1. A 5d/2d/4d correspondence Babak Haghighat, Jan Manschot, S.V., to appear; B. Haghighat and S.V., arXiv:1107.2847

2.  The (0,4) elliptic genus of the magnetic monopole moduli space equals the partition function of N=4 SYM on the del Pezzo surface.

3.  Our conjecture follows from a variant of the 2d/4d correspondence a la AGT:

4.  Maldacena, Strominger, Witten (‘97)  Minasian, Moore and Tsimpis (‘99)  Gaiotto, Strominger and Yin (‘06)  Minahan, Nemeschansky, Vafa and Warner (‘98)  Alim, Haghighat, Hecht, Klemm, Rauch, Wotschke (‘10)  De Boer, Cheng, Dijkgraaf, Manschot, Verlinde (‘06) (Useful for us, but different set-up)

5.  The (0,4) elliptic genus of the magnetic monopole moduli space equals the partition function of N=4 SYM on the del Pezzo surface.

6.  (0,4) Sigma model  Target space: moduli space of magnetic monopoles (hyperkahler) with addition of adjoint fermionic zero modes and N f flavor fermionic zero modes;

7.  This is actually the lift of the quantum mechanics description of magnetic monopoles in SU(2) N=2 D=4 Seiberg-Witten with N f massless hypermultiplets [Sethi, Stern & Zaslow ’95; Cederwall, Ferretti, Nilsson & Salomonson ’95; Gauntlett & Harvey ’95] and [Gauntlett, Kim, Lee, Yi, ’00].

8.  Uplifting the dynamics of the magnetic monopole from d=1 to d=2 amounts to embedding the monopole in 5d gauge theory, where it becomes a BPS magnetic string.  For N f ≤8 massless flavors in 5d SU(2) gauge theory on the coulomb branch, the tension can be computed to be

9.  Study of 5d N=1 susy gauge theories was initiated by Seiberg ‘96.  Nonrenormalizable theories that should be embedded in string theory:  Geometric engineering (Douglas, Katz & Vafa ‘96; Morrison & Seiberg ‘96; Intrilligator, Morrison & Seiberg ’97)  (p,q) branes in IIB (Aharony, Hanany & Kol ‘97)

10.  M-theory on local CY 3 : canonical line bundle over del Pezzo,  In our conventions,  This engineers 5d N=1 SU(2) gauge theory with N f flavors.

11.  Magnetic string is M5 brane wrapping del Pezzo. Its tension precisely matches the volume of the del Pezzo!

12.  Using the connection to 5d gauge theory, we know what the (0,4) CFT is: 5d gauge theory tells us that N f ≤8

13.  The (0,4) elliptic genus of the magnetic monopole moduli space equals the partition function of N=4 SYM on the del Pezzo surface.

14.  r=1, N f =0: Free CFT, 3 non- compact and 1 compact scalars + 4 right- moving fermions. Elliptic genus:

15.  U(1) N=4 SYM partition function on  Localizes on instantons (Vafa & Witten ’94). Result is (Gottsche ’90)  This matches the 2d CFT side since and

16.  r=1, N f ≠0, massless charged flavors. Flavor group SO(2N f )  but 2N f extra left-moving fermions. Moebius bundle; Manton & Schroers ’93)  Quantum mechanics of dyonic monopole must satisfy (Seiberg & Witten ’94, Gauntlett & Harvey ’96)

17.  In the CFT, this is lifted to an orbifold action with  Elliptic genus yields

18.  One can treat the compact boson and flavors separately with twisted and untwisted sectors:

19.  Del Pezzo = P 1 x P 1 with N f blow-ups.  Choose basis in for which the intersection matrix displays SO(2N f ) symmetry :  Lattice instead of usual unimodular lattice with intersection matrix

20.  Partition function has theta-function decomposition (Manschot ’11,…)  For rank one, r=1,

21.  If one chooses the restriction of the Kahler class to vanish along the D-lattice, one has  with

22.  The four terms correspond to the four sectors in the orbifold (0,4) CFT.  The theta functions of the D Nf lattice correspond to the flavor fermions with current algebra SO(2N f ).  The contributions from the A-lattice correspond to the contribution of the compact scalar with shifted momentum and winding modes.  It is a miracle that (if) this works!

23.  We found an interesting new 5d/2d/4d correspondence and provided non-trivial tests for rank r=1.  We have some more results for massive flavors.  For r=2, the monopole moduli space is that of Atiyah-Hitchin. We cannot compute its elliptic genus directly, but we have the answer from the 4d side.