Snapshots of Conformal Field Theory

Abstract

In snapshots, this exposition introduces conformal field theory, with a focus on those perspectives that are relevant for interpreting superconformal field theory by Calabi-Yau geometry. It includes a detailed discussion of the elliptic genus as an invariant which certain superconformal field theories share with the Calabi-Yau manifolds. K3 theories are (re)viewed as prime examples of superconformal field theories where geometric interpretations are known. A final snapshot addresses the K3-related Mathieu Moonshine phenomena, where a lead role is predicted for the chiral de Rham complex.

Appendix—Proof of Proposition 2 in Sect. 3

The entire proof of Proposition 2 rests on the study of the $+1$-eigenspace of the linear operator $\overline{J}_0$ on the subspace $\overline{W}_{1/2}$ of the vector space $\overline{W}$ underlying the chiral algebra. First, one shows that this eigenspace is either trivial or two-dimensional, and from this one deduces claim 1. of the proposition. One direction of claim 2. is checked by direct calculation, using the defining properties of toroidal $N=(2,2)$ superconformal field theories. To obtain the converse, one shows that $\mathcal E(\tau , z)\equiv 0$ implies that an antiholomorphic counterpart of the conformal field theoretic elliptic genus must vanish as well, from which claim 2 is shown to follow.

• 1. Assume that the space $\overline{W}_{1/2}$ contains an eigenvector of $\overline{J}_0$ with eigenvalue $+1$. We denote the field associated to this state by $\overline{\psi }^+_{1}(\overline{z})$. The properties of the real structure on the space of states ${\mathbb {H}}$ of our CFT imply that there is a complex conjugate state with $\overline{J}_0$-eigenvalue $-1$ whose associated field we denote by $\overline{\psi }^-_{1}(\overline{z})$. The properties of unitary irreducible representations of the Virasoro algebra imply that these fields form a Dirac fermion [see the discussion around (15)], and that therefore $\overline{J}_1(\overline{z}):={1\over 2}:\!\!\overline{\psi }^+_{1}\overline{\psi }^-_{1}\!\!\!:\!\!(\overline{z})$ is a $U(1)$-current as in Example 1 in Sect. 2.1. By a procedure known as GKO-construction [54], one obtains $\overline{J}(\overline{z})=\overline{J}_1(\overline{z})+\overline{J}_2(\overline{z})$ for the field $\overline{J}(\overline{z})$ in the $N=2$ superconformal algebra (9)–(10), and $\overline{J}_k(\overline{z})=i\partial \overline{H}_k(\overline{z})$ with $\overline{\psi }^\pm _{1}(\overline{z})=:\!e^{\pm i\overline{H}_1}\!\!:\!\!(\overline{z})$. The fields of twofold right-handed spectral flow, which by assumption are fields of the theory, are moreover given by $\overline{J}^\pm (\overline{z})=:\!e^{\pm i(\overline{H}_1+\overline{H}_2)}\!\!:\!\!(\overline{z})$. Their OPEs with the $\overline{\psi }^\pm _{1}(\overline{z})$ yield an additional Dirac-fermion, with fields $\overline{\psi }^\pm _{2}(\overline{z}):=:\!e^{\pm i\overline{H}_2}\!\!:\!\!(\overline{z})$ in the CFT. This proves that the $\pm 1$-eigenspaces of $\overline{J}_0$ on $\overline{W}_{1/2}$ each are precisely two-dimensional, since by the same argument no further Dirac fermions can be fields of the theory. Note that by definition, the corresponding states belong to the sector $\mathbb H_f\cap \mathbb H^{NS}\subset \mathbb H$ of the space of states of our theory. In summary, the $+1$-eigenspace of the linear operator $\overline{J}_0$ on $\overline{W}_{1/2}$ is either trivial or two-dimensional.

  We now study the leading order contributions in the conformal field theoretic elliptic genus $\mathcal E(\tau ,z)$ of our theory. From (20) and by the very Definition 6 we deduce that $2ay^{-1}$ counts states in the subspace $V\subset \mathbb H^R$ where $L_0,\, \overline{L}_0$ both take eigenvalue ${c\over 24}={1\over 4}={\overline{c}\over 24}$ and $J_0$ takes eigenvalue $-1$. More precisely,

  $$ 2a=\text{ Tr }_V\left( (-1)^{J_0-\overline{J}_0} \right) =-\text{ Tr }_V\left( (-1)^{\overline{J}_0} \right) . $$

  As follows from properties of the so-called chiral ring, see e.g. [92, Sect. 3.1.1], a basis of $V$ is obtained by spectral flow $\varTheta$ (see Ingredient IV in Sect. 2.2) from (i) the vacuum $\varOmega$, (ii) the state whose corresponding field is $\overline{J}^+(\overline{z})$, and (iii) a basis of the $+1$-eigenspace of the linear operator $\overline{J}_0$ on $\overline{W}_{1/2}$. Since according to (13), the eigenvalues of $\overline{J}_0$ after spectral flow to $V$ are (i) $-1$, (ii) $+1$, (iii) $0$, the above trace vanishes if the $+1$-eigenspace of the linear operator $\overline{J}_0$ on $\overline{W}_{1/2}$ is two-dimensional, implying $2a=0$, and if this eigenspace is trivial, then we obtain $2a=2$. In conclusion, the conformal field theoretic elliptic genus of our theory either vanishes, in which case the $+1$-eigenspace of the linear operator $\overline{J}_0$ on $\overline{W}_{1/2}$ is created by two Dirac fermions, or $\mathcal E(\tau , z) = \mathcal E_\mathrm{K3}(\tau , z)$. $\square$

• 2.

  • a. Using the details of toroidal $N=(2,2)$ superconformal field theories that are summarized in Sect. 2.3, one checks by a direct calculation that the conformal field theoretic elliptic genus of all such theories vanishes. $\square$

  • b. To show the converse, first observe that in our discussion of $N=(2,2)$ superconformal field theories, the two commuting copies of a superconformal algebra are mostly treated on an equal level. However, the Definition 6 breaks this symmetry, and

    $$ \overline{\mathcal E}(\overline{\tau },\overline{z}) := \text{ Tr }_{\mathbb H^{R}} \left( (-1)^{J_0-\overline{J}_0}\overline{y}^{\overline{J}_0} q^{L_0-c/24} \overline{q}^{\overline{L}_0-\overline{c}/24}\right) $$

    should define an equally important antiholomorphic counterpart of the conformal field theoretic elliptic genus. In our case by the same reasoning as for $\mathcal E(\tau ,z)$, it must yield zero or $\overline{\mathcal E_\mathrm{K3}(\tau , z)}$. Note that Proposition 1 implies that

    $$ {\mathcal E}(\tau , z=0) = \overline{\mathcal E}(\overline{\tau },\overline{z}=0) $$

    is a constant, which in fact is known as the Witten index [95–97]. In particular, by (18) we have $\mathcal E_\mathrm{K3}(\tau ,z=0)=24$, hence $\mathcal E(\tau , z)\equiv 0$ implies $\overline{\mathcal E}(\overline{\tau },\overline{z})\equiv 0$. It remains to be shown that our theory is a toroidal theory according to Definition 5 in this case. But Step 1. of our proof then implies that the $+1$-eigenspace of the linear operator $\overline{J}_0$ on $\overline{W}_{1/2}$ is created by two Dirac fermions and that the analogous statement holds for the $+1$-eigenspace of the linear operator $J_0$ on $W_{1/2}$.

    Hence we have Dirac fermions $\psi _k^\pm (z)$ and $\overline{\psi }_k^\pm (\overline{z})$, $k\in \{1,\,2\}$, with OPEs as in (15). Compatibility with supersymmetry then implies that the superpartners of these fields yield the two $\mathfrak u(1)^4$-current algebras, as is required in order to identify our theory as a toroidal one. $\square$