Quantum dilogarithm

introduction

• 양자 다이로그 함수(quantum dilogarithm)

 

 

근사 공식

• \(q=e^{-t}\) and as the t goes 0 (i.e. as q goes to 1) \[\sum_{n=0}^{\infty}\frac{q^{\frac{A}{2}n^2+cn}}{(q)_n}\sim\exp(\frac{C}{t})\]

여기서 C는 로저스 다이로그 함수 (Roger's dilogarithm) 의 어떤 값에서의 합

 

 

 

Knot and invariants from quantum dilogarithm

• [Kashaev1995] 
  
• a link invariant, depending on a positive integer parameter N, has been defined via three-dimensional interpretation of the cyclic quantum dilogarithm
  
• The construction can be considered as an example of the simplicial (combinatorial) version of the three-dimensional TQFT
  
• this invariant is in fact a quantum generalization of the hyperbolic volume invariant.
  
• It is possible that the simplicialTQFT, defined in terms of the cyclic quantum dilogarithm, can be associated with quantum 2 + 1-dimensional gravity.
  

• [Kashaev1995]A link invariant from quantum dilogarithm
  
  • Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418

 

 

 

 

 

 

 

related items

• Fermionic summation formula
  
• asymptotic analysis of basic hypergeometric series
  
• Quantum groups
  
• Kashaev's volume Conjecture
  

 

expositions

• Quiver mutations and quantum dilogarithm identities, presentation, Isle of Skye, June 27, 2011
• Quantum dilogarithm identities from quiver mutations, video of a talk given at Banff, September 9, 2010.

 

 

articles

• Keller, http://arxiv.org/abs/1102.4148

• Kashaev, http://arxiv.org/abs/1104.4630