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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.

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magnetic strings m5 branes and n 4 sym on del pezzo surfaces

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

conjecture
Conjecture
  • The (0,4) elliptic genus of the magneticmonopolemodulispaceequals the partitionfunction of

N=4 SYM on the del Pezzosurface .

2 d 4d correspondence
2d/4d correspondence
  • Ourconjecturefollowsfrom a variant of the 2d/4d correspondence a la AGT:
some important references
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)

conjecture1
Conjecture
  • The (0,4) elliptic genus of the magneticmonopolemodulispaceequals the partitionfunction of

N=4 SYM on the del Pezzosurface .

the 0 4 cft
The (0,4) CFT
  • (0,4) Sigma model
  • Target space: modulispace of magneticmonopoles (hyperkahler) withaddition of adjointfermionic zero modes and Nfflavorfermionic zero modes;
the 0 4 cft1
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].

5d gauge theory
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
5d gauge theory1
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)
geometric engineering
Geometric engineering
  • M-theoryonlocal CY3: canonicallinebundle over del Pezzo,
  • In ourconventions,
  • Thisengineers 5d N=1 SU(2) gaugetheorywithNfflavors.
geometric engineering1
Geometric engineering
  • Magneticstring is M5 branewrapping del Pezzo. Itstensionprecisely matches the volume of the del Pezzo!
5d 2d 4d correspondence
5d/2d/4d correspondence
  • Using the connection to 5d gaugetheory, we knowwhat the (0,4) CFT is:

5d gaugetheorytellsusthatNf≤8

conjecture2
Conjecture
  • The (0,4) elliptic genus of the magneticmonopolemodulispaceequals the partitionfunction of

N=4 SYM on the del Pezzosurface .

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

Elliptic genus:

test 1
Test 1
  • U(1) N=4 SYM partitionfunctionon
  • Localizesoninstantons (Vafa & Witten ’94). Result is (Gottsche ’90)
  • This matches the 2d CFT sidesince and
a more complicated test
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)
test 2 2d cft calculation
Test 2: 2d CFT calculation
  • In the CFT, this is lifted to anorbifoldactionwith
  • Elliptic genus yields
test 2 2d cft calculation1
Test 2: 2d CFT calculation
  • Onecantreat the compact boson and flavorsseparatelywithtwisted and untwisted sectors:
test 2 4d calculation
Test 2: 4d calculation
  • Del Pezzo = P1x P1withNfblow-ups.
  • Choose basis in forwhich the intersection matrix displays SO(2Nf) symmetry :
  • Latticeinstead of usualunimodularlatticewithintersection matrix
test 2 4d calculation1
Test 2: 4d calculation
  • Partitionfunction has theta-functiondecomposition (Manschot ’11,…)
  • For rank one, r=1,
test 2 4d calculation2
Test 2: 4d calculation
  • Ifonechooses the restriction of the Kahlerclass to vanishalong the D-lattice, one has
  • with
test 2 the 4d calculation
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!
conclusion
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.