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On cluster theory and quantum dilogarithm identities

Bernhard Keller

TL;DR

This survey unifies quantum dilogarithm identities, cluster algebras, and Donaldson-Thomas theory by tracing identities from Dynkin quivers to quivers with potential. It develops a mutation-compatible framework: stability data yield Z-independent products for Dynkin cases (Reineke identities), which extend to refined DT-invariants for quivers with potential, and are transported across mutations via intertwiners and derived-category equivalences. A groupoid formalism for compositions of mutations (cluster collections and tilting sequences) shows that the induced automorphisms on quantum tori depend only on the target collection, with a tropical groupoid offering a combinatorial realization. The work connects to periodicity phenomena in Y-/T-systems, higher Teichmüller theory, and string-theoretic contexts, providing both categorical and combinatorial tools to compute and understand wall-crossing and mutation invariants. Overall, it provides a cohesive framework linking algebraic, geometric, and physical perspectives on quantum dilogarithm identities and cluster dynamics.

Abstract

These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin-Zelevinsky's theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock-Goncharov, Kontsevich-Soibelman and Nagao.

On cluster theory and quantum dilogarithm identities

TL;DR

This survey unifies quantum dilogarithm identities, cluster algebras, and Donaldson-Thomas theory by tracing identities from Dynkin quivers to quivers with potential. It develops a mutation-compatible framework: stability data yield Z-independent products for Dynkin cases (Reineke identities), which extend to refined DT-invariants for quivers with potential, and are transported across mutations via intertwiners and derived-category equivalences. A groupoid formalism for compositions of mutations (cluster collections and tilting sequences) shows that the induced automorphisms on quantum tori depend only on the target collection, with a tropical groupoid offering a combinatorial realization. The work connects to periodicity phenomena in Y-/T-systems, higher Teichmüller theory, and string-theoretic contexts, providing both categorical and combinatorial tools to compute and understand wall-crossing and mutation invariants. Overall, it provides a cohesive framework linking algebraic, geometric, and physical perspectives on quantum dilogarithm identities and cluster dynamics.

Abstract

These are expanded notes from three survey lectures given at the 14th International Conference on Representations of Algebras (ICRA XIV) held in Tokyo in August 2010. We first study identities between products of quantum dilogarithm series associated with Dynkin quivers following Reineke. We then examine similar identities for quivers with potential and link them to Fomin-Zelevinsky's theory of cluster algebras. Here we mainly follow ideas due to Bridgeland, Fock-Goncharov, Kontsevich-Soibelman and Nagao.

Paper Structure

This paper contains 24 sections, 17 theorems, 98 equations, 5 figures.

Table of Contents

  1. Quantum dilogarithm identities from Dynkin quivers, after Reineke
  2. The pentagon identity
  3. Reineke's identities
  4. The Kronecker quiver
  5. Sketch of proof of Reineke's theorem
  6. Quantum dilogarithm identities from quivers with potential, after Kontsevich-Soibelman
  7. DT-invariants for quivers with potential
  8. Remark on the scaling factor
  9. DT-invariants and mutations
  10. The comparison problem
  11. The setup
  12. Comparison of categories
  13. Comparison of the invariants
  14. The rational case
  15. The interwiners
  16. ...and 9 more sections

Key Result

Theorem 1.2

For two indeterminates $y_1$ and $y_2$ which $q$-commute in the sense that we have the equality

Figures (5)

  • Figure 1: Image of a non zero object under the central charge
  • Figure 2: Two different generic stability functions for $A_2$
  • Figure 3: Right and left mutation of a heart
  • Figure 4: Stable objects in a heart and its right and left mutations
  • Figure 5: The two maximal green sequences for $A_2$

Theorems & Definitions (19)

  • Theorem 1.2: Schützenberger Schuetzenberger53, Faddeev-Volkov FaddeevVolkov93, Faddeev-Kashaev FaddeevKashaev94
  • Proposition 1.4: King King94
  • Definition 1.5
  • Theorem 1.6: Reineke Reineke10
  • Corollary 1.7
  • Lemma 2.1
  • Conjecture 3.2
  • Lemma 4.6
  • Lemma 4.9
  • Theorem 5.2
  • ...and 9 more