Conformal and Topological QFT Axioms¶

Job: develop the Segal axioms for 2D CFT, the vertex operator algebra (VOA) reformulation, the Atiyah–Segal TFT axioms and the Baez–Dolan–Lurie cobordism hypothesis, and the Costello–Gwilliam factorization-algebra framework — with the Ising minimal model and Dijkgraaf–Witten theory as the worked examples.

Core¶

Conformal field theory axioms (Segal)¶

A 2D CFT (Segal 1991/2004) is a symmetric monoidal functor $$ \mathcal{C}:(\mathrm{Riem}^{\mathrm{pb}},\,\coprod)\longrightarrow(\mathrm{Hilb},\,\otimes) $$ from the category of (parametrized-boundary) Riemann surfaces with disjoint union as the monoidal product, to Hilbert spaces (or topological vector spaces):

to a circle $S^1$ (a closed 1-manifold with parametrization) the functor assigns the state space $\mathcal{H}=\mathcal{C}(S^1)$;
to an interval $I$ (with parametrized endpoints) an operator $\mathcal{C}(I)\in B(\mathcal{H})$;
to a Riemann surface $\Sigma$ with parametrized boundary circles $\partial\Sigma=\coprod_i S^1_i$ a trace / amplitude $\mathcal{C}(\Sigma):\bigotimes_{\mathrm{in}}\mathcal{H}\to\bigotimes_{\mathrm{out}}\mathcal{H}$;
sewing / factorization: gluing two boundary circles of $\Sigma$ (composing the parametrized surfaces) corresponds to composing the operators / taking the trace — the functorial gluing axiom.

Segal axioms (precise). (i) Covariance under the conformal group (Möbius / $\mathrm{PSL}(2,\mathbb{R})$ on the boundary, the Virasoro algebra on the interior); (ii) vacuum: the unit $\mathbf{1}\in\mathcal{H}$ corresponding to the disk; (iii) energy positivity $L_0\ge0$ (the conformal Hamiltonian); (iv) sewing: the gluing axiom above; (v) continuity in the topology of surfaces.

Vertex operator algebras (VOAs)¶

The algebraic reformulation of a 2D chiral CFT. A VOA (Borcherds, Frenkel–Lepowsky–Meurman) is:

a $\mathbb{Z}_{\ge0}$-graded vector space $V=\bigoplus_{n\ge0}V_n$ (the state space, with $\dim V_n<\infty$);
a vacuum $|0\rangle\in V_0$;
a conformal vector $\omega\in V_2$ with $Y(\omega,z)=\sum_n L_n z^{-n-2}$ giving the Virasoro action ($[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m,-n}$, central charge $c$);
a vertex operation $Y(v,z)\in\mathrm{End}(V)[[z,z^{-1}]]$ (the state-field correspondence), satisfying:
vacuum: $Y(|0\rangle,z)=\mathrm{id}_V$, $Y(v,z)|0\rangle\in V[[z]]$ with $Y(v,z)|0\rangle|_{z=0}=v$;
locality: $Y(u,z)Y(v,w)\sim Y(v,w)Y(u,z)$ (commuting up to poles at $z=w$);
Borcherds / Jacobi identity (the vertex-algebra Jacobi relation encoding the OPE).

Theorem (Borcherds). A VOA is the chiral algebra of a (rational) CFT; the Borcherds / Frenkel–Lepowsky–Meurman VOA $V^\natural$ of the Monster CFT has the Monster group as its automorphism group, with the modular invariance of its character giving the Monster module structure.

The CFT ↔ VOA ↔ bootstrap dictionary¶

CFT object	VOA object	Bootstrap object
state-field map $v\mapsto Y(v,z)$	vertex operation $Y(v,z)$	OPE $v(z)v'(w)\sim\sum_n C_n(v,v')(z-w)^{-n}$
character $\chi_V(q)=\mathrm{Tr}_V q^{L_0-c/24}$	graded character	modular form of weight 0 (Zhu)
conformal blocks	vertex-operator correlators	bootstrap equations (crossing)
OPE coefficients $C_{ij}^k$	structure constants of $V$	the fusion / crossing data
central charge $c$	central charge of the Virasoro	the conformal anomaly

Theorem (Zhu). For a rational VOA ($V$ has finitely many irreducible modules, the fusion ring is finite), the characters $\chi_{V_i}(q)$ of the irreducible modules are modular invariant — they form a vector-valued modular form of $\mathrm{SL}_2(\mathbb{Z})$, the modular invariance of characters (the VOA proof of the CPT / modular invariance of the partition function).

The conformal bootstrap and minimal models¶

The BPZ construction (Belavin–Polyakov–Zamolodchikov) of the minimal models $\mathcal{M}(p,p')$ (coprime $p,p'\ge2$): the Virasoro algebra with central charge $$ c=1-\frac{6(p-p')^2}{pp'} $$ has degenerate representations at weights $h_{r,s}=\frac{(rp'-sp)^2-(p-p')^2}{4pp'}$; the minimal model $\mathcal{M}(p,p')$ has primary fields $\{\phi_{r,s}:1\le r\le p-1,1\le s\le p'-1\}$ with these weights and the fusion rules $$ \phi_{r,s}\times\phi_{r',s'}=\bigoplus_{r''=|r-r'|+1,\,2}^{\min(r+r'-1,\,2p-r-r'-1)}\bigoplus_{s''=|s-s'|+1,\,2}^{\min(s+s'-1,\,2p'-s-s'-1)}\phi_{r'',s''}. $$

The conformal bootstrap: the four-point function $\langle\phi_i\phi_j\phi_k\phi_l\rangle$ satisfies the crossing symmetry (the bootstrap equation) — the equality of the OPE channel $(ij)(kl)$ and $(ik)(jl)$. The bootstrap equation, combined with the spectrum (the allowed $h_{r,s}$), determines the OPE coefficients and hence the CFT. The Moore–Seiberg polynomial equations: the braiding / fusion / duality data of the conformal blocks of a rational CFT form a modular tensor category (the braiding = the statistics, the fusion = the ring, the duality = the Frobenius-Schur indicators).

Topological QFT axioms (Atiyah–Segal)¶

A $d$-dim TFT (Atiyah 1988, building on Segal) is a symmetric monoidal functor $$ Z:(\mathrm{Bord}_d,\,\sqcup)\longrightarrow(\mathrm{Vect},\,\otimes) $$ from the bordism category of $d$-manifolds (objects: closed $(d-1)$-manifolds $\Sigma$; morphisms: $d$-cobordisms $M:\Sigma_0\to\Sigma_1$; monoidal structure: disjoint union $\sqcup$) to vector spaces: - to a closed $(d-1)$-manifold $\Sigma$ a vector space $Z(\Sigma)$ (the state space); - to a cobordism $M:\Sigma_0\to\Sigma_1$ a linear map $Z(M):Z(\Sigma_0)\to Z(\Sigma_1)$; - gluing / composition: $Z(M'\circ M)=Z(M')\circ Z(M)$ (functoriality); - monoidal: $Z(\Sigma_0\sqcup\Sigma_1)=Z(\Sigma_0)\otimes Z(\Sigma_1)$, $Z(\varnothing)=\mathbb{C}$ (the unit); - closed $d$-manifold $M$ (a cobordism $\varnothing\to\varnothing$): $Z(M)\in\mathbb{C}$ (a number, the partition function).

Cobordism hypothesis (Baez–Dolan, Lurie). Fully extended TFTs — functors from the $(\infty,d)$-category $\mathrm{Bord}_d^{\mathrm{fr}}$ of framed bordisms to an $(\infty,1)$-category $\mathcal{C}$ — are classified by fully dualizable objects of $\mathcal{C}$: the value $Z$ is determined by $Z(\mathrm{pt})\in\mathcal{C}$ (a fully dualizable object). For oriented / unoriented / $G$-structured TFTs, the classification is by homotopy-fixed points of the $\mathrm{SO}(d)$ / $O(d)$ / $G$ action on the space of fully dualizable objects.

TFT type	Object classified	Examples
1D framed TFT	finite-dim vector space	the partition function = the trace
2D TFT	commutative Frobenius algebra	Dijkgraaf–Witten (finite group), Turaev–Viro
2D fully extended	fully dualizable object of $\mathcal{C}$	Dijkgraaf–Witten (the finite group $G$)
3D	modular tensor category	Reshetikhin–Turaev (quantum group), Turaev–Viro
$d$-D oriented	fully dualizable + $\mathrm{SO}(d)$-homotopy-fixed data	Dijkgraaf–Witten with $G\to\mathrm{SO}(d)$

Factorization algebras (Costello–Gwilliam)¶

The modern / derived-geometric reformulation. A (pre)factorization algebra $\mathcal{F}$ on a space $X$ assigns to each open $U\subset X$ a chain complex (or vector space) $\mathcal{F}(U)$, and to disjoint opens $U_1,\dots,U_n\subset U$ a map $$ \mu_{U_1,\dots,U_n}:\mathcal{F}(U_1)\otimes\cdots\otimes\mathcal{F}(U_n)\to\mathcal{F}(U) $$ compatible with refinement and composition (a "colored operad" structure on the opens of $X$). A factorization algebra is a prefactorization algebra satisfying descent (the local-to-global / sheaf-like condition for the factorization cover).

Theorem (Costello–Gwilliam). The factorization algebra of observables of a perturbative QFT (constructed via the BV / Batalin–Vilkovisky formalism on the derived stack of fields) recovers the operator product expansion (OPE) and the Wightman / Haag–Kastler structures in the appropriate (perturbative, Euclidean) setting. The OPE coefficients are the structure constants of the factorization algebra; the Haag–Kastler net is recovered from the factorization algebra on spacetime opens.

Framework	Object	Geometric input
AQFT (Haag–Kastler)	net $\mathcal{O}\mapsto\mathcal{A}(\mathcal{O})$ of von Neumann algebras	spacetime regions
Factorization algebra (Costello–Gwilliam)	prefactorization $\mathcal{F}$ with OPE maps	opens of $X$ (Euclidean or derived)
TFT (Atiyah–Segal)	symmetric monoidal functor $Z:\mathrm{Bord}\to\mathrm{Vect}$	bordism category
Segal CFT	functor from Riemann surfaces to Hilbert spaces	conformal surfaces

See it¶

Ising CFT minimal model $\mathcal{M}(3,4)$. The critical Ising model is the minimal model with $p=3,p'=4$, central charge $$ c=1-\frac{6(3-4)^2}{3\cdot4}=1-\frac{6}{12}=\frac{1}{2}. $$ The primary fields are $\{1=\phi_{1,1},\,\sigma=\phi_{1,2},\,\varepsilon=\phi_{2,1}\}$ with conformal weights $$ h_{1,1}=0,\quad h_{1,2}=\frac{1}{16},\quad h_{2,1}=\frac{1}{2}. $$ The fusion rules (from the BPZ formula) are $$ \sigma\times\sigma=1+\varepsilon,\qquad\sigma\times\varepsilon=\sigma,\qquad\varepsilon\times\varepsilon=1. $$ The four-point function $\langle\sigma(z_1)\sigma(z_2)\sigma(z_3)\sigma(z_4)\rangle$ satisfies the crossing / bootstrap equation: the OPE in the $(12)(34)$ channel gives $\langle\sigma\sigma\rangle\langle\sigma\sigma\rangle+\langle\sigma\sigma\rangle\langle\sigma\sigma\rangle_{\varepsilon}$ (the conformal blocks), and crossing symmetry (equality with the $(13)(24)$ channel) determines the OPE coefficient $C_{\sigma\sigma}^1=C_{\sigma\sigma}^{\varepsilon}=1/\sqrt{2}$ up to normalization. The conformal bootstrap determines the Ising model from the central charge $c=1/2$ + the spectrum $\{0,1/16,1/2\}$ — the minimal model $\mathcal{M}(3,4)$ is the unique CFT with this data.

Contrast: Dijkgraaf–Witten 2D TFT for a finite group $G$. The 2D TFT $Z_G:\mathrm{Bord}_2^{\mathrm{or}}\to\mathrm{Vect}$ assigns to a closed 1-manifold $\Sigma=\coprod S^1$ the vector space $Z_G(\coprod S^1)=\mathbb{C}[G]^{\otimes n}=\mathbb{C}[\mathrm{Hom}(\pi_1(\Sigma),G)]$ (the space of flat $G$-bundles on $\Sigma$, i.e. conjugacy classes of homomorphisms $\pi_1\to G$), and to a closed 2-manifold $M$ the partition function $$ Z_G(M)=\frac{1}{|G|}#{\text{flat }G\text{-bundles on }M}=\frac{|\mathrm{Hom}(\pi_1(M),G)|}{|G|}. $$ This is a fully extended TFT classified by the finite group $G$ as a fully dualizable object of the $(\infty,2)$-category of spans — the cobordism hypothesis identifies $G$ (with the canonical duality data) as the classifying object.

Check¶

Verify that the Ising fusion rules $\sigma\times\sigma=1+\varepsilon$ are consistent with the conformal weights $h_\sigma=1/16$, $h_\varepsilon=1/2$ via the BPZ fusion formula.

The BPZ fusion $\phi_{r,s}\times\phi_{r',s'}=\bigoplus\phi_{r'',s''}$ has $r''\in\{|r-r'|+1,\dots,\min(r+r'-1,2p-r-r'-1)\}$ step 2 and similarly for $s''$. For $\sigma=\phi_{1,2}\times\sigma=\phi_{1,2}$: $r''\in\{1,2\}\cap\{1,1\}=\{1\}$ (only $r''=1$ since $2p-r-r'-1=6-1-1=4$ but $\min(r+r'-1,2p-r-r'-1)=\min(1,4)=1$ so step 2 gives only $r''=1$); $s''\in\{|2-2|+1,\dots,\min(2+2-1,2\cdot4-2-2-1)\}=\{1,3\}$ step 2, i.e. $s''\in\{1,3\}$. So $\phi_{1,2}\times\phi_{1,2}=\phi_{1,1}+\phi_{1,3}$; but $\phi_{1,3}=\phi_{2,1}=\varepsilon$ by the minimal-model identification ($\phi_{r,s}=\phi_{p-r,p'-s}$, $\phi_{1,3}=\phi_{2,1}$), giving $\sigma\times\sigma=1+\varepsilon$ with weights $0$ and $1/2$.

Show that the Dijkgraaf–Witten 2D TFT for $G=\mathbb{Z}/2$ assigns to the torus $T^2$ the partition function $Z(T^2)=2$.

$\pi_1(T^2)=\mathbb{Z}^2$, $\mathrm{Hom}(\mathbb{Z}^2,\mathbb{Z}/2)=(\mathbb{Z}/2)^2$ has $|\mathrm{Hom}|=4$ homomorphisms; $Z(T^2)=|\mathrm{Hom}|/|G|=4/2=2$. (The two flat bundles on $T^2$ for $G=\mathbb{Z}/2$, counted up to gauge / divided by $|G|$.)

Pitfalls¶

The Segal CFT axioms are for 2D chiral / full CFT; the chiral / VOA side captures only the holomorphic sector, and the full CFT is a sum / extension of chiral sectors (the Cardy case).
The VOA ↔ CFT correspondence is for RATIONAL / chiral CFT — the general / irrational / full CFT is subtler (no finite module category, no Zhu modular invariance in the simple form).
The cobordism hypothesis classifies FULLY EXTENDED framed TFTs — the oriented / unoriented / with-structure cases are refinements (homotopy-fixed points of the $\mathrm{SO}(d)$ action); the unextended Atiyah–Segal TFT has more data (a symmetric Frobenius algebra in 2D, not just a fully dualizable object).
The Atiyah–Segal TFT is topological (no metric / conformal structure); the Segal CFT is conformal (the Virasoro / conformal group acts). Do not conflate the two.
The Dijkgraaf–Witten theory requires a finite group (or finite 2-group) for the finiteness / dualizability; infinite groups do not give fully dualizable objects directly.
The Costello–Gwilliam factorization algebra is perturbative (around a classical solution / free theory); the non-perturbative / interacting factorization algebra is the open problem (the constructive program in the factorization-algebra language).

Refs¶

Segal, "The definition of conformal field theory" (preprint 1988, in Geom., Topol. and Quantum Field Theory, 2004)
Borcherds, "Vertex algebras, Kac–Moody algebras, and the Monster" (PNAS 1986); Frenkel–Lepowsky–Meurman, Vertex Operator Algebras and the Monster (Acad. Press 1988)
Belavin–Polyakov–Zamolodchikov, "Infinite conformal symmetry in 2D QFT" (Nucl. Phys. B 1984); Di Francesco–Mathieu–Sénéchal, Conformal Field Theory
Atiyah, "Topological quantum field theories" (Publ. Math. IHÉS 1988); Lurie, "On the classification of TFTs" (2009)
Costello–Gwilliam, Factorization Algebras in QFT (Cambridge 2017); Moore–Seiberg, "Classical and quantum conformal field theory" (CMP 1989)

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