1 / 23

Magnetic strings , M5 branes , and N=4 SYM on del Pezzo surfaces:

Magnetic strings , M5 branes , and N=4 SYM on del Pezzo surfaces:. A 5d/2d/4d correspondence Babak Haghighat , Jan Manschot, S.V., to appear ; B. Haghighat and S.V., arXiv :1107.2847. Conjecture.

jamar
Download Presentation

Magnetic strings , M5 branes , and N=4 SYM on del Pezzo surfaces:

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Magneticstrings, M5 branes, and N=4 SYM on del Pezzo surfaces: A 5d/2d/4d correspondence BabakHaghighat, Jan Manschot, S.V., to appear; B. Haghighat and S.V., arXiv:1107.2847

  2. Conjecture • The (0,4) elliptic genus of the magneticmonopolemodulispaceequals the partitionfunction of N=4 SYM on the del Pezzosurface .

  3. 2d/4d correspondence • Ourconjecturefollowsfrom a variant of the 2d/4d correspondence a la AGT:

  4. Some important references • 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) (Usefulforus, but different set-up)

  5. Conjecture • The (0,4) elliptic genus of the magneticmonopolemodulispaceequals the partitionfunction of N=4 SYM on the del Pezzosurface .

  6. The (0,4) CFT • (0,4) Sigma model • Target space: modulispace of magneticmonopoles (hyperkahler) withaddition of adjointfermionic zero modes and Nfflavorfermionic zero modes;

  7. The (0,4) CFT • This is actually the lift of the quantummechanicsdescription of magneticmonopoles in SU(2) N=2 D=4 Seiberg-WittenwithNfmasslesshypermultiplets [Sethi, Stern & Zaslow ’95; Cederwall, Ferretti, Nilsson & Salomonson ’95; Gauntlett & Harvey ’95] and [Gauntlett, Kim, Lee, Yi, ’00].

  8. 5d GaugeTheory • Uplifting the dynamics of the magneticmonopolefromd=1 to d=2 amounts to embedding the monopole in 5d gaugetheory, whereitbecomes a BPS magneticstring. • For Nf≤8masslessflavors in 5d SU(2) gaugetheoryon the coulomb branch, the tensioncanbecomputed to be

  9. 5d GaugeTheory • Study of 5d N=1 susygaugetheories was initiatedbySeiberg ‘96. • Nonrenormalizabletheoriesthatshouldbeembedded in stringtheory: • Geometric engineering (Douglas, Katz & Vafa ‘96; Morrison & Seiberg ‘96; Intrilligator, Morrison & Seiberg ’97) • (p,q) branes in IIB (Aharony, Hanany & Kol ‘97)

  10. Geometric engineering • M-theoryonlocal CY3: canonicallinebundle over del Pezzo, • In ourconventions, • Thisengineers 5d N=1 SU(2) gaugetheorywithNfflavors.

  11. Geometric engineering • Magneticstring is M5 branewrapping del Pezzo. Itstensionprecisely matches the volume of the del Pezzo!

  12. 5d/2d/4d correspondence • Using the connection to 5d gaugetheory, we knowwhat the (0,4) CFT is: 5d gaugetheorytellsusthatNf≤8

  13. Conjecture • The (0,4) elliptic genus of the magneticmonopolemodulispaceequals the partitionfunction of N=4 SYM on the del Pezzosurface .

  14. Tests • r=1, Nf=0: Free CFT, 3 non-compact and 1 compact scalars + 4 right-movingfermions. Elliptic genus:

  15. Test 1 • U(1) N=4 SYM partitionfunctionon • Localizesoninstantons (Vafa & Witten ’94). Result is (Gottsche ’90) • This matches the 2d CFT sidesince and

  16. A more complicated test • r=1, Nf ≠0, masslesschargedflavors. Flavorgroup SO(2Nf) • but 2Nf extra left-movingfermions. Moebiusbundle; Manton & Schroers ’93) • Quantummechanics of dyonicmonopole must satisfy(Seiberg & Witten ’94, Gauntlett & Harvey ’96)

  17. Test 2: 2d CFT calculation • In the CFT, this is lifted to anorbifoldactionwith • Elliptic genus yields

  18. Test 2: 2d CFT calculation • Onecantreat the compact boson and flavorsseparatelywithtwisted and untwisted sectors:

  19. Test 2: 4d calculation • Del Pezzo = P1x P1withNfblow-ups. • Choose basis in forwhich the intersection matrix displays SO(2Nf) symmetry : • Latticeinstead of usualunimodularlatticewithintersection matrix

  20. Test 2: 4d calculation • Partitionfunction has theta-functiondecomposition (Manschot ’11,…) • For rank one, r=1,

  21. Test 2: 4d calculation • Ifonechooses the restriction of the Kahlerclass to vanishalong the D-lattice, one has • with

  22. Test 2: the 4d calculation • The fourtermscorrespond to the four sectors in the orbifold (0,4) CFT. • The thetafunctions of the DNflatticecorrespond to the flavorfermionswithcurrent algebra SO(2Nf). • The contributionsfrom the A-latticecorrespond to the contribution of the compact scalarwithshiftedmomentum and winding modes. • It is a miraclethat (if) thisworks!

  23. Conclusion • We foundaninterestingnew 5d/2d/4d correspondence and providednon-trivial tests for rank r=1. • We have some more resultsformassiveflavors. • For r=2, the monopolemodulispace is that of Atiyah-Hitchin. We cannotcomputeitselliptic genus directly, but we have the answerfrom the 4d side.

More Related