Generalized fiber sum formula for free hypermultiplet theories

Let $X_1$ and $X_2$ be smooth, spin, closed, oriented four-manifolds, and let

$$X=X_1\mathbin{\#_f}X_2$$

be their generalized fiber sum, also smooth, spin, closed, and oriented. Let $d_1$ and $d_2$ be their TMF degrees, and let $H(X)$ denote the generator of the associated $cmathbb{Z}$-polynomial-ring summand of $cpi_*\operatorname{TMF}$. Generalized fiber sum conjecture. For a six-dimensional $(1,0)$ free hypermultiplet SCFT compactified on $X_i$, $i=1,2$, if $d_1+d_2\notin24\mathbb{Z}$, then

$$H(X_1\mathbin{\#_f}X_2)\circeq \Delta^{\left\lfloor\frac{w_1+w_2}{2}\right\rfloor}H(X_1)\ast H(X_2),$$

with products determined by

$$E_4^{p_1}\ast E_4^{p_2}=E_4^{p_1+p_2},\qquad \text{if }p_1+p_2\geq3,\ \text{use }E_4^3=j\Delta,$$ $$E_6^{w_1}\ast E_6^{w_2}=E_6^b,\qquad w_1+w_2\equiv b\pmod 2,\quad b=0,1,$$

and

$$\Delta^{m_1}\ast\Delta^{m_2}=\Delta^{m_1+m_2}.$$

Here $E_4$, $E_6$, $\Delta$, and $j$ are the standard modular-form generators and invariant, while $p_i$, $w_i$, $b$, and $m_i$ are the exponents appearing in the formula. This conjecture proposes an additive and multiplicative rule for TMF generators under generalized fiber sum, extending the observed additivity of TMF degrees. Its validity is asserted only when $d_1+d_2$ is not divisible by $24$; the source provides no resolution of the conjecture.