f-theory, seiberg-witten curves, and n = 2 dualitiestheory.uchicago.edu/~sethi/great lakes...

F-Theory, Seiberg-Witten Curves, and N = 2Dualities

Keshav Dasgupta

Department of PhysicsMcGill University

Montréal, QC, CANADA

Dasgupta (McGill) String Theory Chicago2011 1 / 24

In this talk I’ll basically summarise some of the older works onSeiberg-Witten theories, F-theories and some new work that Ihave started. The talk will be based on the following papers.

A Novel Method for locating Argyres-Douglas Loci in N = 2Theories, Jihye Seo, and KD To appear

F-Theory, Seiberg-Witten Curves and N = 2 Dualities: I,Alisha Wissanji, Jihye Seo, and KD To appear

F-Theory, Seiberg-Witten Curves and N = 2 Dualities: II,Alisha Wissanji Work in progress


Outline of the talk

A brief summary of the SU(2) Seiberg-Witten theory, and ababy derivation of the simplest F-theory model from SU(2)Seiberg-Witten theory

How to generalise the model to incorporate multiple probesthat would reproduce the curves and monodromiesdiscussed by Jihye Seo.

Further generalisation of the F-theory model that wouldincorporate many of the Gaiotto’s model including ways tosee the underlying N = 2 dualities.


Outline of the talk


How to generalise the model to incorporate multiple probes

that would reproduce the curves and monodromiesdiscussed by Jihye Seo.



Outline of the talk





Outline of the talk



Further generalisation of the F-theory model that wouldincorporate many of the Gaiotto’s model

including ways tosee the underlying N = 2 dualities.


Outline of the talk





Yet another possible generalisation of the F-theoryconstruction that may incorporate certain N = 2 theories thatcould have heterotic duals with small instantons.

Finally, by slightly modifying the above constructions, I willdiscuss some possible ways to go to the N = 1 theories.


N = 2 supersymmetric Yang-Mills theory is at the borderbetween trivial and not fully solvable, in that it is in thelow-energy limit exactly solvable. It is governed by aholomorphic function, the prepotential F , for which theperturbative quantum corrections are under completecontrol, ie., occur just to one loop order.

The prepotential is typically given by:

F(A) =12τ0A2 +

iπ

A2 log[A2

Λ2

]+

12πi

A2∞∑L=1

cL(Λ

A

)4L

By taking two derivatives, F gives rise to the effectivecoupling. Note that indeed for large a ≡ A|θ=0, the instantonsum converges well, and the theory is dominated bysemi-classical, one-loop physics.


The transition from the classical to the exact quantum theoryinvolves splitting and shifting of the strong couplingsingularity away from u = 0 to u = ±Λ2.


The exact quantum moduli space is covered by three distinctregions, in the center of each of which the theory is weaklycoupled when choosing suitable local variables. A localeffective lagrangian exists in each coordinate patch,representing a particular perturbative approximation. None ofsuch lagrangians is more fundamental than the other ones,and no local lagrangian exists that would be globally validthroughout the moduli space.


On the other hand the F-theory story is rather simple.Imagine we start with a single D3-brane probing the followingbackground:

R2

Ω · (−1)FL · I45× R4 × R0123

X6789

X 0123

X 45

D3!brane Mirror D3!brane

Orientifold 7!planeHiggs branch

Coulomb branch


On the other hand the F-theory story is rather simple.Imagine we start with a single D3-brane probing the followingbackground:

R2

Ω · (−1)FL · I45× R4 × R0123

X6789

X 0123

X 45

D3!brane

Higgs branch

Coulomb branch

(0, 1) seven!brane(1, !1) seven!brane


Which means: on the D3-brane world-volume we see anN = 2 Sp(2) SW theory, and therefore under quantumcorrections we expect the O7-plane to split into two (p,q)seven-branes.

X6789

X 0123

X 45

D3!brane

Higgs branch

Coulomb branch

(0, 1) seven!brane(1, !1) seven!brane


The above picture when compactified leads to F-theory, i.e

Type IIB onT2 × R4 × R0123

Ω · (−1)FL · I45= F Theory on K 3× R4 × R0123

Which is of course the statement that F-theory is nothing bySW theory in disguise. The Weierstrass equation of F-theoryis the corresponding SW curve; and so on.

So what happens if we now probe the background withmultiple D3-branes? We will get Sp(2r) SW theory. Thenwhats the big deal about this?

Let us take r = 2 and see what happens.






So what happens if we now probe the background withmultiple D3-branes?

We will get Sp(2r) SW theory. Thenwhats the big deal about this?







So what happens if we now probe the background withmultiple D3-branes? We will get Sp(2r) SW theory.

Thenwhats the big deal about this?







So what happens if we now probe the background withmultiple D3-branes? We will get Sp(2r) SW theory. Thenwhats the big deal about this?



This would be the Sp(4) theory whose curves, cycles andArgyres-Douglas points were discussed by Jihye.

For thosewho overslept, here it is again.


This would be the Sp(4) theory whose curves, cycles andArgyres-Douglas points were discussed by Jihye. For thosewho overslept, here it is again.


From the gauge theory side, there are now multiplemonopole/dyon points.

There are also new Argyres-Douglaspoints.

If, like the Sp(2) case, the monopole/dyon points map to thenumber of (p,q) seven-branes appearing from an orientifoldseven-plane, do we get the O7 plane to split into many (p,q)seven-branes?

But from F-theory why would the physics of O7 plane beaffected by how many probe D3-branes that we put in thetheory? Then this means that for any number of probes, theO7-plane will split into only two (p,q) seven-branes.

Then where did all the monopole/dyon points go? Where arethe AD points now? How do we see the multi-dimensionalCoulomb branch from F-theory? For r >> 2 the picture getseven more complicated, as three or more cycles cometogether to give generalised AD points. Can we ever hope tosee all these from F-theory??


From the gauge theory side, there are now multiplemonopole/dyon points. There are also new Argyres-Douglaspoints.







But from F-theory why would the physics of O7 plane beaffected by how many probe D3-branes that we put in thetheory?

Then this means that for any number of probes, theO7-plane will split into only two (p,q) seven-branes.






Then where did all the monopole/dyon points go?

Where arethe AD points now? How do we see the multi-dimensionalCoulomb branch from F-theory? For r >> 2 the picture getseven more complicated, as three or more cycles cometogether to give generalised AD points. Can we ever hope tosee all these from F-theory??





Then where did all the monopole/dyon points go? Where arethe AD points now?

How do we see the multi-dimensionalCoulomb branch from F-theory? For r >> 2 the picture getseven more complicated, as three or more cycles cometogether to give generalised AD points. Can we ever hope tosee all these from F-theory??





Then where did all the monopole/dyon points go? Where arethe AD points now? How do we see the multi-dimensionalCoulomb branch from F-theory?

For r >> 2 the picture getseven more complicated, as three or more cycles cometogether to give generalised AD points. Can we ever hope tosee all these from F-theory??





Then where did all the monopole/dyon points go? Where arethe AD points now? How do we see the multi-dimensionalCoulomb branch from F-theory? For r >> 2 the picture getseven more complicated, as three or more cycles cometogether to give generalised AD points.

Can we ever hope tosee all these from F-theory??





Then where did all the monopole/dyon points go? Where arethe AD points now? How do we see the multi-dimensionalCoulomb branch from F-theory? For r >> 2 the picture getseven more complicated, as three or more cycles cometogether to give generalised AD points. Can we ever hope tosee all these from F-theory??Dasgupta (McGill) String Theory Chicago2011 13 / 24

Unfortunately I’ll not have time to answer all these questions.

We have some answers to these questions, but none toosatisfactory! But let me discuss one possible way to see theAD loci.

monopole dyon elsewhere

2 0 00 2 01 1

10

11 10

00 0 2


Unfortunately I’ll not have time to answer all these questions.We have some answers to these questions, but none toosatisfactory!

But let me discuss one possible way to see theAD loci.


2 0 00 2 01 1

10

11 10

00 0 2


Unfortunately I’ll not have time to answer all these questions.We have some answers to these questions, but none toosatisfactory! But let me discuss one possible way to see theAD loci.


2 0 00 2 01 1

10

11 10

00 0 2


With a similar table for the Sp(2r) case


rr0

000

1 r!1.... ....r!1 1

0

0....

.... .... ....r!p q p!q

These would be the most likely way to look for AD points/locifrom F-theory.


Let me change gear a little bit and ask the following question.

In the duality



can we replace the R4 by another four-dimensional manifoldwithout breaking any supersymmetry?

The answer is yes. There are two non-trivial choices:Taub-NUT, multi Taub-NUT and K3 manifold. With theTaub-NUT spaces, we will call them Model 2 and with K3manifold, we wil call them Model 3. This means the rD3-branes are probing seven-brane backgrounds wrapped onTN or K3 spaces.

But there are more! The r D3-branes could probe intersectingseven-brane backgrounds! The susy there could be N = 1.


Let me change gear a little bit and ask the following question.In the duality











The answer is yes. There are two non-trivial choices:Taub-NUT, multi Taub-NUT and K3 manifold.

With theTaub-NUT spaces, we will call them Model 2 and with K3manifold, we wil call them Model 3. This means the rD3-branes are probing seven-brane backgrounds wrapped onTN or K3 spaces.







The answer is yes. There are two non-trivial choices:Taub-NUT, multi Taub-NUT and K3 manifold. With theTaub-NUT spaces, we will call them Model 2

and with K3manifold, we wil call them Model 3. This means the rD3-branes are probing seven-brane backgrounds wrapped onTN or K3 spaces.







The answer is yes. There are two non-trivial choices:Taub-NUT, multi Taub-NUT and K3 manifold. With theTaub-NUT spaces, we will call them Model 2 and with K3manifold, we wil call them Model 3.

This means the rD3-branes are probing seven-brane backgrounds wrapped onTN or K3 spaces.








But there are more!

The r D3-branes could probe intersectingseven-brane backgrounds! The susy there could be N = 1.







But there are more! The r D3-branes could probe intersectingseven-brane backgrounds!

The susy there could be N = 1.


Claim: The case where the seven-branes wrap multiTaub-NUT spaces lead to the Gaiotto type models!

X0123

X6789

X 45

D3!Branes along X0123

Taub!NUT space along X6789

Higgs branch

Coulomb branch

Seven!Branes wrapping multi


The above constructions have distinct advantages over theM-theory or IIA D4/NS5 constructions, as both conformal andnon-conformal cases can be easily studied using theseven-branes.

Choosing the appropriate Tate’s algorithm onemay study any global symmetries for the system.

The Gaiotto dualities may be understood in our D3/D7/TNmodel from chiral anomaly cancellation and brane creationmechanism.

The multi-dimensional Coulomb branch appears easily fromthe dissociation of every D3-branes to D5 anti-D5 pairs at theTaub-NUT singularities.

This dissociation happens when the system is away from theorientifold point, and therefore gauge groups likeSU(2)k ,SU(3)k etc could be realised.


The above constructions have distinct advantages over theM-theory or IIA D4/NS5 constructions, as both conformal andnon-conformal cases can be easily studied using theseven-branes. Choosing the appropriate Tate’s algorithm onemay study any global symmetries for the system.

The Gaiotto dualities may be understood in our D3/D7/TNmodel from chiral anomaly cancellation and brane creationmechanism.

The multi-dimensional Coulomb branch appears easily fromthe dissociation of every D3-branes to D5 anti-D5 pairs at theTaub-NUT singularities.

This dissociation happens when the system is away from theorientifold point, and therefore gauge groups likeSU(2)k ,SU(3)k etc could be realised.


The tachyons on the D5 anti-D5 pairs are cancelled byworld-volume fluxes using the technique calledmode-flipping.

Thus this dissociation don’t break susy, andin particular, remain stable.

E

!0

"1/2

"1/4

11/2

Supersymmetry, or the so-called s-rule, is perfectlymaintained by making the D5 anti-D5 pairs to wrap differentcycles of the multi Taub-NUT space. Thus susy requires: rD3-branes to probe r -centered Taub-NUT space.


The tachyons on the D5 anti-D5 pairs are cancelled byworld-volume fluxes using the technique calledmode-flipping. Thus this dissociation don’t break susy, andin particular, remain stable.

E

!0

"1/2

"1/4

11/2




E

!0

"1/2

"1/4

11/2

Supersymmetry, or the so-called s-rule, is perfectlymaintained by making the D5 anti-D5 pairs to wrap differentcycles of the multi Taub-NUT space.

Thus susy requires: rD3-branes to probe r -centered Taub-NUT space.



E

!0

"1/2

"1/4

11/2



Thus our model 2 is described by:

Type IIB on P1 × R4

Z2× R0123 = F Theory on K 3× R4

Z2× R0123

Switching on a time varying self-dual connection along theboundary S1 of the TN space, creates an anomaly like term ofthe form: ∫

d2x ωεab∂aAb

This is exactly cancelled by the seven-brane Chern-Simonsterm:S =

∫G5 ∧ A1 ∧ F1 −→ −

∫dG5 ∧ (ωF1)

when dG5 = delta function, i.e there is a D3-brane source.





Z2× R0123


d2x ωεab∂aAb


∫G5 ∧ A1 ∧ F1

−→ −∫

dG5 ∧ (ωF1)when dG5 = delta function, i.e there is a D3-brane source.





Z2× R0123


d2x ωεab∂aAb


∫G5 ∧ A1 ∧ F1 −→ −

∫dG5 ∧ (ωF1)

when dG5 = delta function, i.e there is a D3-brane source.


I would like to discuss more on this, but due to lack of timelets go to the next model, i.e model 3.

Its easy to see that now we have F-theory on K3 × K3 whichcan only be probed by 24 D3-branes. This model doesn’thave a brane creation process. Instead ther is a heteroticdual given by 24 small instantons wrapped on a T 2 andprobing conformally Kahler K3 manifold. This model hasbeen studied long ago.

If we switch on fluxes, the number r of D3-branes could bereduced i.e r < 24. This model is dual to r heterotic smallinstantons wrapped on the two-cycle of a non-Kählersix-manifold with torsion!

But there is more.



Its easy to see that now we have F-theory on K3 × K3 whichcan only be probed by 24 D3-branes.

This model doesn’thave a brane creation process. Instead ther is a heteroticdual given by 24 small instantons wrapped on a T 2 andprobing conformally Kahler K3 manifold. This model hasbeen studied long ago.


But there is more.



Its easy to see that now we have F-theory on K3 × K3 whichcan only be probed by 24 D3-branes. This model doesn’thave a brane creation process.

Instead ther is a heteroticdual given by 24 small instantons wrapped on a T 2 andprobing conformally Kahler K3 manifold. This model hasbeen studied long ago.


But there is more.



Its easy to see that now we have F-theory on K3 × K3 whichcan only be probed by 24 D3-branes. This model doesn’thave a brane creation process. Instead ther is a heteroticdual given by 24 small instantons wrapped on a T 2 andprobing conformally Kahler K3 manifold.

This model hasbeen studied long ago.


But there is more.





But there is more.




If we switch on fluxes, the number r of D3-branes could bereduced i.e r < 24.

This model is dual to r heterotic smallinstantons wrapped on the two-cycle of a non-Kählersix-manifold with torsion!

But there is more.





But there is more.


Using similar argument of the stability of the D5 anti-D5system, one may allow D3 anti-D3 branes by making thetachyon massless.

This would mean that we can studyr >> 24 D3 anti-D3 branes in the K3 × K3 set-up. The systemis known to be supersymmetric, but could be N = 1, as wedon’t know if there is a N = 2 story for this configuration.

Then there is a whole class of F-theory models withintersecting O7 planes and D7-branes.


Using similar argument of the stability of the D5 anti-D5system, one may allow D3 anti-D3 branes by making thetachyon massless. This would mean that we can studyr >> 24 D3 anti-D3 branes in the K3 × K3 set-up.

The systemis known to be supersymmetric, but could be N = 1, as wedon’t know if there is a N = 2 story for this configuration.



Using similar argument of the stability of the D5 anti-D5system, one may allow D3 anti-D3 branes by making thetachyon massless. This would mean that we can studyr >> 24 D3 anti-D3 branes in the K3 × K3 set-up. The systemis known to be supersymmetric, but could be N = 1, as wedon’t know if there is a N = 2 story for this configuration.



These are the Gimon-Polchinski type models.

What happenswhen we probe them with multiple D3-branes? The F-theorylift of this with studied briefly by Sen.

the susy on the D3-branes would be N = 1 now, so it wouldbe interesting to see what comes out from the here. Thiscould be related to some of the recent work ofTachikawa-Vafa-Wecht.

Can we wrap some directions of the seven-branes withoutbreaking further susy, and study the probe physics? This,and many more, are for part II of our work.


These are the Gimon-Polchinski type models. What happenswhen we probe them with multiple D3-branes?

The F-theorylift of this with studied briefly by Sen.




These are the Gimon-Polchinski type models. What happenswhen we probe them with multiple D3-branes? The F-theorylift of this with studied briefly by Sen.





the susy on the D3-branes would be N = 1 now, so it wouldbe interesting to see what comes out from the here.

Thiscould be related to some of the recent work ofTachikawa-Vafa-Wecht.





Can we wrap some directions of the seven-branes withoutbreaking further susy, and study the probe physics?

This,and many more, are for part II of our work.


Conclusions

We have shown how, starting from a simple D3-branesprobing quantum corrected orientifold backgrounds, one maystudy many more non-trivial models.

The recently studied Gaiotto type models and the assocoateddualities may also be analysed from this F-theoreticapproach.

Our model clearly generalises to incorporate conformal,non-conformal, as well as lower susy cases.

FIN!


f-theory, seiberg-witten curves, and n = 2 dualitiestheory.uchicago.edu/~sethi/great lakes...

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