1. Refined Cigar and Ω-deformed conifold Yu Nakayama (Berkeley) arxiv: 1004.2986

2. What is a Refined Cigar?

3. Refined Topological String Theory for Conifold (= Cigar)

4. What is a Refined Cigar? Not Marijuana! (even though I live in Berkeley…)

5. Arena for 21st century mathematics? “Non-commutative” quantization? Khovanov homology, categorification of CS-theory Beta deformed ensemble Doubly periodic q-deformed double Gamma function Geomertic Langlands program Refined topological amplitudes

6. Ω-deformation and topological string Take N=2 gauge theory, uplifted to 5d. Compute Witten index (5D partition function): Twisted by Lorentz Group SO(4) = SU(2) X SU(2)  Twisting can be identified with the garviphoton correction.

7. Topological string Consider type II string theory to compute Graviphton insertion  Topological twist

8. Refined topological amplitude I When Ω-deformation is self-dual, Nekrasov’s partition function = topological string partition function What about the ``anti-self-dual amplitude”? F + is anti-self-dual field strength for a vector multiplet (Antoniadis et al)

9. Conjecture: F g,n gives Nekrasov’s partition function with 2-parameter.

10. Check for Conifold Use triality: topological string on conifold, c=1 non-critical string, SL(2,R)/U(1) coset (cigar) by Ghoshal-Mukhi-Vafa Ω-deformation gives change of the radius By using the triality, we would like to connect refined topological amplitudes and the radius deformation directly.

11. Ghoshal-Mukhi-Vafa Triality topological string on conifold  c=1 non-critical string,  SL(2,R)/U(1) topological coset (cigar)

12. Refined topological amplitude becomes To obtain 1PI effective action, we need to subtract the leading singularity: Not 1 PI Subtract leading singularity

13. This gives the desired radius deformation: The total amplitude becomes One can show that higher order terms in OPE does not contribute in this case. The amplitude is holomorphic!

14. Discussions F g,n are typically non-holomorphic How can I choose F + among many vector multiplets? Why not graviphoton?? Better one for computability? Maybe the total volume moduli (= dilaton in heterotic dual)? In IIB side, there is no natural choice… And most importantly, why?