in the gauge group. In our case,N is the same for each segment, so the gauge group isU(N)s, but one could also consider a more general configuration.
• For eachN S5 brane, which bounds two such segments, there is a bifundamental hypermultiplet between the corresponding gauge group factors. This comes from massless modes coming of fundamental strings stretching between theD3 branes on either side of theN S5 branes.
• On each such segment, any D5 branes intersecting theD3 brane segments contribute a fun- damental hypermultiplet in the corresponding gauge group factor, and is associated to funda- mental strings stretching between theD3 andD5 branes.
We can also consider deformations of this setup, corresponding to moving the various branes:
• The transverse positions of theD3 branes are parametrized by scalar fields in the gauge theory, so moving the branes corresponds to changing the position in moduli space. As usual, we will be interested in the origin of moduli space, where theD3 branes are coincident
• Moving the N S5 branes gives rise to anF I termR
d4θ(Di−Dj).
• Moving the D5 branes corresponds to a real mass to the corresponding hypermultiplet, i.e., real masses of opposite signs to the two chiral multiplets it contains.
With this construction, mirror symmetry follows directly fromS-duality in type IIB string theory.
This duality exchangesN S5 branes withD5 branes, while preservingD3 branes. Thus it maps one configuration of the type above to another. In theIR, the two gauge theories must be equivalent.
Typically theU V description of dual theories will look very different from the point of view of gauge theory, and there is not convenient IR description, so the duality is very nontrivial.
Y
i
1
2 cosh(πλi) (6.9)
• A bifundamental hypermuliplet between the U(N) factor corresponding to λi and another factor corresponding to ˆλi contributes:
Y
i,j
1
2 coshπ(λi−ˆλj) (6.10)
One multiplies all the appropriate 1-loop factors, and integrates over theλi for eachU(N) factor in the gauge group.
In principle, this prescription allows us to write down the matrix model for any of the theories considered above, given only the sequenceαa. However, to prove the duality, it will be convenient to organize these in a coherent way.
• For each D5 brane, there is a factor of:
Y
i
1
2 cosh(πλi) (6.11)
whereλicorresponds to the gauge group in which the hypermultiplet sits. Note that there is a set of integration variablesλi for eachN S5 brane, but not for theD5 branes. To try to make the situation more symmetric, we can introduce variablesλia for each five-brane, and then if theath five-brane is aD5 brane, we can simply write its contribution as:2.
Z dNλa
Y
i
δ(λai
−λa+1i
)
2 cosh(πλi) (6.12)
We will find it convenient to introduce an auxiliary variable τi to enforce the delta function constraint:
Z
dNλadnτa
Y
i
e2πiτai(λai−λa+1i)
2 cosh(πλi) (6.13)
• If the ath five-brane is an N S5 brane, it will contribute a bifundamemental hypermultiplet between the two neighboring gauge groups, giving a contribution:
Y
i,j
1 2 coshπ(λai
−λa+1j
) (6.14)
2Although we write this as an integral, this expression is to be inserted into a larger integral, with other factors that may depend onλaandλa+1, so the integral should not be evaluated at this point
In addition, we must account for gauge group 1-loop determinants. Since the gauge groups are associated to a pair of consecutive branes, we can use the fact that the 1-loop determinant is a perfect square to split it up into two factors, and associate each factor to the two bounding N S5 branes. Thus the total contribution of theathN S5 brane can be written:
1 N!
Z dNλa
Q
i<j(2 sinhπ(λai−λaj))(2 sinhπ(λa+1i−λa+1j)) Q
i,j2 coshπ(λai
−λa+1j
) (6.15)
The real reason for writing the contribution this way is because this can be rewritten using the Cauchy determinant formula as:
1 N!
Z
dNλaX
σ
(−1)σY
i
1
2 coshπ(λai−λa+1σ(i)) (6.16) where σruns over the permutations inSN. Again, we introduce an auxiliary variableτa and use the fact that 2 cosh(πx)1 is its own Fourier transform, as we saw in the abelian duality above, to write this as:
1 N!
Z
dNλadNτaX
σ
(−1)σY
i
e2πiτai(λai−λa+1σ(i))
2 cosh(πτai) (6.17)
With this in mind, let us define
Iα(λai, τai) =
Y
i
1
2 cosh(πσai) α=D5 Y
i
1
2 cosh(πτai) α=N S5
(6.18)
Then we claim the partition function of the brane configuration corresponding to the sequenceαa
is given by:
Y
a
Z
dNλadNτaX
σ
(−1)σY
i
e2πiτai(λai−λa+1σ(i))Iαa(λai, τai) (6.19) We have essentially shown this above. The only thing to check is that the antisymmetrization we have added in theD5 contribution does not affect anything. But, provided there is at least oneN S5 brane3, one can see that the integral will always be antisymmetric in all of theλai, so this is true.
In this form, the duality is essentially manifest. Namely, upon exchanging the variables λa and τa, the contributions ofN S5 andD5 branes are exchanged, and we get the partition function of the
3We can assume this without loss, since the matter content of a theory withoutN S5 branes is the same as one with a singleN S5 brane added. The naive dual of the original, pureD5 brane theory would have onlyN S5 branes, and is not well-defined for the same reason asN= 8 SYM, which we will see below.
dual theory.4
Finally, we consider the effect of adding deformations.
• For each D5 brane, we can add a real massm for the corresponding hypermultiplet, which simply gives:
Y
i
1
2 coshπ(λi+m) (6.20)
• For eachN S5 brane, the corresponding operation is to add an FI termη to the two adjacent gauge groups with opposite signs, so that the contribution is modified to:
1 N!
Z
dNλadNτa
X
σ
(−1)σY
i
e2πi(τai+η)(λai−λa+1σ(i))
2 cosh(πτai) (6.21)
By shiftingτai, one can move this factor inside the cosh.
Thus we see the effect of the two deformations is identical: they shift the argument inside the cosh in Iα. Thus the deformed partition function will be unchanged provided that when we exchanged λ and τ, we simultaneously exchange m and η. This demonstrates the correct mapping of the symmetries.