In this section, we introduce various Hodge-Elliptic genera for K3 surfaces, following [32, 46], and discuss some of their properties. The Hodge-Elliptic genus is invariant under complex structure deformations but depends sensitively on the Kähler structure. The orbifold Hodge-Elliptic genus [32] corresponds to \(\textrm\) surfaces which are resolutions of a certain quotient, while the generic conformal field theoretic Hodge-Elliptic genus [46] is conjectured to capture the large radius conformal field theory. Both are explicitly computable. We also discuss the complex Hodge-Elliptic genus [32] and express it in closed form by using representation theory arguments. In passing we also clarify the structure of each contribution to the Elliptic genus from bundles on \(\textrm\).

3.1 Refined Partition Functions

Counting functions of BPS states can usually be refined by keeping track of additional quantum numbers. For example, the function \(1/ \varphi _\) can be interpreted as counting 1/4-BPS states in type II string compactifications on \(\textrm \times E\) with fixed electric charge and transforming in a certain spin representation of \(\textrm(2)_L\). The function \(\varphi _\) admits a refinement which was derived by Katz, Klemm and Pandharipande [36]

$$\begin \phi _ (\tau , z , \nu )=&\, q \, (y^\frac u^\frac - y^} u^}) (y^} u^\frac - y^} u^}) \nonumber \\&\quad \times \prod _ (1-q^m)^ (1-q^m u\, y) (1-q^m u \, y^) \nonumber \\&\quad \times (1-q^m y \, u^) (1 - q^m u^ y^) \, , \end$$

(3.1)

where we have introduced the new counting parameter \(u = }^\,}\nu }\). Physically \(1 / \phi _ (\tau , z , \nu )\) refines the counting of \(1/ \varphi _\) in the sense that the fugacity u keeps track of the spin representation now in \(\textrm(2)_R\). This function has also a geometrical interpretation as

$$\begin \sum _ \chi _} \left( \textsf^n \textrm \right) q^&= \frac+u^)} (\tau , z , \nu )} \nonumber \\&= \prod _^ (1-q^k)^ (1-u \,y \,q^k)^ (1-u^ y\, q^k)^ \nonumber \\&\quad \times (1- u \,y^q^k)^ (1-u^ y^q^k)^ \end$$

(3.2)

where the Hodge polynomial for a variety Y is given by

$$\begin \chi _ (Y) = u^ \, y^ \sum _ \, (-u)^q \, (-y)^p \, h^ (Y) \end$$

(3.3)

For a \(\textrm\) surface, \(\chi _} (\textrm) = 1/(u y) +y/u + 20 + u/y + y \, u\), which reduces to \(\chi _ (\textrm) = 2 \, y+20+2/y\) for \(u=1\).

Precisely as the generating function \(\varphi _\) admits the refinement \(\phi _\), also the elliptic genus and the associated generating functions can be refined. Kachru and Tripathy [32] introduce the conformal field theory Hodge-Elliptic genus as a refinement of the Elliptic genus

$$\begin \mathsf Ell}(\tau , z , \nu )= & \textrm\left( (-1)^_0} \, y^ u^_0} q^} \right) \nonumber \\= & \sum _ \, c \left( n,\ell ,m \right) q^n y^\ell u^m \, . \end$$

(3.4)

The trace is taken only over states which are arbitrary in the left-moving sector but in a Ramond ground state in the right-moving sector. Such a quantity takes its name from the fact that its first term in the q expansion is the Hodge polynomial of the \(\textrm\). It is, however, not a genus in the strict mathematical sense. (For example, it is non-vanishing on the torus.) This quantity is moduli dependent and jumps over the Calabi–Yau moduli space, for example, at those points where the left chiral CFT algebra is enhanced.

At first sight, (3.4) is similar to (2.8) and the variable u appears to play a role similar to the variable y. Therefore, it is tempting to promote it to an elliptic variable and try to interpret (3.4) as a Jacobi form with two elliptic variables. However, this symmetry is not really there, due to the fact that the trace is taken only over those states which are in a Ramond ground state in the right-moving sector. As a consequence, the range of values of the variable m in (3.4) is restricted to a finite number (we will see that \(m=-1,0,1\) with the appropriate normalization) and (3.4) is only a Laurent polynomial in u. This argument, however, does not rule out that u could transform in a more complicated way under the modular group. In this paper, we take the simplest route and assume that u is invariant under the modular group.

The Hodge-Elliptic genus can also be given a geometric definition on a variety Y using the geometric realization of the Elliptic genus as the Euler characteristic of the sheaf (2.20) [32]:

$$\begin \mathsf Ell}^c (Y ; \tau , z , \nu )&= \,}^ q^ \, y^ \, u^ \nonumber \\&\sum _^r (-u)^j \dim H^j \left( Y , \bigotimes _^ \, \Lambda _} }}\otimes \bigotimes _^ \Lambda _ q^n} }}^ \otimes \bigotimes _^\right. \nonumber \\&\quad \left. }}_ }}_X \otimes \bigotimes _^\infty }}_ \, }}^*_X \right) \, . \end$$

(3.5)

As for the Elliptic genus, we will only consider the case where Y is a \(\textrm\) surface and \(}}= }}\) its holomorphic tangent bundle. We will refer to this as the complex Hodge-Elliptic genus. In general, a direct computation of the Hodge-Elliptic genus will give different results when carried out at different points of the moduli space, as we will discuss momentarily. Remarkably in certain cases the Hodge-Elliptic genus is known in closed form. A direct computation of (3.4) was done in Kachru and Tripathy [32] at a certain orbifold point and in Wendland [46] in the strict large radius limit. We will refer to these as \(\mathsf Ell}^}\) and \(\mathsf Ell}^g\); when writing \(\mathsf Ell}\) without any further specification we refer to properties which hold for each of the Hodge-Elliptic genera described so far.

The main interest of this paper is how Hodge-Elliptic genera are related to enumerative geometry. As shown in Kachru and Tripathy [32], essentially following the argument of Dijkgraaf et al. [20] one can define a generating function

$$\begin \sum _^ p^k \, \mathsf Ell}\left( \textrm^ (\textrm) \right) & = \prod _ \frac}\nonumber \\ & = \frac (\tau , z , \nu )}} (\tau , z , \nu , \sigma ) } \, . \end$$

(3.6)

The authors of Kachru and Tripathy [32] conjecture that this \(\Phi ^} (\tau , z , \nu , \sigma )\) is the generating function of motivic/refined Donaldson–Thomas invariants. Since it is defined in terms of the Hodge-Elliptic genus, its precise form will depend on the Calabi–Yau moduli. In this note, we will study various \(\Phi ^} (\tau , z , \nu , \sigma ) \) corresponding to different version of the Hodge-Elliptic genus and use it to make prediction for enumerative invariants. These matters will be discussed in Sect. 4.

3.2 BPS Jumping Loci

The flavored partition function (3.4) is not an index and can exhibit jumping behavior at certain points in the moduli space [33, 34]. This behavior is generically different from wall-crossing and is due to the unprotected nature of the new partition function. For example, the indexed count can differ from the flavored partition function at generic points in the moduli space due to the presence of boson-fermion pairs (where the statistics refers to the fermion number appearing in the index, so typically we really are talking about different multiplets) whose contribution to the Elliptic genus cancels exactly. Additionally, an extra chiral current can appear at special points in the moduli space, leading to more state appearing (always in Bose–Fermi pairs to leave the indexed count invariant). A physical interpretation could be that for special loci in the moduli space some particles in the full spectrum are now annihilated by a certain supercharge (or a combination thereof). Another difference with wall-crossing is that jumping loci are typically of higher codimension.

For example, the moduli space of type IIA compactification on K3 surfaces has the form of a locally symmetric space

$$\begin }}(p,q) = O(p,q ; }}) \backslash O(p,q ; }}) / \left( O(p) \times O(q) \right) \end$$

(3.7)

which can be understood as the moduli space of lattices \(\Gamma ^\) of signature (p, q). By adopting this perspective, the appearance of extra chiral currents corresponds to loci in the moduli space where the lattice generated by a collection of k vectors becomes purely left moving, corresponding to subvarieties of the form \(}}(p,q-k) \subset }}(p,q)\) and called special cycles. In algebraic geometry, such loci where the rank of the lattice changes abruptly are known as Noether–Lefschetz loci, or generalization thereof, and are often Shimura varieties. Remarkably the formal generating functions of these loci, that is sums whose coefficients are special cycles, are in certain cases (mock) modular forms valued in \(H^\bullet (}}(p,q))\) [34].

3.3 The Orbifold Hodge-Elliptic Genus

In Kachru and Tripathy [32], the Hodge-Elliptic genus is computed at a certain point in the moduli space where the \(\textrm\) is a resolution of the quotient of \(}}^4\) (a product of two square tori with unit volume) by the \(}}_2\) inversion. Using CFT techniques, one finds explicitlyFootnote 1

$$\begin \mathsf Ell}^\textrm (K3)&= 8 \left[ - \left( \frac \frac-u^} \right) ^2 + \left( \frac \frac+u^} \right) ^2 \right. \nonumber \\&\left. + \left( \frac \right) ^2 + \left( \frac \right) ^2 \right] \end$$

(3.8)

with \(\theta _1^* (\tau , 0 ) = - 2 q^ \prod _^\infty (1-q^n)^3\). We can rewrite this as

$$\begin \mathsf Ell}^\textrm (K3)&= 2 \left( \frac + u - 2 \right) \left( - \left( \frac \right) ^2 + \left( \frac \right) ^2 \right) + 2 \, \varphi _ (\tau , z) \nonumber \\&= 2 \left( \frac + u - 2 \right) \left( \frac \varphi _ (\tau , z) + \left( \frac \right) ^2 \right) + 2 \, \varphi _ (\tau , z) \end$$

(3.9)

Define the functionFootnote 2\(\Lambda _N (\tau ) \in M_2 (\Gamma _0 (N))\) as

$$\begin \Lambda _N (q) = N \frac}q} \log \left( \frac\right) = \frac \left( N E_2 (N \tau )-E_2 (\tau ) \right) \end$$

(3.10)

Then, using the identity [21]

$$\begin \left( \frac \right) ^2 = \frac \, \varphi _ (\tau , z) +2 \, \Lambda _2 (\tau ) \, \varphi _ (\tau , z) \end$$

(3.11)

one can write

$$\begin \mathsf Ell}^\textrm (K3)&= \left( \frac + \frac + \frac \right) \varphi _ (\tau ,z) \nonumber \\&\quad - \left( 1 - \frac - \frac \right) \left( 1 + 8 \Lambda _2 (\tau ) \right) \varphi _ (\tau , z) \end$$

(3.12)

$$\begin&=\frac \left( \frac + 20 + 2 u \right) \textsf (\tau , z) \nonumber \\&\quad + \left( 1 - \frac - \frac \right) \frac \left( 1 + 8 \Lambda _2 (\tau ) \right) \end$$

(3.13)

which has the structure of the sum of two Jacobi forms with coefficients which are u-dependent and a weight 2 modular form on \(\Gamma _0 (2)\).

3.4 The Generic Hodge-Elliptic Genus

The generic conformal field theory Hodge-Elliptic genus corresponds to the partition function (3.4) computed in the infinite volume limit. It was computed in Wendland [46] by a careful analysis of the space of ground states and of the representation theory of the superconformal algebra. The result is

$$\begin \mathsf Ell}^g (\tau , z , \nu ) = \textsf (\tau , z ) + \left( 2 - \frac - u \right) \frac \left[ q^ - 2 \mu (\tau , z) \right] \nonumber \\ \end$$

(3.14)

where the second term on the right-hand side is proportional to the character \(\textrm_ , \frac} (\tau , z)\) of the superconformal algebra, which will, however, appear later on. The latter can also be written as the quantity \(\eta (\tau )^3 \left[ q^ - 2 \mu (\tau , z) \right] \) multiplying the Jacobi form \(\varphi _ (\tau , z) = \theta _1 (\tau , z)^2 / \eta (\tau )^6\). Therefore, also the generic Hodge-Elliptic genus has the structure of the sum of two Jacobi forms with u-dependent coefficients. However, now the u-dependent part spoils the modular properties, due to the presence of the term \(q^\) and of the mock modular form \(\mu (\tau , z)\).

Remarkably (3.14) has a geometric interpretation in terms of the chiral de Rham complex \(\Omega ^\) of \(\textrm\) introduced in Malikov [40]. Recall that the chiral de Rham complex is a sheaf of vertex operator algebras obtained by gluing together local \((b c - \beta \gamma )\)-systems. Taking the sheaf homology \(H^\bullet (\textrm , \Omega ^)\) provides a model for the sigma model Hilbert space of states. Then, it is shown in Wendland [46] that

$$\begin \mathsf Ell}^g (\tau , z , \nu ) = (y \, u)^ \sum _^2 (- u )^j \, \textrm_ , \Omega ^)} \left( (-1)^ y^ q^ J_0} \right) \nonumber \\ \end$$

(3.15)

where the combination \(L_0 - \frac J_0\) signals that the chiral de Rham complex carries the action of a topologically twisted superconformal algebra.

By using (2.10), we can write (3.14) in a form similar to (3.13)

$$\begin&\mathsf Ell}^ (\tau , z, \nu ) = \frac \left( 20 + \frac + 2 u \right) \textsf (\tau , z) \nonumber \\&\quad + \left( 2 - \frac - u \right) \frac \left[ \frac H (\tau ) + q^} \right] \, . \end$$

(3.16)

3.5 The Complex Hodge-Elliptic Genus and Representation Theory

We will now outline a procedure to compute (3.5) directly from the definition, as an expansion in q. The idea is to reduce the computation to a sum of factors which can be read of from the ordinary elliptic genus expansion, with different weights.

A \(\textrm\) surface has strict \(\mathrm \) holonomy and therefore \(\mathrm \) acts on the tangent space \(}}\). By using the splitting principle, we can write the character \(\textrm(}}) = t + 1/t\) which is then identified with the character of the fundamental representation of SU(2). In the case of SU(2) characters, one can compute explicitly the generating function

$$\begin \sum _ s^n \chi _n (t) = \frac)} \, . \end$$

(3.17)

We can think of t as a one-dimensional module under the action of the diagonal \(\textrm(1) \subset \textrm(2)\). Here \(\chi _n (t) = \frac - t^}} = \sum _^n t^\).

Now, we look explicitly at the bundle \(\mathbb \) in (2.20). It has a form of a direct sum of bundles whose coefficients are weighted by \(q^n\). The antisymmetric factors only contain a finite number of terms. We can then write

$$\begin \mathbb &= \bigotimes _^ \left( }}- y q^ }}+ y^2 q^ }}\right) \left( }}- \frac }}+ \frac}} }}\right) \nonumber \\&\quad \left( \bigoplus _ q^ }}^k (}}) \right) \left( \bigoplus _ q^ }}^l (}}) \right) \, . \end$$

(3.18)

Under the SU(2) action \(}}^k (}})\) corresponds to the character \(\chi _k\). The reason why it is useful to think in this terms is that \(H^0 (X , }}^m }}) = 0\) \(\forall m >0\), by a classic result of Kobayashi. Therefore to compute the Hodge-Elliptic genus, it is sufficient to single out the terms which admit global sections. The terms which do not admit global sections have trivial \(H^0\) and therefore by Serre’s duality trivial \(H^2\). The strategy of the computation is to use the representation theory of SU(2) to decompose a generic term \(}}^k\) into terms which admit global sections (that are given by \(}}\) and correspond to the trivial representation) and terms which do not (which have the form \( }}^m }}\) and correspond to non-trivial characters). Note that products of terms of the form \(}}^m }}\) may contain a copy of the trivial bundle in their decomposition.

In order to extract the full contribution of the trivial bundle, we will formally write

$$\begin \bigoplus _^\infty q^ }}^k }}= \frac}}q^ + q^} = \exp \left( - \log (1-}}q^ + q^) \right) \, . \end$$

(3.19)

More precisely, we formally identify \(}}\) with the character of the fundamental representation of SU(2) and interpret the above formula as a formal power series in its generator. This rewriting is convenient since now by expanding we have succeeded in writing \(\mathbb \) as a direct sum whose summands are all of the form \(}}^k\) for some \(k \in \mathbb \).

Now to extract the contribution of the trivial bundle \(}}\), we have to use repeatedly the tensor product decomposition rules, recalling that \(}}\) transforms as the character of the fundamental representation under the SU(2) action. One starts by \(}}\otimes }}= }}\oplus }}^2 }}\), where the second factor does not admit global sections and can therefore be discarded. Tensoring again by \(}}\) one gets \(}}^3 = 2 }}\oplus }}^3 }}\). Similarly, \(}}^4 = 2 }}\oplus 3 }}^2 }}\oplus }}^4 }}\). It is easy to see that any time we tensor \(}}^k\) with k odd with \(}}\), and the resulting decomposition of \(}}^\) has one \(}}\) factor whose coefficient is the same as the coefficient of the \(}}\) factor in the decomposition of \(}}^k\). Similarly, any time we tensor \(}}^n\) with n even with \(}}\), the decomposition \(}}^\) has one \(}}\) factor whose coefficient is the sum of the coefficients of \(}}\) and \(S^2 }}\) in \(}}^n\). The coefficient of \(}}^2 }}\) in \(}}^n\) is, however, the sum of the coefficients of \(}}\) and \(}}^3 }}\) in \(}}^\). All these facts follow immediately from the tensor product decomposition of products of the fundamental representation. In summary we see that

$$\begin }}^&= C_i \ }}\oplus \cdots \nonumber \\ }}^&= C_ \ }}\oplus \cdots \end$$

(3.20)

where \(i \in \mathbb \) and \(C_i = \frac\) is the \(i^\textrm\) Catalan number. The dots denote terms which are sums of factors of the form \(}}^k }}\) with \(k>1\).

We are finally ready to put all of our results together. The Hodge-elliptic genus has two contributions: the contribution from the \(}}^ }}_X\) bundles, which is equal to their contribution to the elliptic genus (the u factor cancels with the overall 1/u normalization of (3.5)) since for these bundles only \(H^1\) is non-trivial; and the contribution from the \(}}_X\) factors which by Serre duality is equal to their contribution to the elliptic genus weighted by \(\frac \left( u + \frac \right) \), since for these bundles only \(H^0 \) and \(H^2\) are non-trivial and they are weighted, respectively, by 1/u and u.

Therefore, an equivalent and perhaps simpler way of computing the Hodge-elliptic genus is to compute the ordinary elliptic genus and add the contribution of the trivial bundles weighted by \(\frac \left( u + \frac \right) -1\). We formally express this by writing

$$\begin \mathsf Ell}^c (\tau , z, \nu ) = \textsf (\tau , z ) - \left[ 1- \frac \left( u + \frac \right) \right] \textsf (\tau , z ) \bigg \vert _}}_X} \end$$

(3.21)

where \( \textsf (\tau , z ) \big \vert _}}_X}\) is the contribution to the Elliptic genus from the flat bundle \(}}_X\), and this equation is intended as an equivalence between formal power series.

This strategy can be easily implemented to compute \(\mathsf Ell}^c (\tau , z, \nu )\) as a power series in q

$$\begin&\mathsf Ell}^c (\tau , z, \nu ) = \left( u y+\frac+\frac+\frac+20 \right) \nonumber \\&\quad + q \left( u y+\frac+\frac+\frac-2 u-\frac\right. \nonumber \\&\quad \left. +20 y^2+\frac-130 y-\frac+220 \right) \nonumber \\&\quad + q^2 \left( u y^3+\frac+\frac+\frac-2 u y^2-\frac\right. \nonumber \\&\quad -\frac-\frac+4 u y+\frac+\frac+\frac \nonumber \\&\quad \left. -6 u-\frac+220 y^2+\frac-1034 y-\frac+1628 \right) \nonumber \\&\quad + q^3 \left( u y^3+\frac+\frac+\frac-6 u y^2-\frac-\frac-\frac\right. \nonumber \\&\quad +13 u y+\frac+\frac+\frac-16 u-\frac-130 y^3-\frac\nonumber \\&\quad \left. +1628 y^2+\frac-5530 y-\frac+8064 \right) + \cdots \end$$

(3.22)

We will show momentarily how to obtain \(\mathsf Ell}^c (\tau , z, \nu )\) in closed form, by finding a way to implement the above arguments systematically. Before that we have to revisit the computation of the Elliptic genus.

3.6 The Elliptic Genus Revisited

The Elliptic genus integrand is determined, up to a normalization, by the Chern character of the bundle (2.20). By using (3.19), we can write

$$\begin \frac \textrm (\mathbb ) = \frac \prod _^\infty \frac q^ \right) \left( 1 - \frac q^ \right) \left( 1 - \frac q^ \right) \left( 1 - y \, \zeta q^ \right) } \right) ^2 \left( 1 - \frac q^ \right) ^2 }\qquad \end$$

(3.23)

where we have used the splitting principle to write \(\textrm (}}) = \zeta + 1/\zeta \), where \(\zeta = }^x\) and x is the integration variable. Now, following [17] we use the following denominator formulas from Kac and Wakimoto [31] (see Example 4.1)

$$\begin \sum _}}} \frac}&= \frac \prod _^\infty \frac q^n \right) \left( 1 - \frac q^ \right) } \, q^n \right) \left( 1 - \frac \, q^n \right) \left( 1 - y \, q^ \right) }\nonumber \\ \sum _}}} \frac&= \frac \prod _^\infty \frac \right) \left( 1 - \frac q^ \right) } \, q^n \right) \left( 1 - \frac \, q^n \right) \left( 1 - y \, q^ \right) } \end$$

(3.24)

Noting that

$$\begin \theta _1^2 (\tau , z)&= - q^\frac \frac \prod _^\infty \left( 1 - q^n \right) ^2 \left( 1 - y \, q^ \right) ^2 \left( 1 - \frac q^n \right) ^2 \nonumber \\ \eta ^6 (\tau )&= q^\frac \prod _^\infty \left( 1 - q^n \right) ^6, \end$$

(3.25)

we can now write (3.23) as

$$\begin \frac \textrm (\mathbb )&= -\frac \ \frac \ \sum _}}} \frac} q^i)} = - \frac \nonumber \\&\quad \sum _}}} \sum _}}} \frac- 2 \zeta ^N+ \zeta ^} q^)} \nonumber \\&= - \frac \sum _}}} \zeta ^N (s_(\tau , z) - 2 s_(\tau , z) + s_(\tau , z)) \end$$

(3.26)

by changing summation variable \(i+j=N\). In the last step, we have introduced the functions

$$\begin s_N (\tau , z) = \frac \sum _}}} \frac q^)} \, . \end$$

(3.27)

As we have discussed, we would like to rewrite the Chern character of \(\mathbb \) in a form where the contribution from each \(S^N }}\) bundle is highlighted:

$$\begin \frac \textrm(\mathbb ) = \sum _^\infty \textrm(S^N }}) \mathscr _N (\tau , z) \end$$

(3.28)

in terms of certain functions \(\mathscr _N\) to be determined by comparing with (3.28). In order to do so note that on the right-hand side of (3.28) the coefficient of \(\zeta ^k\) is of the form \(\sum _^\infty \mathscr _\). Therefore, it follows that we can obtain \(\mathscr _N\) by taking the difference between the coefficient of \(\zeta ^N\) and the coefficient of \(\zeta ^\) in (3.26), since all the other terms cancel. Therefore,

$$\begin & \mathscr _N (\tau , z)= - \frac \nonumber \\ & \quad \left( s_ (\tau , z) - 2 s_(\tau , z) + 2 s_ (\tau , z)- s_ (\tau , z)\right) \end$$

(3.29)

In particular, now we know how to isolate the trivial character which corresponds to the contribution of the trivial bundle.

We prove in Appendix A that

$$\begin s_N (\tau , z) = \frac }_N (\tau ) + \delta _ \, \theta _1 (\tau ,z) \, \mu (\tau ,z) \end$$

(3.30)

with

$$\begin }_N (\tau ) = \frac & \text \ N=0 \, ,\\ 2 F_2^ (\tau ) & \text \ N=1 \, ,\\ \frac} & \text \end\right. } \end$$

(3.31)

Therefore, we can write the full Elliptic genus as

$$\begin \textsf (\tau , z) = \frac \int \textrm(\mathbb ) \textrm (}}) = \sum _^\infty \mathscr _N (\tau , z) \int \textrm(S^N }}) \textrm (}}) \, .\nonumber \\ \end$$

(3.32)

The integration is now elementary:

$$\begin&\int \textrm(S^N }}) \textrm (}}) = \int \frac}^ - }^}}^x - }^} \frac}^x)(1-}^)} \nonumber \\&\quad = \frac (2 N^3 + 6 N^2 + 3 N -1) \int x^2 = -2 (2 N^3 + 6 N^2 + 3 N -1) \end$$

(3.33)

where we have used (2.7).

We conclude that the Elliptic genus can be written as

$$\begin \textsf (\tau , z) =- \sum _^\infty 2 (2 N^3 + 6 N^2 + 3 N -1) \ \mathscr _N (\tau , z) \, . \end$$

(3.34)

It is convenient to rearrange terms as

$$\begin&\sum _^\infty 2 (2 N^3 + 6 N^2 + 3 N -1) \ (s_N (\tau , z) - 2 s_ (\tau , z) + 2 s_ (\tau , z) - s_ (\tau , z) ) \nonumber \\&\quad = - 2 s_0 (\tau , z) + 24 s_1 (\tau , z) + 50 s_2 (\tau , z) + 48 \sum _^\infty s_N (\tau , z) \, . \end$$

(3.35)

The first three contributions can be checked directly. All the other follow from some boring but straightforward algebra

$$\begin&\left( 2 (2 N^3 + 6 N^2 + 3 N -1) \right) - 2 \left( 2 (2 (N-1)^3 + 6 (N-1)^2 + 3 (N-1) -1) \right) \nonumber \\&\quad + 2 \left( 2 (2 (N-3)^3 + 6 (N-3)^2 + 3 (N-3) -1) \right) \nonumber \\&\quad - \left( 2 (2 (N-4)^3 + 6 (N-4)^2 + 3 (N-4) -1) \right) = 48 \, . \end$$

(3.36)

To summarize, we have shown

$$\begin \textsf (\tau , z) =&\frac\qquad \nonumber \\&\left[ \frac \left( - 2 }_0 + 24 }_1 + 50 }_2 + 48 \sum _^ }_N \right) + 24 \mu (\tau , z) \right] \, .\qquad \end$$

(3.37)

Furthermore, the combination

$$\begin - 2 }_0 + 50 }_2 + 48 \sum _^ }_N = 2 + 48 \frac + 48 \sum _^\infty \frac = - 2 E_ (\tau ) \, , \end$$

(3.38)

gives the Eisenstein series \(E_2 (\tau )\). Indeed the latter can be written asFootnote 3

$$\begin E_2&= 1 - 24 \sum _^\infty \frac = 1 - 24 \sum _^\infty \left( \frac - n \frac \right) \nonumber \\&= 1 - 24 \left( \sum _^\infty \frac + \frac \right) \nonumber \\ &= -1 -24 \sum _^\infty \frac \end$$

(3.39)

where we have used \(\sum _^ n = \zeta (-1) = - \frac\) in terms of the analytically continued Riemann zeta function \(\zeta (s)\). By putting everything together, we finally have

$$\begin \textsf (\tau , z) = \frac \left( \frac (\tau )} + 24 \mu (\tau , z) \right) \end$$

(3.40)

as expected.

The above result was obtained by using the zeta function regularization. One may wonder if this procedure introduces ambiguities in the final expression; for example, a different prescription could shift the Eisenstein function by a constant. However, such shifts are not compatible with modular invariance, which apparently forces this specific form of regularization. Nevertheless, it would be desirable to find a better argument which does not assume modular invariance, for example, by avoiding divergent expressions altogether.

3.7 The Complex Hodge-Elliptic Genus

We have just shown that

$$\begin \textsf (\tau , z)&= \frac \chi (\mathbb ) = \frac \sum _^2 (-1)^j \dim H^j (\mathbb )\nonumber \\&= \sum _^ \mathscr _N (\tau , z) \sum _^2 (-1)^j \dim H^j (S^N }}) \, . \end$$

(3.41)

Similarly, we can write for the complex Hodge-Elliptic genus

$$\begin&\mathsf Ell}^c (\tau , z , \nu ) = \frac \sum _^2 (-u)^j \dim H^j (\mathbb ) \nonumber \\ &\quad = \sum _^ \mathscr _N (\tau , z) \left( \frac \dim H^0 \left( S^N }}\right) - \dim H^1 \left( S^N }}\right) + u \dim H^2 \left( S^N }}\right) \right) \nonumber \\ &\quad = \frac \left( u + \frac \right) \mathscr _0 (\tau , z) \, \chi (}}) + \sum _^\infty \mathscr _N (\tau , z) \, \chi (S^N }}) \nonumber \\ &\quad = \textsf (\tau , z) - \left[ 1 - \frac \left( u + \frac \right) \right] \mathscr _0 (\tau , z) \, \chi (}}) \end$$

(3.42)

where we have used

$$\begin & \frac \dim H^ \left( S^ }}\right) - \dim H^ \left( S^ }}\right) + u \dim H^ \left( S^ }}\right) \nonumber \\ & \quad = \left\ \frac \left( \frac + u \right) \chi (}}) & \text N=0 \\ & \\ \chi (S^ }}) & \text N \ne 0 \end \right. \end$$

(3.43)

Now, we can compute the trivial bundle contribution

$$\begin&\left[ 1 - \frac \left( u + \frac \right) \right] \mathscr _0 (\tau , z) \, \chi (}})\nonumber \\&\quad = \left[ 1 - \frac \left( u + \frac \right) \right] \left[ \left( - 2 \frac \right) \left( \frac + \frac - \frac - 4 F_2^ (\tau ) \right) \right. \nonumber \\&\qquad \left. + 4 \frac \, \mu (\tau , z) \right] \nonumber \\&\quad = \left[ -2 + u \quad + \frac \right] \frac \left( \frac - 4 F_2^ (\tau ) \right) \nonumber \\&\qquad + \left[ 4 - 2 \left( u + \frac \right) \right] \frac \, \mu (\tau , z) \end$$

(3.44)

Finally, putting everything together we conclude that

$$\begin&\mathsf Ell}^c (\tau , z, \nu ; K3) = \frac \left[ \left( 20 + 2 \left( \frac + u \right) \right) \mu (q,y) + H(q) \right] \nonumber \\&\quad + \left( 2 - \left( \frac + u \right) \right) \frac \left( \frac - 4 F_^ (\tau )\right) \nonumber \\ \end$$

(3.45)

Equivalently, we can use the definition (2.14) to write

$$\begin&\mathsf Ell}^c (\tau , z, \nu ; K3) = \frac \left( 20 + 2 \left( u + \frac \right) \right) \textsf (\tau , z) \nonumber \\ &\quad + \left( 2 - \left( \frac + u \right) \right) \frac \left( \frac- \frac E_2 (\tau ) \right) \nonumber \\ \end$$

(3.46)

Recall that \(\mu (\tau ,z)\) and \(H (\tau )\) are mock modular forms with shadows \(- \eta (\tau )^3\) and \(24 \eta (\tau )^3\). Remarkably the u-dependence factors out in the first line and again the two shadows cancel exactly to give a Jacobi form with a u-dependent coefficient which reduces to one in the limit \(u \rightarrow 1\). In the second line, \(\varphi _ = - \theta _1^2 / \eta ^6\) is also a Jacobi form, and \(E_2 (\tau )\) is a quasi-modular form, and it transforms as a modular form of weight 2 when we add \(-3/ \pi \textrm (\tau )\). Modular properties are, however, spoiled by the presence of the rational function.

Note that the form (3.46) of the u-dependent factor \( \frac \left( 20 + 2 u + \frac \right) \) multiplying the Elliptic genus plus a correction is common to all Hodge-Elliptic genera.

3.8 Character Decomposition and Mathieu Moonshine

Among the connections between string theory and number theory, perhaps one of the most striking is the observation by Eguchi, Ooguri and Tachikawa [23] that certain coefficients in the character expansion of the Elliptic genus are twice the dimensions of certain irreducible representations of the largest Mathieu group \(\mathbb _\). We would like to investigate how this statement is modified when the Elliptic genus is refined into its Hodge-Elliptic counterpart. The main motivation in doing so is that often in string theory refined enumerative invariant helps to understand the Hilbert space of BPS statesFootnote 4 and could help in identifying the physical and geometrical reason the Mathieu group appears. We will now show directly that all the Hodge-Elliptic genera we have seen can be decomposed as a sum over the same superconformal characters as the Elliptic genus with u dependent coefficients.

The Elliptic genus admits the following decomposition [23]

$$\begin \textsf (\tau ; z) = 20 \, \textrm_,0} (\tau ; z) - 2 \, \textrm_,\frac} (\tau ; z) + \sum _^ \, c_H (n) \, \textrm_ + n , \frac} (\tau , z)\qquad \end$$

(3.47)

in terms of the characters of the superconformal algebra. These have the form

$$\begin \textrm_ (\tau , z) = \textrm_} \left( (-1)^ \, y^ q^} \right) \, , \end$$

(3.48)

where an irreducible representation \(V_\) of the \(}}=4\) algebra is labeled by the quantum numbers h and \(\ell \), the eigenvalues of \(L_0\) and \(J_0^3\), respectively. For central charge \(c=6\), the massless representations have quantum numbers \((h,\ell ) = (\frac , 0)\) and \((h,\ell ) = (\frac , \frac)\), while the massive representations have \((h,\ell ) = (\frac + n , 0)\) with \(n=1,2,\dots \). Their characters are [24,25,26]

$$\begin \textrm_,0} (\tau ; z)&= \frac \ \mu (\tau , z) \, , \nonumber \\ \textrm_,\frac} (\tau ; z)&= q^} \frac - 2 \frac \ \mu (\tau , z) \, , \nonumber \\ \textrm_ + n , \frac} (\tau ; z)&= q^ + n} \frac \, . \end$$

(3.49)

In (3.47), the coefficients \(c_H (n)\) are defined via (2.14), or equivalently

$$\begin \sum _^ \, c_H (n) \, \textrm_ + n , \frac} (\tau , z) = \frac H (\tau ) + 2 q^} \frac \, . \end$$

(3.50)

Now, let us consider the Hodge-Elliptic genera (3.46), (3.13) and (3.14). Differently from (3.47), the Hodge-Elliptic genus will involve also right-moving characters of the form

$$\begin \textrm_ (} , \nu ) = \textrm_} \left( (-1)^_0} \, u^_0} \overline^_0 - \frac}} \right) \, . \end$$

(3.51)

However, since the sum is constrained to states with \(\overline_0 = \frac}\), the net effect is that we expect a decomposition similar to that of (3.47) whose coefficients depend now on the fugacity u. In the following, we will exhibit explicitly this structure for the three Hodge-Elliptic partition functions we know in closed form. The results have the form of a refined Mathieu moonshine. It is natural to hope that the u-dependent coefficients can now be interpreted as a refined trace over the \(\mathbb _\) module which appear in the decomposition (3.47). Indeed from the geometrical point of view in the complex and generic definitions (3.5) and (3.16), the refinement consists in an extra parameter