Compact Wess–Zumino–Witten model and its left-right completion

Use a connected oriented smooth Riemann surface $\mathrm{\Sigma }$ with dimensionless local coordinate $z=x+iy$, as in the chosen full local CFT data package. For the action data choose a smooth dimensionless Riemannian metric $h$ in that conformal class and a smooth field $g:\mathrm{\Sigma }\to G$, where $G$ is a finite-dimensional smooth Lie group: multiplication and inversion are smooth, $G$ is compact, connected and simply connected, and its real Lie algebra ${\mathfrak{g}}_{\mathbb{R}}=T_{e}G$ is a compact real form of the nonabelian simple complex Lie algebra $\mathfrak{g}={\mathfrak{g}}_{\mathbb{R}}{\otimes }_{\mathbb{R}}\mathbb{C}$. The Lie bracket on $T_{e}G$ is the bracket of the corresponding left-invariant vector fields. Compactness and simple connectivity here are topological properties of the group manifold. Include only fields for which the nonnegative kinetic integrand has finite integral; this is automatic for smooth fields on a closed compact worldsheet.

Let $B_0$ be the normalized invariant complex-bilinear form on $\mathfrak{g}$ used in def-residue-two-cocycle-on-a-loop-algebra and def-untwisted-affine-central-extension; its long roots have squared length $2$. Its restriction to the compact real form ${\mathfrak{g}}_{\mathbb{R}}$ is real, negative-definite by def-compact-real-form-of-a-complex-semisimple-lie-algebra, and invariant, with $B_0([x,y],z)=B_0(x,[y,z])$. Define the positive-definite real form $Q_0=-B_0{\mathrm{\mid }}_{{\mathfrak{g}}_{\mathbb{R}}}$ for the kinetic metric. Let $C_{B_0}$ be the adjoint Casimir for $B_0$ and write $C_{B_0}={\gamma }_{B_0}{I}_{\mathfrak{g}}$; the dual Coxeter number is $h^{\vee }={\gamma }_{B_0}\mathrm{/}2$, following def-dual-coxeter-number-for-normalized-form. The basic complex form $B_0$ is used for the affine current modes and Sugawara coefficient; the positive real form $Q_0$ is used in the Euclidean kinetic action.

Write ${\vartheta }_L$ for the left Maurer–Cartan form on $G$, defined at $g_0\in G$ by $({\vartheta }_L{)}_{g_0}(v)=d(L_{g_0^{-1}}{)}_{g_0}v$. For a smooth field $g$, put $J_{\mu }=(g^{\ast }{\vartheta }_L)({\mathrm{\partial }}_{\mu })$, where ${\mathrm{\partial }}_{\mu }$ is a coordinate tangent vector on $\mathrm{\Sigma }$. At positive integer level $k$, the Euclidean kinetic action is $S_{\mathrm{k}\mathrm{i}\mathrm{n}}[g;h]=\frac{k}{8\pi }{\int }_{\mathrm{\Sigma }}d{\mathrm{v}\mathrm{o}\mathrm{l}}_h{h}^{\mu \nu }{Q}_0(J_{\mu },J_{\nu })\mathrm{.}$ Here $k$, $g$, and the worldsheet coordinates are dimensionless, and $\mathrm{\hslash }=1$, so $S_{\mathrm{k}\mathrm{i}\mathrm{n}}$ is dimensionless.

For each smooth field $g:\mathrm{\Sigma }\to G$, let ${\tilde{E}}_g$ be the set of quadruples $(B,\mathrm{ȷ},c_B,\tilde{g})$ where $B$ is an oriented smooth three-manifold, possibly noncompact, $\mathrm{ȷ}:\mathrm{\Sigma }\to \mathrm{\partial }B$ is an orientation-preserving diffeomorphism, and $c_B:\mathrm{\partial }B\times [0,ϵ)\hookrightarrow B$ is a smooth boundary collar for some $ϵ>0$. A boundary collar is a smooth embedding with $c_B(p,0)=p$ and image a neighborhood of $\mathrm{\partial }B$. Require $\tilde{g}\in C^{\mathrm{\infty }}(B,G)$ to satisfy $\tilde{g}\circ \mathrm{ȷ}=g$ and $\tilde{g}(c_B(\mathrm{ȷ}(s),t))=g(s)$ for every $s\in \mathrm{\Sigma }$ and $0\le t<ϵ$.

Define the real Cartan three-form $H_{B_0}$ by its values on left-invariant vector fields: for $X,Y,Z\in {\mathfrak{g}}_{\mathbb{R}}$, $(H_{B_0}{)}_e(X^L,Y^L,Z^L)=B_0(X,[Y,Z])$. Invariance of $B_0$ makes this alternating; the Maurer–Cartan equation and the Jacobi identity make it closed, and invariance under the adjoint action makes it bi-invariant. The Wess–Zumino three-form is ${\eta }_G={\lambda }_G{H}_{B_0}$ for a scalar ${\lambda }_G\in i\mathbb{R}$. Let $I_g\subseteq {\tilde{E}}_g$ consist of the extensions for which ${\tilde{g}}^{\ast }{\eta }_G$ is absolutely integrable, meaning its associated absolute density has finite integral. The model supplies a nonempty admissible extension family $E_g\subseteq I_g$ for every included field $g$. Membership in $E_g$ is additional Wess–Zumino data; it is not defined by absolute integrability alone. For $e=(B,\mathrm{ȷ},c_B,\tilde{g})\in E_g$, put ${\mathrm{\Gamma }}_g(e)={\int }_B{\tilde{g}}^{\ast }{\eta }_G$.

Specify the family's boundary collars and, for noncompact extensions, fixed end data and identifications under which the admissible extensions agree on those ends. For each pair $e_0,e_1\in E_g$, supply compact comparison data: finitely many compact boundaryless oriented smooth three-manifolds $N_a$, smooth maps $u_a:N_a\to G$, and integers $n_a$, with the comparison identity ${\mathrm{\Gamma }}_g(e_1)-{\mathrm{\Gamma }}_g(e_0)={\sum }_a{n}_a{\int }_{N_a}{u}_a^{\ast }{\eta }_G\mathrm{.}$ The comparison is obtained by gluing along the common boundary and cancelling the identified matching ends; the displayed equality is part of the required comparison data. For compact extensions the ordinary collar gluing gives a compact closed comparison directly. A boundaryless noncompact component with a new unmatched end is not automatically admissible. Neither end cancellation nor the comparison identity follows from finite absolute integral alone.

Choose ${\lambda }_G$ and these data with the integral-period normalization ${\int }_N{u}^{\ast }{\eta }_G\in 2\pi i\mathbb{Z}$ for every compact boundaryless oriented smooth three-manifold $N$ and smooth map $u:N\to G$. Require an actual admissible pair for some included field to have difference exactly $2\pi i$. Consequently every difference within each supplied $E_g$ is in $2\pi i\mathbb{Z}$. The integral normalization, nonempty admissible families, fixed ends and primitive witness are requirements on the chosen model data, not a theorem that arbitrary finite-action fields or all absolutely integrable noncompact extensions have such data.

For each allowed smooth field choose a $U(1)$-valued phase ${\chi }_k(g)$ and, for every admissible extension $e=(B,\mathrm{ȷ},c_B,\tilde{g})$, require ${\chi }_k(g)=\exp \text{\hspace{0.17em}⁣}(-k{\int }_B{\tilde{g}}^{\ast }{\eta }_G)\mathrm{.}$ The boundary orientation is the one induced from $B$. Since ${\eta }_G$ is imaginary-valued, ${\chi }_k(g)$ lies in $U(1)$ for real $k$. By the extension-phase lemma, the displayed phase is independent of the chosen extension exactly for integral $k$. The compact model selects $k\in {\mathbb{Z}}_{>0}$. The Euclidean weight is $e^{-S_{\mathrm{k}\mathrm{i}\mathrm{n}}[g;h]}{\chi }_k(g)$; this notation specifies no measure on the space of fields.

Define the right Maurer–Cartan form by $({\vartheta }_R{)}_{g_0}(v)=d(R_{g_0^{-1}}{)}_{g_0}v$, where $R_a(g)=ga$, and retain the left Maurer–Cartan form ${\vartheta }_L$ defined above. Extend these one-forms complex-linearly when evaluating on complex tangent vectors. For $z=x+iy$, put $\mathrm{\partial }=\frac{1}{2}({\mathrm{\partial }}_x-i{\mathrm{\partial }}_y)$ and $\bar{\mathrm{\partial }}=\frac{1}{2}({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y)$. The classical left and right action-current observables on the field space $C^{\mathrm{\infty }}(\mathrm{\Sigma },G)$ are the $\mathfrak{g}$-valued maps $j_L[g](z)=-k(g^{\ast }{\vartheta }_R)(\mathrm{\partial }),j_R[g](\bar{z})=k(g^{\ast }{\vartheta }_L)(\bar{\mathrm{\partial }})\mathrm{.}$ For $x\in \mathfrak{g}$, their components are the functions ${\mathcal{J}}_L^x[g](z)=B_0(x,j_L[g](z))$ and ${\mathcal{J}}_R^x[g](\bar{z})=B_0(x,j_R[g](\bar{z}))$, using the complex-bilinear extension of $B_0$. In a matrix realization these reproduce $-k\mathrm{\partial }g{g}^{-1}$ and $k{g}^{-1}\bar{\mathrm{\partial }}g$, respectively.

Choose a full local two-dimensional CFT data package exactly as in def-cft-r5-full-local-cft-data, and denote its state space, common dense invariant domain, field-label space, state-field map, local fields, and Virasoro modes by $\mathcal{H}$, $\mathcal{D}$, $\mathcal{F}$, $\sigma$, $Y$, and $L_n^{\mathrm{l}\mathrm{o}\mathrm{c}},{\bar{L}}_n^{\mathrm{l}\mathrm{o}\mathrm{c}}$. As additional model data, choose linear assignments $Q_L,Q_R$ from the spans of $\{{\mathcal{J}}_L^x:x\in \mathfrak{g}\}$ and $\{{\mathcal{J}}_R^x:x\in \mathfrak{g}\}$ to labels $a_L^x,a_R^x\in \mathcal{F}$, linear in $x$, with $Q_L({\mathcal{J}}_L^x)=a_L^x$ and $Q_R({\mathcal{J}}_R^x)=a_R^x$. These assignments identify the classical current components with the corresponding quantum local fields. On $\mathcal{D}$, the local-field distributions and their radial mode expansions must be $Y(a_L^x;z,\bar{z})=J^x(z)={\sum }_{n\in \mathbb{Z}}{J}_n^x{z}^{-n-1},Y(a_R^x;z,\bar{z})={\bar{J}}^x(\bar{z})={\sum }_{n\in \mathbb{Z}}{\bar{J}}_n^x{\bar{z}}^{-n-1}\mathrm{.}$ The expansions have the weak distributional and radial-domain meaning required by the chosen full-local package. The correspondence from the classical current components to these local fields is adopted model data; it is not derived from a formal path integral.

In the Lie-algebra-valued affine convention, the specified current modes satisfy $[J_m^x,J_n^y]=J_{m+n}^{[x,y]}+mk{B}_0(x,y){\delta }_{m+n,0}I,$ and the antiholomorphic modes satisfy the same bracket. On the common invariant domain the left and right current modes commute, $[J_m^x,{\bar{J}}_n^y]=0$ for every $x,y\in \mathfrak{g}$ and $m,n\in \mathbb{Z}$. Use the central-generator convention $K_{\mathrm{a}\mathrm{f}\mathrm{f}}=iK$ of the stated WZW/KZ interface, with $K_{\mathrm{a}\mathrm{f}\mathrm{f}}$ acting as $kI$; this is the convention that identifies the compact-current cocycle with the normalized untwisted affine bracket. The corresponding formal current fields are $J^x(z)={\sum }_{n\in \mathbb{Z}}{J}_n^x{z}^{-n-1}$ and ${\bar{J}}^x(\bar{z})={\sum }_{n\in \mathbb{Z}}{\bar{J}}_n^x{\bar{z}}^{-n-1}$; their singular OPE is $J^x(z)J^y(w)\sim k{B}_0(x,y)I\mathrm{/}(z-w{)}^2+J^{[x,y]}(w)\mathrm{/}(z-w)$, with the analogous barred OPE. The record includes an identification of these affine fields with the displayed action-current expressions as model data. It does not derive the quantum OPE from a formal path integral.

Choose restricted affine-current modules of level $k$ for the left and right copies on the common invariant dense domain $\mathcal{D}$. In addition, supply ordinary module structures for the universal affine vertex algebra of thm-universal-affine-vertex-algebra on each left and right sector used in the state-space pairing below, in the sense of def-vertex-algebra-module. The current-generator fields of these modules must coincide with the local fields $Y(a_L^x;z,\bar{z})$ and $Y(a_R^x;z,\bar{z})$ under the assignments above, and their mode actions on $\mathcal{D}$ must be exactly $J_n^x$ and ${\bar{J}}_n^x$. Thus the module, local-field, and current-mode data are required to agree; the algebraic current bracket alone does not assert that compatibility.

For each $B_0$-orthonormal basis $(e_a)$, the Sugawara modes are $L_n=\frac{1}{2(k+h^{\vee })}{\sum }_{a,r}:J_{e_a}(r)J_{e_a}(n-r):,$ where the normal-ordering split is $r<0$ versus $r\ge 0$; define ${\bar{L}}_n$ from the barred modes in the same way. Since $k>0$ and $h^{\vee }>0$, the denominator is nonzero. On each restricted module the Sugawara theorem gives Virasoro central charge $c=k\dim \mathfrak{g}\mathrm{/}(k+h^{\vee })$. This algebraic consequence does not establish unitarity, integrability, or a left-right local theory.

The Virasoro modes $L_n^{\mathrm{l}\mathrm{o}\mathrm{c}},{\bar{L}}_n^{\mathrm{l}\mathrm{o}\mathrm{c}}$ in the chosen full-local package are required to be exactly these Sugawara modes on $\mathcal{D}$: $L_n^{\mathrm{l}\mathrm{o}\mathrm{c}}=L_n,{\bar{L}}_n^{\mathrm{l}\mathrm{o}\mathrm{c}}={\bar{L}}_n(n\in \mathbb{Z})\mathrm{.}$ In particular both package central charges equal $k\dim \mathfrak{g}\mathrm{/}(k+h^{\vee })$. The Sugawara theorem proves the mode relations under restrictedness and noncriticality; this equality with the selected full-local Virasoro actions is separate model data.

Specify the full state-space pairing as data. In the discrete case, give an index set $\mathrm{\Lambda }$ of actual left and right restricted affine modules $M_{\lambda }^L,M_{\mu }^R$, the nonnegative integer multiplicities $N_{\lambda \mu }$, the unique vacuum pair with multiplicity one, the common domain, and the topology/completion in $\mathcal{H}={\widehat{⨁}}_{\lambda ,\mu \in \mathrm{\Lambda }}({\mathbb{C}}^{N_{\lambda \mu }}\otimes M_{\lambda }^L\otimes \overline{M_{\mu }^R})\mathrm{.}$ If a different summation type is used, specify it completely. If integrability, unitarity, reflection positivity, rationality, or a particular modular invariant is claimed, supply its exact module/category and positivity data; none follows merely from compact $G$ or positive integer $k$. The selected $\mathcal{H}$, vacuum, local fields, convergent radial OPE domains, locality and crossed-channel equalities must satisfy the full local state-space/OPE package. Higher-genus sewing and factorization are separate data or theorems.

Full local two-dimensional CFT data

Restricted affine-current module

Sugawara modes and their normal ordering

Untwisted affine central extension

Residue two cocycle on a loop algebra

Compact real form of a complex semisimple Lie algebra

Dual Coxeter number for the normalized invariant form

Restricted affine-current fields are mutually local

Extension phases with period 2πi are independent exactly at integral level

Universal affine vertex algebras exist at every complex level

Module for an ordinary vertex algebra

Sugawara modes satisfy Virasoro at every noncritical complex level

Integrable affine modules and conformal-block completion contract

The Wess–Zumino phase, affine current module, and full left-right state-space/OPE data are explicit inputs of this model record. The action alone is not used to infer an affine vertex algebra, a state-space pairing, modular invariance, crossing, or a path-integral measure. The universal-affine theorem supplies only its stated chiral vertex-algebra structure, and the Sugawara theorem applies only to restricted modules at noncritical level. A KZ connection or a chiral current algebra does not by itself prove the existence of a full local WZW theory.