# Second quantization of SCFT and Geometric Engineering

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I'm sure I'm not using these terms in their full scope of generality, but when I say "second quantization of a CFT" I will mean in the following instance.  Given a compact Calabi-Yau manifold $X$ of complex dimension $d$, the elliptic genus

$$\text{Ell}_{q,y}(X) = \sum_{n \geq 0,l \in \mathbb{Z}} c(n,l)q^{n} y^{l}$$

is a weak Jacobi form of weight 0 and index $d/2$ which is some partition function in a SCFT with target space $X$.  The second quantization of the elliptic genus is the generating function of the elliptic genera of the symmetric products, which we can write as an infinite product by DMVV

$$\sum_{m=0}^{\infty} Q^{m} \text{Ell}_{q,y}\big(\text{Sym}^{m}(X)\big) = \prod_{m > 0, n \geq 0, l \in \mathbb{Z}}\big( 1- Q^{m}q^{n}y^{l}\big)^{-c(nm, l)}$$

Noting that the elliptic genus is an automorphic object of weight zero, the infinite product form of the DMVV formula is very much aligned with the work of Borcherds on "lifting" weight zero automorphic forms to infinite products.

By geometric engineering, I am referring to the work of Nekrasov which identifies the instanton partition function in some supersymmetric theory with the topological string partition function on a threefold.  Noting that the infinite product in the DMVV formula looks perhaps like a Donaldson-Thomas, or topological string partition function, basically my question is the following.  Does every instance of geometric engineering involve the second quantization of some weight zero object?  Perhaps some object encoding BPS numbers?  And visa versa, does every second quantization of a SCFT partition function give rise to some topological string partition function via geometric engineering?

From a purely mathematical point of view, I have an example which involves both and I'm wondering if this is interesting, or totally expected/accidental etc from a physics perspective.  Sparing the details, I have an equivariant elliptic genus whose second quantization (more precisely, the Borcherds lift as it has an extra prefactor) gives rise to Donaldson-Thomas/topological string partition function of a Calabi-Yau threefold.  The equivariant parameter $t$ in the elliptic genus is precisely the DT/GW parameter on the string theory side $t = e^{i \lambda}$.  This seems very much aligned with Nekrasov's geometric engineering picture.

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