Holomorphic stress-tensor data in a plane CFT

Let $U\subseteq \mathbb{C}$ be a nonempty open patch in one fixed Euclidean coordinate chart, with dimensionless coordinate $z=x+iy$, area form $d^{2}z=dxdy$, and metric $d{x}^2+d{y}^2$. Write ${\mathrm{\partial }}_z=\frac{1}{2}({\mathrm{\partial }}_x-i{\mathrm{\partial }}_y)$ and ${\mathrm{\partial }}_{\bar{z}}=\frac{1}{2}({\mathrm{\partial }}_x+i{\mathrm{\partial }}_y)$. Use the test-function space $\mathcal{D}(U)=C_c^{\mathrm{\infty }}(U;\mathbb{C})$ with its locally convex inductive-limit topology from the compact-support seminorms $p_{K,m}(\varphi )={\max }_{\mathrm{\mid }\alpha \mathrm{\mid }\le m}{\sup }_K\mathrm{\mid }{\mathrm{\partial }}^{\alpha }\varphi \mathrm{\mid }$ for compact $K⋐U$, as specified by the primary-field convention. Fix a complex vector space $\mathcal{D}$ (the common field domain) and its algebraic dual ${\mathcal{D}}^{\ast }={\mathrm{Hom}}_{\mathbb{C}}(\mathcal{D},\mathbb{C})$. Put $\mathcal{W}(U,\mathcal{D})$ for the complex-linear maps $A:\mathcal{D}(U)\to {\mathrm{Hom}}_{\mathbb{C}}(\mathcal{D},{\mathcal{D}}^{\ast })$ for which every scalar coefficient $A_{u,v}(\varphi ):=(A(\varphi )u)(v)$, $u,v\in \mathcal{D}$, is continuous. Such a map is a weak operator-valued distribution; point evaluation need not define an operator. For $A\in \mathcal{W}(U,\mathcal{D})$ define distributional derivatives coefficientwise by $({\mathrm{\partial }}_{j}A{)}_{u,v}(\varphi ):=-A_{u,v}({\mathrm{\partial }}_j\varphi )$. If a global-plane weak coefficient extends continuously from compactly supported tests to Schwartz space, it is called tempered in the supplied sense; temperateness is not required of a local field.

A stress-tensor datum on this chart consists of components $T_{ab}\in \mathcal{W}(U,\mathcal{D})$, $a,b\in \{x,y\}$, symmetric in the sense $T_{ab}=T_{ba}$. The labels $a,b$ refer to the fixed Cartesian coordinate basis; no coordinate-transition law or global tensor-section assertion is included. For a specified closed discrete contact set $P\subset U$, let $C_b,C\in \mathcal{W}(U,\mathcal{D})$ be weak contact terms supported on $P$, meaning that each coefficient $(C_b{)}_{u,v}$ and $C_{u,v}$ vanishes on every test function supported in $U\setminus P$. Conservation and trace data mean the coefficientwise distributional identities ${\mathrm{\partial }}^a{T}_{ab}=C_b$ and ${\delta }^{ab}{T}_{ab}=C$, where ${\mathrm{\partial }}^a={\delta }^{ac}{\mathrm{\partial }}_c$ and ${\delta }^{ab}$ is the Euclidean inverse metric. Thus the components are conserved and traceless away from $P$. In complex coordinates put $T_{zz}=\frac{1}{4}(T_{xx}-T_{yy}-2i{T}_{xy})$, $T_{\bar{z}\bar{z}}=\frac{1}{4}(T_{xx}-T_{yy}+2i{T}_{xy})$, and $T_{z\bar{z}}=\frac{1}{4}(T_{xx}+T_{yy})$. Write $T(z):=T_{zz}$ and, when a barred sector is included, $\bar{T}(\bar{z}):=T_{\bar{z}\bar{z}}$. Require ${\mathrm{\partial }}_{\bar{z}}T=0$ on $U\setminus P$ coefficientwise; for an included barred sector also require ${\mathrm{\partial }}_z\bar{T}=0$ there. These are explicit regularity clauses; holomorphy is not inferred from conservation and trace without further hypotheses.

The mode structure is additional data, not a residue operation on a weak distribution. Specify an injective complex-linear map $j:\mathcal{D}\to {\mathcal{D}}^{\ast }$, and endomorphisms $t_m\in {\mathrm{End}}_{\mathbb{C}}(\mathcal{D})$ for every $m\in \mathbb{Z}$. The formal representative is the coefficient array $\mathsf{T}(z)={\sum }_{m\in \mathbb{Z}}{t}_m{z}^{-m-2}$; no sum in $\mathrm{End}(\mathcal{D})$ is intended. Its compatibility with the weak field is part of the data: for each $u,v\in \mathcal{D}$, the scalar series ${\sum }_m(j(t_{m}u))(v)z^{-m-2}$ converges locally uniformly on a specified annulus $A=\{r<\mathrm{\mid }z\mathrm{\mid }<R\}$, $0\le r<R\le \mathrm{\infty }$, to a holomorphic function admitting a specified holomorphic continuation $t_{u,v}$ to $U\setminus P$. The annulus and $U\setminus P$ lie in a specified connected continuation domain. For every $\varphi \in \mathcal{D}(U\setminus P)$ require

$(T(\varphi )u)(v)={\int }_{U\setminus P}\varphi (z)t_{u,v}(z)d^{2}z\mathrm{.}$

Thus the modes realize the same weak field away from the declared contacts; the weak extension across $P$ remains separately specified. This requires convergence only of the stated scalar one-field coefficients, not of an operator series or a general OPE tail. A barred sector supplies its own endomorphisms and the same compatibility in the independent variable $\xi =\bar{z}$. In a purely formal use of the displayed OPE conventions, one may retain only the formal representatives; such a use does not assert that weak stress data have been constructed.

Products of formal representatives mean coefficientwise composition on $\mathcal{D}$: the coefficient of $z^{-m-2}{w}^{-n-2}$ in $\mathsf{T}(z)\mathsf{T}(w)$ is $t_m{t}_n$. Any further expansion or iterated residue used below must be supplied by the radial-product prescription as a well-defined linear operation on these arrays with values in $\mathrm{End}(\mathcal{D})$. It must agree, after applying $j$ and evaluating on $v$, with the corresponding specified weak meromorphic product wherever an analytic representative is used. The same requirement applies to products with primary representatives when they are used; weak smearings into ${\mathcal{D}}^{\ast }$ are never composed. In the OPE and charge formulas below, $T(z)$ denotes $\mathsf{T}(z)$ or its specified weak analytic representative, and the scalar central term means a multiple of $I_{\mathcal{D}}$. Radial ordering means that a factor at larger $\mathrm{\mid }z\mathrm{\mid }$ is placed to the left. The notation $A(z)B(w)\sim {\sum }_{k=1}^N{C}_k(w)(z-w{)}^{-k}$ denotes only the principal part of the weak radially ordered product: it is either a formal Laurent expansion in the chosen radial region or the principal part of a meromorphic weak coefficient on a stated analytic domain. It asserts neither convergence of an infinite regular OPE tail nor equality at the coincident diagonal as distributions. The normalized residue ${\mathrm{Res}}_{z}F(z)$ is the coefficient of $z^{-1}$ in a formal Laurent series; when an analytic contour exists it equals $(2\pi i{)}^{-1}\oint F(z)dz$ on a positively oriented contour.

The holomorphic sector fixes a dimensionless central scalar $c\in \mathbb{C}$ and the principal part

A left-right datum separately fixes a dimensionless scalar $\bar{c}\in \mathbb{C}$ and the principal part

No equality between $c$ and $\bar{c}$, and no mixed $T\bar{T}$ principal part, is implicit. The chiral-only specialization omits the barred sector (equivalently, it sets $\bar{T}=0$ and $\bar{c}=0$). For each declared primary field ${\mathcal{O}}_i$ of weights $(h_i,{\bar{h}}_i)$ in the passive-coordinate convention supplied by the primary-field definition, the holomorphic singular datum is

$T(z){\mathcal{O}}_i(w_i,{\bar{w}}_i)\sim \frac{h_i{\mathcal{O}}_i(w_i,{\bar{w}}_i)}{(z-w_i{)}^2}+\frac{{\mathrm{\partial }}_{w_i}{\mathcal{O}}_i(w_i,{\bar{w}}_i)}{z-w_i}\mathrm{.}$

When a barred sector and its primary-field data are included, its singular datum is

$\bar{T}(\bar{z}){\mathcal{O}}_i(w_i,{\bar{w}}_i)\sim \frac{{\bar{h}}_i{\mathcal{O}}_i(w_i,{\bar{w}}_i)}{(\bar{z}-{\bar{w}}_i{)}^2}+\frac{{\mathrm{\partial }}_{{\bar{w}}_i}{\mathcal{O}}_i(w_i,{\bar{w}}_i)}{\bar{z}-{\bar{w}}_i}\mathrm{.}$

The holomorphic mode charges are the endomorphisms $L_m=t_m={\mathrm{Res}}_z{z}^{m+1}T(z)$ for $m\in \mathbb{Z}$; for a barred sector define ${\bar{L}}_m={\mathrm{Res}}_{\bar{z}}{\bar{z}}^{m+1}\bar{T}(\bar{z})$. A formal or analytic contour-product prescription is part of the data for each included sector. In particular, the holomorphic prescription states on the common domain that

with the nested residue taken in the specified radial region and all contact and boundary terms recorded. The barred sector has the analogous prescription when its modes are used. A stress-tensor datum is a mathematical interface, not an existence theorem: the word “CFT” alone supplies neither these fields nor their OPEs, domains, contour rules, positivity, nor convergence.

Primary field in a holomorphic coordinate

Tempered distribution

The local weak-distribution clause uses compactly supported tests. The tempered-distribution supplier is used only for the optional global-plane extension. All data above are confined to one chart; the components are not asserted to glue as an ordinary covariant tensor across conformal charts. For comparison, at the level of the usual formal or meromorphic stress-field representatives, the passive CFT law for a biholomorphic change $w=f(z)$ is $T_w(w)=(f^{\mathrm{\prime }}(z){)}^{-2}(T_z(z)-\frac{c}{12}\{f,z\}{I}_{\mathcal{D}})$, where $I_{\mathcal{D}}$ is the identity on $\mathcal{D}$ and $\{f,z\}=f^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)-\frac{3}{2}(f^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z){)}^2$; this chart-local definition neither imposes nor uses that transition law. A model may include both chiral sectors, whose central scalars remain independent, without asserting unitarity or physical realization.