1}{g^2}{\int }_{X}B(F_{\omega }\wedge {\ast }_g{F}_{\omega })-i\theta k,\text{weight }{e}^{-S_E^{\mathrm{g}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{e}}[\omega ]}\mathrm{.}$

For a self-dual connection, $F_{\omega }={\ast }_g{F}_{\omega }$ and $k=(8{\pi }^2{)}^{-1}{\int }_{X}B(F_{\omega }\wedge {\ast }_g{F}_{\omega })$, so its classical action weight is $e^{-8{\pi }^{2}k\mathrm{/}{g}^2+i\theta k}=q^k$. For an anti-self-dual connection, $F_{\omega }=-{\ast }_g{F}_{\omega }$, $k=-k_{\mathrm{A}}$, and $k_{\mathrm{A}}=(8{\pi }^2{)}^{-1}{\int }_{X}B(F_{\omega }\wedge {\ast }_g{F}_{\omega })$, so its classical action weight is $e^{-8{\pi }^2{k}_{\mathrm{A}}\mathrm{/}{g}^2-i\theta k_{\mathrm{A}}}={\bar{q}}^{k_{\mathrm{A}}}$, where $q=e^{2\pi i\tau }$. These are classical action factors for the specified Euclidean fields; no measure or full partition-function contribution is asserted. A local ADHM/Nekrasov pole orientation and weight identification is separate data; it is not obtained by equating these signed charge symbols. The corresponding source-reported $S$ and $T$ parameter maps are $S:\tau \mapsto -1\mathrm{/}\tau$ and $T:\tau \mapsto \tau +1$. The global form and line-operator spectrum are part of the theory data, not determined by the Lie algebra alone. In the $\mathfrak{s}\mathfrak{u}(2)$ center-charge sector define $\mathcal{C}={\mathbb{Z}}_2\times {\mathbb{Z}}_2$ with charge $(e,m)$ and mutual-locality pairing

$⟨(e,m),(e^{\mathrm{\prime }},m^{\mathrm{\prime }})⟩=e{m}^{\mathrm{\prime }}-m{e}^{\mathrm{\prime }}(\mathrm{m}\mathrm{o}\mathrm{d}2)\mathrm{.}$

An electric Wilson line is parallel transport traced in a representation of the global gauge group; a magnetic 't Hooft line is specified by a magnetic singularity around the support curve; a dyonic line carries both charges. Their center-charge classes are $(e,m)\in {\mathbb{Z}}_2\times {\mathbb{Z}}_2$. An allowed center-charge set is a subgroup $L\subseteq \mathcal{C}$ on which the displayed pairing vanishes; the three maximal such subgroups are the line-spectrum data for $SU(2)$, $SO(3{)}_{+}$, and $SO(3{)}_{-}$. The source-reported charge maps are $S:(e,m)\mapsto (m,-e)$ and $T:(e,m)\mapsto (e+m,m)$, so these transformations may change the global form. Kapustin and Witten's 2007 source explicitly describes quantum $S$-duality as a conjecture; Aharony, Seiberg, and Tachikawa's 2013 article gives the source-reported line spectra and charge transformations, including their global-form distinctions. This dated source status is recorded without an exhaustive literature-status review, an open-problem classification, or a proof of equivalence between the theories.

David Tong, Supersymmetric Field Theory, §§2.2–2.4, 4.4.2, and 6.4.2

Kinney, Maldacena, Minwalla, and Raju, An Index for 4 Dimensional Super Conformal Theories, §2.1, §3, and §4.1

Kapustin and Witten, Electric-Magnetic Duality and the Geometric Langlands Program, §§2.1–2.2 and §4.2

Aharony, Seiberg, and Tachikawa, Reading between the Lines of Four-Dimensional Gauge Theories, §§1.1–1.2 and 2.4

Rastelli and Razamat, §2.1, equations (2.2)–(2.3)

Four-dimensional $N=4$ SU(2) super-Yang-Mills model

Trace-normalized instanton number and self-dual sectors

Full Axiom of Choice for the integral Chern-class comparison