$(d,\sigma)$-twisted Affine-Virasoro superalgebras

For any finite dimensional Lie superalgebra $\dot{\mathfrak{g}}$ (maybe a Lie algebra) with an even derivation $d$ and a finite order automorphism $\sigma$ that commutes with $d$, we introduce the $(d,\sigma)$-twisted Affine-Virasoro superalgebra $\mathfrak{L}=\mathfrak{L}(\dot{\mathfrak{g}},d,\sigma)$ and determine its universal central extension $\hat{\mathfrak{L}}=\hat{\mathfrak{L}}(\dot{\mathfrak{g}},d,\sigma)$. This is a huge class of infinite-dimensional Lie superalgebras. Such Lie superalgebras consist of many new and well-known Lie algebras and superalgebras, including the Affine-Virasoro superalgebras, the twisted Heisenberg-Virasoro algebra, the mirror Heisenberg-Virasoro algebra, the W-algebra $W(2,2)$, the gap-$p$ Virasoro algebras, the Fermion-Virasoro algebra, the $N=1$ BMS superalgebra, the planar Galilean conformal algebra. Then we give the classification of cuspidal $A\mathfrak{L}$-modules by using the weighting functor from $U(\mathfrak{h})$-free modules to weight modules. Consequently, we give the classification of simple cuspidal $\mathfrak{L}$-modules by using the $A$-cover method. Finally, all simple quasi-finite modules over $\mathfrak{L}$ and $\hat{\mathfrak{L}}$ are classified. Our results recover many known Lie superalgebra results from mathematics and mathematical physics, and give many new Lie superalgebras.