Full local two-dimensional CFT data

Fix a connected worldsheet $\mathrm{\Sigma }$: a Hausdorff, second-countable, oriented smooth real two-manifold without boundary, equipped with an atlas of charts to open subsets of $\mathbb{C}$ whose transition maps are biholomorphic and preserve the orientation. This is the Riemann-surface structure used here; it specifies a conformal structure and no metric length scale. Write each local coordinate as $z=x+iy$ with conjugate $\bar{z}$; coordinates are dimensionless.

A full local two-dimensional CFT data package in the state-space/OPE presentation is the following mathematical data satisfying the conditions below. It is one chosen axiom package, not a theorem asserting that such data exist for a given model.

1. State space and spectrum. Choose a complete Hausdorff locally convex complex vector space $\mathcal{H}$: addition and scalar multiplication are continuous, its topology is generated by specified continuous seminorms, and it is complete for the resulting uniform structure. Choose a dense subspace $\mathcal{D}\subseteq \mathcal{H}$ invariant under both Virasoro actions and all smeared fields below, so the declared operator products are defined on one common domain. Present $\mathcal{H}$ either as a completed direct sum ${\widehat{⨁}}_{\alpha \in I}{\mathcal{H}}_{\alpha }$ of declared joint left/right sectors (each ${\mathcal{H}}_{\alpha }$ is invariant under both actions), or as a specified direct integral ${\int }_X^{\oplus }{\mathcal{H}}_{x}d\mu (x)$. In the direct-sum case, give the index set, sector spaces, topology and completion. In the direct-integral case, give the measurable space $X$, the $\sigma$-finite measure $\mu$ and its normalization, the measurable field of fibers, the fiber category, and the direct-integral construction; if it is not a Hilbert direct integral, define that construction explicitly. The common dense domain, topology and any multiplicity spaces are part of the data.

2. Left and right Virasoro actions. Give representations of two copies of the Virasoro algebra on $\mathcal{D}$, with modes $L_n$ and ${\bar{L}}_n$ and scalar central charges $c_L,c_R\in \mathbb{C}$. Thus, for $m,n\in \mathbb{Z}$, $[L_m,L_n]=(m-n)L_{m+n}+\frac{c_L}{12}(m^3-m){\delta }_{m+n,0}I,[{\bar{L}}_m,{\bar{L}}_n]=(m-n){\bar{L}}_{m+n}+\frac{c_R}{12}(m^3-m){\delta }_{m+n,0}I,$ and $[L_m,{\bar{L}}_n]=0$ on $\mathcal{D}$. Positive energy means the declared spectral decomposition of each of $L_0$ and ${\bar{L}}_0$ has real weights bounded below. For a discrete spectrum, specify each joint generalized-weight space (vectors killed by some power of both $L_0-hI$ and ${\bar{L}}_0-\bar{h}I$) and every nilpotent Jordan part; for a continuous spectrum, specify the spectral resolution and the measure support. Bounded below does not mean that all weights are nonnegative.

3. Vacuum and correlator functional. Choose a nonzero vacuum $\mathrm{\Omega }\in \mathcal{D}$ and a continuous linear functional $ϵ\in {\mathcal{H}}^{\mathrm{\prime }}$ with $ϵ(\mathrm{\Omega })=1$, where ${\mathcal{H}}^{\mathrm{\prime }}$ is the continuous linear dual. Require $L_n\mathrm{\Omega }={\bar{L}}_n\mathrm{\Omega }=0$ for every integer $n\ge -1$, and require the common global-vacuum subspace $\{v\in \mathcal{D}:L_{n}v={\bar{L}}_{n}v=0\text{ for all }n\ge -1\}$ to equal $\mathbb{C}\mathrm{\Omega }$. The vacuum correlation of insertions is the matrix coefficient obtained by applying $ϵ$ to their radially ordered field product acting on $\mathrm{\Omega }$.

4. Local fields and state-field correspondence. Choose a complex vector space $\mathcal{F}$ of field labels and a bijective linear state-field map $\sigma :\mathcal{F}\to \mathcal{D}$ with an identity field $\mathbf{1}$ satisfying $\sigma (\mathbf{1})=\mathrm{\Omega }$. For each $a\in \mathcal{F}$ and coordinate chart $U\subseteq \mathrm{\Sigma }$, a field $Y(a;z,\bar{z})$ is a weak operator-valued distribution: for every test function $f\in C_c^{\mathrm{\infty }}(U)$ it defines a linear map $Y(a;f):\mathcal{D}\to \mathcal{D}$, and for every $u\in \mathcal{D}$ and $\lambda \in {\mathcal{H}}^{\mathrm{\prime }}$, the map $f\mapsto \lambda (Y(a;f)u)$ is a continuous distribution on $C_c^{\mathrm{\infty }}(U)$, the complex smooth functions of compact support, with the topology generated on each compact support by uniform seminorms of all derivatives. Smearing uses test densities: in a coordinate $z$ write a density as $f(z)d^{2}z$, where $d^{2}z=dxdy$; the identity field smeared against it is $(\int f{d}^{2}z)I$.

   A centered insertion chart is a chart $i=(U_i,z_i,p)$ with $p\in U_i$ and $z_i(p)=0$. Supply such charts at every insertion, allowing restriction to smaller neighborhoods of the center. Supply a bijective linear map ${\sigma }_i:\mathcal{F}\to \mathcal{D}$ and chart-vacuum data ${\mathrm{\Omega }}_i\in \mathcal{D}$, ${ϵ}_i\in {\mathcal{H}}^{\mathrm{\prime }}$ with ${\sigma }_i(\mathbf{1})={\mathrm{\Omega }}_i\mathrm{\ne }0$ and ${ϵ}_i({\mathrm{\Omega }}_i)=1$. One reference centered presentation has $({\sigma }_i,{\mathrm{\Omega }}_i,{ϵ}_i)=(\sigma ,\mathrm{\Omega },ϵ)$. Other presentations also carry their explicitly transported Virasoro actions and satisfy clause 3's vacuum condition for those actions. Correlators in a presentation use its vacuum and functional. Only in a centered chart require $Y_i(a;z,\bar{z}){\mathrm{\Omega }}_i$ to have a continuous vector representative near $0$, with radial limit ${\sigma }_i(a)$ in $\mathcal{H}$. There is no limit-at-zero requirement for a chart whose image misses zero.

   Finite coordinate transport. Supply the following finite data in addition to the infinitesimal identities below. Give $\mathcal{F}$ a specified Hausdorff locally convex topology. For every coordinate change $g_{ji}=z_j\circ z_i^{-1}$ on an overlap, supply a smooth family of continuous linear automorphisms $R_{ji}(z)$ of $\mathcal{F}$, fixing $\mathbf{1}$, and a continuous linear automorphism $W_{ji}$ of $\mathcal{H}$, preserving $\mathcal{D}$, with $W_{ji}{\mathrm{\Omega }}_i={\mathrm{\Omega }}_j$ and ${ϵ}_j{W}_{ji}={ϵ}_i$. Uncentered charts have the same chart-vacuum presentation data, but no creation limit at zero. Restricting a chart preserves its presentation. The reference vacuum functional need not be invariant under all finite coordinate changes: the bra as well as the fields is transported. Smoothness here means that every partial derivative in the two real coordinate variables of $R_{ji}(z)a$ exists to every finite order in $\mathcal{F}$, and that each resulting map of $(z,a)$ is jointly continuous; require the same of the inverse family. On triple overlaps require the actual finite cocycles $R_{ki}(z)=R_{kj}(g_{ji}(z))R_{ji}(z),W_{ki}=W_{kj}{W}_{ji}\mathrm{.}$ Identity transitions have $R=I,W=I$, inverse transitions have the inverse maps, and all these assignments commute with restrictions. These are normalized passive changes of presentation; no representation-integration theorem for an arbitrary locally convex space is asserted. Scalar projective lifts of $W$ may also be recorded, but their scalar cocycle must cancel on passage to these vacuum-normalized maps.

   Specify spaces $E_i(V)$ of compactly supported smooth $\mathcal{F}$-valued test densities on each coordinate open set $V$, with locally convex topologies, containing the elementary densities $af(z)d^{2}z$. Specify linear extensions ${\mathcal{Y}}_i:E_i(V)\to \mathrm{End}(\mathcal{D})$ of $af{d}^{2}z\mapsto Y_i(a;f)$, continuous in every weak matrix coefficient. Require the finite linear span of elementary densities to be dense in each $E_i(V)$, so weak continuity determines each extension uniquely. In a centered neighborhood write $V_i(a,z)=Y_i(a;z,\bar{z}){\mathrm{\Omega }}_i$ for its jointly continuous vacuum representative. For every test density $s(z)d^{2}z$ supported there and every $\lambda \in {\mathcal{H}}^{\mathrm{\prime }}$, require $\lambda ({\mathcal{Y}}_i(s){\mathrm{\Omega }}_i)=\int \lambda (V_i(s(z),z))d^{2}z\mathrm{.}$ This scalar integral identity specifies compatibility with variable labels; it is not inferred from pointwise smoothness of $R$ alone. Their restriction rule is extension of compactly supported tests by zero. Require the following pushforwards to be well-defined continuous maps between the specified test spaces, with their inverses, commuting with extension by zero: $(P_{ji}s)(w)=\mathrm{\mid }{g}_{ji}^{\mathrm{\prime }}(z){\mathrm{\mid }}^{-2}{R}_{ji}(z)s(z),z=g_{ji}^{-1}(w)\mathrm{.}$ Here $s(z)$ denotes the coefficient of $d^{2}z$; the output is the coefficient of $d^{2}w$. Require the finite overlap identity $W_{ji}{\mathcal{Y}}_i(s)W_{ji}^{-1}={\mathcal{Y}}_j(P_{ji}s)\mathrm{.}$ This is a transformation of field-valued test densities, not scalar gluing of every field component. For example a primary component can have $R_{ji}(z)a=(g_{ji}^{\mathrm{\prime }}(z){)}^h(\overline{g_{ji}^{\mathrm{\prime }}(z)}{)}^{\bar{h}}a$ with specified branches; a stress component may mix with the identity through a Schwarzian term. Spin branches must obey clause 6. All mixing and branches, including those of descendants, are actual supplied data. Along every admissible differentiable one-parameter coordinate flow for which a Virasoro generator is specified, require the derivative of this finite transport in weak matrix coefficients to equal the corresponding infinitesimal covariance identity below, with the declared coordinate and label variation. The existence and differentiability of these finite maps are required data, not inferred from a Lie algebra action.

   For two centered charts at the same insertion, require compatibility $W_{ji}{\sigma }_i(a)={\sigma }_j(R_{ji}(0)a)\mathrm{.}$ Require the vacuum representatives to be jointly continuous in the field label and coordinate near the center, so this identity is also the limit of the overlap identity on the vacuum. Supply the finite transport of OPE coefficient families and their convergence domains, so transporting either side of each OPE by the stated maps gives the same weak distribution; require the same for iterated OPEs and analytic continuation. For a polynomial $F$ of fields in one presentation put $F_j=W_{ji}{F}_i{W}_{ji}^{-1}$, using $P_{ji}$ on every smearing. Then ${ϵ}_j(F_j{\mathrm{\Omega }}_j)={ϵ}_i(F_i{\mathrm{\Omega }}_i)$. Thus local fields and correlators descend with their actual field-label transition maps and transported vacuum data. These are conditions on a package, not deductions from formal mode commutators.

   In the reference centered presentation, state-field covariance is part of the package. With $L_{j-1}a$ denoting ${\sigma }^{-1}(L_{j-1}\sigma (a))$ (and similarly for $\bar{L}$), require, in matrix coefficients on the common domain, $[L_n,Y(a;z,\bar{z})]={\sum }_{j\ge 0}\left(\genfrac{}{}{0px}{}{n+1}{j}\right)z^{n+1-j}Y(L_{j-1}a;z,\bar{z}),[{\bar{L}}_n,Y(a;z,\bar{z})]={\sum }_{j\ge 0}\left(\genfrac{}{}{0px}{}{n+1}{j}\right){\bar{z}}^{n+1-j}Y({\bar{L}}_{j-1}a;z,\bar{z})\mathrm{.}$ Here $\left(\genfrac{}{}{0px}{}{n+1}{j}\right)$ is the generalized integer binomial coefficient. Each sum must be locally finite on a field label or converge in the declared topology. In other presentations use its finite transport above; if local mode operators are written there, they are transported operators on the transported common domain. Transport includes the coordinate and field-label changes; modes in a new coordinate are its Laurent coefficients, not an assertion that identically indexed modes are merely conjugated. Require every local presentation to satisfy the same Virasoro and vacuum conditions, with its declared mode coefficients and central charges.

5. Radial OPE data and convergence. A Virasoro descendant is a state obtained by a finite word in the modes $L_n,{\bar{L}}_n$ from a state-field label. For every ordered pair of fields $a,b\in \mathcal{F}$, specify a nonempty radial domain, for example $0<\mathrm{\mid }z-w\mathrm{\mid }<R_{a,b}$ with $R_{a,b}>0$ for $Y(a;z,\bar{z})Y(b;w,\bar{w})$. On that domain specify an expansion into a declared family of local fields and descendants; no linear independence is assumed, and any nonuniqueness of coefficients is part of the data, $Y(a;z,\bar{z})Y(b;w,\bar{w})={\sum }_{\alpha \in I_{a,b}}{C}_{a,b}^{\alpha }(z-w,\bar{z}-\bar{w})Y({\varphi }_{\alpha };w,\bar{w}),$ or the corresponding explicitly normalized integral in a direct-integral model. On each declared radial domain, the weak matrix coefficients of the field products and coefficient functions must have holomorphic/antiholomorphic representatives in the complexified insertion coordinates on the specified branch or cover, whose restrictions give the Euclidean matrix coefficients. The coefficient functions, index set or integration measure, summation order, branches, and topology are data. The equality means that for every $u\in \mathcal{D}$ and $\lambda \in {\mathcal{H}}^{\mathrm{\prime }}$, the resulting matrix coefficients converge absolutely and locally uniformly on compact subsets of the declared domain, in the specified topology; they are not merely formal or asymptotic expansions. If generalized weights produce logarithmic terms, give the log powers and branch choices. For several insertions, state the nested radial domains on which each iterated expansion converges.

6. Bosonic or spin locality. Mark the package as bosonic or spin. For bosonic data, local fields have integral spin $h-\bar{h}\in \mathbb{Z}$ on each joint weight component, correlators are single-valued functions on the configuration space $\{(z_1,\dots ,z_n):z_i\mathrm{\ne }{z}_j\}$, and fields smeared with test functions of disjoint support commute. These are coordinate components: on changing charts they transform by the supplied field-label maps and transported vacuum data. Single-valuedness excludes additional monodromy in those components; it does not assert that nonzero-weight fields or stress fields are scalar functions under coordinate change. For spin data, specify a spin structure (a lift of the oriented orthonormal frame bundle, for a declared metric representative, to its double cover $\mathrm{Spin}(2)$) and a ${\mathbb{Z}}_2$ grading $p$ on fields. On each joint weight component require $e^{2\pi i(h-\bar{h})}=(-1{)}^p$; disjointly supported smeared fields supercommute with this parity. Spin correlators are sections of the declared spin bundles with their spin-structure holonomy, not unqualified scalar functions. State the parity projection. If torus amplitudes are supplied, give their spin-structure vector and modular covariance separately.

7. Crossed-channel equality. For every finite set of insertions, the vacuum correlator computed from any two iterated OPE trees must agree on their common convergence domains and after the declared analytic continuation, with the field order and parity signs fixed above. In particular, for four fields the expansions in the $(12)(34)$, $(13)(24)$, and $(14)(23)$ pairings give the same four-point correlator wherever both channel continuations are defined. The continuation paths, branches, and equality of matrix coefficients are stated as part of the data.

8. Classification flags. Record separately whether the data are unitary, reflection-positive, rational, and discrete-spectrum or continuous-spectrum; do not infer one flag from another. A unitary flag supplies a positive definite Hilbert inner product on $\mathcal{H}$, its adjoint convention, the identities $L_n^{\ast }=L_{-n}$ and ${\bar{L}}_n^{\ast }={\bar{L}}_{-n}$ on the declared domains, and field adjoints. A reflection-positive flag separately supplies a reflection $\theta$ and a test-field domain of field polynomials $F$ supported on one side of a chosen half-surface. Here $\mathrm{\Theta }$ is the antilinear order-reversing map induced by $\theta$ and the declared field adjoints, defined on the algebra generated by the two reflected polynomial domains, with ${\mathrm{\Theta }}^2=I$ and $\mathrm{\Theta }(1)=1$; it exchanges the two domains. Require the sesquilinear form $B(F,G)=ϵ(\mathrm{\Theta }(F)G\mathrm{\Omega })$ to be Hermitian and positive semidefinite: $B(F,G)=\overline{B(G,F)}$ and $Q(F)=B(F,F)\in [0,\mathrm{\infty })$ for all polynomials in this domain. The polynomial domain is a complex vector space containing the identity, each polynomial acts on $\mathcal{D}$, and reflection sends it to polynomials on the opposite side, so every displayed product on $\mathrm{\Omega }$ is defined. If rationality is claimed, name the chiral algebra and module category; here “rational” means that this declared category is semisimple with finitely many irreducible isomorphism classes. A discrete spectrum uses the direct-sum presentation; a continuous spectrum uses a specified direct integral. A nonrational discrete spectrum is allowed.

This package specifies genus-zero local fields and their vacuum correlators. It does not include higher-genus sewing or factorization. A torus amplitude is not assumed: if a trace over $\mathcal{H}$ is used, specify the trace-class or regularized convergence domain. Modular invariance is a separate property and does not follow from this definition.

Virasoro algebra in the $L_m$ convention

The convergence, locality and crossing conditions above are assumptions in this chosen data package. They are not consequences of a formal vertex algebra, Virasoro symmetry, or a modular-invariant character sum. This definition makes no equivalence claim with Osterwalder–Schrader, conformal-net, or Segal formulations and does not identify a formal path integral with a measure. It is a mathematical interface for model data, not a physical action or a claim that any specified model is realized in nature.