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

The (0,4) elliptic genus of the magnetic monopole moduli space equals the partition function of

N=4 SYM on the del Pezzo surface .

Conjecture

M r ,N f

U(r)

dPN f

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

2d/4d correspondence

                             r  M 5 branes

                            [      Tτ2 ⊗ dPN f

 ]  

U(r) N =4SYM on dPN f(0,4)   CFT on T 2

gaugecoupling τ                         cL =r(4 + N f );  cR =6r

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)

Some important references

The (0,4) elliptic genus of the magnetic monopole moduli space equals the partition function of

N=4 SYM on the del Pezzo surface .

Conjecture

M r ,N f

U(r)

dPN f

(0,4) Sigma model

Target space: moduli space of magnetic monopoles (hyperkahler) with addition of adjoint fermionic zero modes and Nf flavor fermionic zero modes;

The (0,4) CFT

S =T d2σ   gmn(∂+Xm∂−X

n + iψ mD+ψn) + iχ AD−χ

A −12

FmnABψmψ nχ Aχ B⎡

⎣⎢⎤⎦⎥T 2

∫

m =1,..., 4r ⇒    cR =4r + 2r =6rA=1,...,2rN f    ⇒   cL =4r + rN f =r(4 + N f )

This is actually the lift of the quantum mechanics description of magnetic monopoles in SU(2) N=2 D=4 Seiberg-Witten with Nf massless hypermultiplets

[Sethi, Stern & Zaslow ’95; Cederwall, Ferretti, Nilsson & Salomonson ’95; Gauntlett & Harvey ’95] and [Gauntlett, Kim, Lee, Yi, ’00].

The (0,4) CFT

S =

12

dt   gmn( &Xm &Xn + iψ mDtψ

n) + iχ ADtχA −

12

FmnABψmψ nχ Aχ B⎡

⎣⎢⎤⎦⎥∫

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 Nf ≤8 massless flavors in 5d SU(2) gauge theory on the coulomb branch, the tension can be computed to be

5d Gauge Theory

T =rφg5

2 + (8 −N f )φ2⎧

⎨⎩

⎫⎬⎭

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)

5d Gauge Theory

M-theory on local CY3: canonical line bundle over del Pezzo,

In our conventions,

This engineers 5d N=1 SU(2) gauge theory with Nf flavors.

Geometric engineering

O(K )→ CY3

                ↓               dPN f

dP0 =F0 =PB1 ⊗ Pf

1

dPN f=F0  with  N f  blow−ups

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

Geometric engineering

Using the connection to 5d gauge theory, we know what the (0,4) CFT is:

5d/2d/4d correspondence

                             r  M 5 branes

                            [      T 2 ⊗ dPN f ]  

U(r) N =4SYM on dPN f(0,4)   CFT on T 2

gaugecoupling τ                        with target space Mr,N f

5d gauge theory tells us that Nf ≤8

The (0,4) elliptic genus of the magnetic monopole moduli space equals the partition function of

N=4 SYM on the del Pezzo surface .

Conjecture

M r ,N f

U(r)

dPN f

r=1, Nf=0: Free CFT, 3 non-compact and 1 compact scalars + 4 right-moving fermions.

Elliptic genus:

Tests

M1,0 =° 3 ⊗S1

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

Test 1

dP0 =F0 =P1 ⊗ P1

χ(F0 ) = 4

r=1, Nf ≠0, massless charged flavors. Flavor group SO(2Nf)

but 2Nf extra left-moving fermions. Moebius bundle; Manton & Schroers ’93)

Quantum mechanics of dyonic monopole must satisfy (Seiberg & Witten ’94, Gauntlett & Harvey ’96)

A more complicated test

M1,0 =° 3 ⊗S1

(−)ne |ψ > =(−)H |ψ >

¢ 2  action

In the CFT, this is lifted to an orbifold action with

Elliptic genus yields

Test 2: 2d CFT calculation

G ={1,g}      g=(−)ne+F

ZCFT (τ ) =ZMB(τ )η(τ )N f +4

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

Test 2: 2d CFT calculation

Del Pezzo = P1x P1 with Nf blow-ups. Choose basis in for which the

intersection matrix displays SO(2Nf) symmetry :

Lattice instead of usual unimodular lattice with intersection matrix

Test 2: 4d calculation

H 2 (dPN f ,Z)

(n =N f )

Λ =A⊕D

Partition function has theta-function decomposition (Manschot ’11,…)

For rank one, r=1,

Test 2: 4d calculation

h1(τ,dPN f) =

1η(τ )b2 +2       b2 =2 + N f

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

with

Test 2: 4d calculation

The four terms correspond to the four sectors in the orbifold (0,4) CFT.

The theta functions of the DNf lattice correspond to the flavor fermions with current algebra SO(2Nf).

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!

Test 2: the 4d calculation

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.

Conclusion