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.
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Acknowledgments
It is my pleasure to thank Ron Donagi for his hospitality and for many inspiring discussions on elliptic genera and Mathieu Moonshine which have greatly influenced this work. My sincere thanks also go to Scott Carnahan, since the final touches to this exposition have benefited much from my discussions with him. Some of the material underlying this exposition arose from lecture courses that I presented at two summer schools, and I am grateful for the invitations to these events: Cordial thanks go to the organizers Alexander Cardona, Sylvie Paycha, Andrés Reyes, Hernán Ocampo and the participants of the 8th Summer School on “Geometric, Algebraic and Topological Methods for Quantum Field Theory 2013” at Villa de Leyva, Columbia, as well as the organizers Martin Schlichenmaier, Pierre Bielavsky, Harald Grosse, Ryoichi Kobayashi, Armen Sergeev, Oleg Sheinman, Weiping Zhang and the participants of the ESI School and Conference “Geometry and Quantization GEOQUANT 2013” at Vienna, Austria, for creating such inspiring events. This work is in part supported by my ERC Starting Independent Researcher Grant StG No. 204757-TQFT “The geometry of topological quantum field theories”.
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Appendix—Proof of Proposition 2 in Sect. 3
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.
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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 \(\mathcal E(\tau , z) = \mathcal E_\mathrm{K3}(\tau , z)\). \(\square \)
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2.
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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. \(\square \)
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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. \(\square \)
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a.
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Wendland, K. (2015). Snapshots of Conformal Field Theory. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_4
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