**Authors:**

(1) Wahei Hara;

(2) Yuki Hirano.

## Table of Links

- Abstract and Intro
- Exchanges and Mutations of modifying modules
- Quasi-symmetric representation and GIT quotient
- Main results
- Applications to Calabi-Yau complete intersections
- Appendix A. Matrix factorizations
- Appendix B. List of Notation
- References

## 4. Main results

4.1. **Wall crossing and tilting equivalence.** This section shows that wall-crossings of magic windows correspond to equivalences that are induced by tilting modules.

Proof. By Teleman’s quantization theorem [Tel], for all k ∈ Z, the natural restriction map induces an isomorphism

*of equivalences is commutative.*

*Proof*. (1) The adjunction gives an isomorphism

Therefore we only need to prove that the right hand sides of (4.E) and (4.F) are isomorphic functors. But this follows from a natural isomorphism

**Lemma 4.8.** *Notation is same as above.*

(2) This also follows from Lemma 3.19 and the fact that µδ,δ′ is a bijection.

(3) This is a consequence of (2).

For each F ∈ F(δ,δ′)

**Theorem 4.9.** *Notation is same as above.*

This paper is available on arxiv under CC0 1.0 DEED license.