#### Duality and enhancement of symmetry in 5d gauge theories

Gauge Field Theories
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1
0
Open Access
,
c The Authors
1
Department of Physics
,
Technion
We study various cases of dualities between N = 1 5d supersymmetric gauge theories. We motivate the dualities using brane webs, and provide evidence for them by comparing the superconformal index. In many cases we find that the classical global symmetry is enhanced by instantons to a larger group including one where the enhancement is to the exceptional group G2.
Contents
1 Introduction
2 The superconformal index
3 Adding more flavor
3.1 Index calculation
5 Inserting an SU(3) group 5.1 5.2 Index calculation
Two extra nodes
6 Conclusions A Determining gauge theory parameters from the web
4 Enhancement of symmetry in SU(2) USp(6)
Introduction
effective coupling takes the following rough form:
of ENf +1 [47].
for SU(3) + 2F and one can continue past the singularity.
massless implying that an instanton of the quiver theory becomes massless.
to exist whenever a is written.
full global symmetry, and compare it between proposed dual theories.
The superconformal index
considered on Sd1 S1.
following [4] the index is:
I = Tr (1)F x2 (j1+R) y2 j2 qQ .
topological symmetries.
and vector supermultiplets which contribute:
wW i=1
of the appropriate flavor representations.
P E[f ()] = exp
"X 1
integrating over the full instanton partition function.
and integrated over the gauge group.
number k:
inst = 1 + aZ1 + a2Z2 + . . . ,
instantons are needed.
not reproduce them here.
its rank grows with k.
will be mentioned shortly.
A thorough discussion of these problems can be found in [57, 11].
associated with these legs which is the state we need to mod out.
is only achieved once these states are modded out.
The removal of these states is generally achieved by:
Zc = P E
x2 P qimi
(1 xy) 1 y
charges respectively.
is valid when all these poles, that are within the contour, are included.
the classical global symmetry.
remnants of this U(1), such as decoupled states, to get the correct result.
Adding more flavor
and 3. Examining their S-duals we conjecture that:
1F + SU(2) SU(2) + 1F SU0(3) + 4F .
baryonic UB(1) and an SU(4) flavor symmetry.
quantum global symmetry is U(1) SU(2) SU(4).
symmetry on one side is realized perturbativly on the other side.
having a U(1)2 SU(4) global symmetry.
flavors [11].
Index calculation
giving further support to the above discussion.
we removed the two decoupled states by:
Zc = P E
(1 xy) 1 y
+z2 + c+
+z2 + c+
+ qt+
calculated to order x5.
to the order we are working in.
branes or signs. We use a for the instanton fugacity, and span the UF (4) by:
0 0 pc 0
IndexpSUet(r3.)+4 = 1 + x2 5 +
at this order for which we find:
IndexiSnUst0.(3)+4 = x
where we labeled the SU(3) instanton fugacity by a.
and one must mod out the decoupled U(1) state by:
(1 xy)(1 x/y) ]ZSU1(3)+4F
can be generated by charge conjugating the result for the positive case).
at x4. We find:
IndexiSnUst1.(3)+4 = x
x3 terms demands q =
to order x5.
q b3ap . With this mapping we find that the two indices match
symmetry U(1) SU(2) SU(4):
ones corresponding to the Cartan weights (0, 2, 0) and (2, 0, 2) respectively.
x2 tzl + qj + jlqt
(1 xy)(1 x/y)
Z1F +SU(2)SU(2)+1F
12 , 12 . There is also another web, not related by an SL(2, Z) transformation
1
2 , 21 in the field theory (this is similar to the flavorless case with
the order we are working in.
from fermionic zero modes.
To order x
5 we get contributions from the (1,0)+(0,1)+(1,1)+(2,0)+(0,2)+(2,1)+
(1,2)+(2,2)-instantons. We find:
+ qtpjl +
+
qtjl
jq
+
!
+pjq + +t l
jq
+ t l
qtjl
lj
lj
the two indices match to order x5.
where it reads:
representations are ambiguous, and are the same as stated above.
Enhancement of symmetry in SU(2) USp(6)
and leave the USp(6)s angle unspecified for the moment.
by correcting the partition function by:
(1 xy)(1 x/y) ZUSp(2)+6F
for USp(2) + 6F as predicted in [4] and evaluated by [57].
(3,0)+(4,0)-instantons. The lowest order terms in the index are:
z3 +z +
that the index can be written in G2 characters as:
q + 1q . Using this it is possible to show
remove these contribution from the Nekrasov partition function.
moduli space. (b) The web deformed as to exhibit the quiver structure.
moduli space. (b) The web deformed as to exhibit the quiver structure.
of symmetry to G2.
postpone this for future study.
Inserting an SU(3) group
possible to read the CS level, as explained in the appendix.
the quiver structure.
physically equivalent.
From these we conjecture the following dualities:5
global symmetry of these theories is U(1)3 SU(2)2.
5We thank Davide Gaiotto for suggesting the first duality to us.
quiver structure.
(which sees effectively 4 flavors).
Overall, the quantum symmetry in both theories is SU(2)3 U(1)2.
two end groups.
Index calculation
sector so as to make the results more presentable. We find:
+ 12 +
for the perturbative part.
the (1,0,0)+(0,1,0)+(0,0,1)-instantons. Their contribution is:
Index1SUin(s2t).SU0(3)SU0(2) = x
+tb 2 +
1+y2 +
+
+tb 2 +
+tb 2 +
+tb 2 +
(2,0,0)+(0,0,2)+(1,1,1) instantons. These provide:
IndexhSiUghe(2r)insStU.0(3)SU0(2) = x
bltz + bltz
qtb
+ bqt
bltz + bltz
+
blqt
+ blqt
blqt
+ blqt
and the line is the bifundamental.
+ 14 +
where the fugacities are allocated as in figure 9.
the (1,0)+(0,1)+(1,1) instantons which contribute:
Indexi1nFst+. SU1(3)SU1(3)+1F = x
3
3
3 +aB 23 p+
ap
ap
ap
3
3 +aB 23 p+
3
aB 23 p
3 +aB 23 p+
aB 23 p
aAp
aAp
3 +aB 23 p+
ap
aB 23 p
+ Bp
aAp
Bp
af p
aB 23 p
aAp
+O x5 . (5.7)
af p
+ aAB2 +
Af
Af
Bp
Bp
matching now requires b 2 t = AB 23
3
completing the matching.
three last U(1)s seem to be l, zb and qt.
one instanton part, including the (1,0,0)+(0,1,0)+(0,0,1) instantons, is:
Index1SUin(s2t).SU1(3)SU(2) = x
z 41 lqt
z 41 lqt
z 41 lqt
6It can also be the other way because of the discrete symmetries.
qt
z 41 lqt
bt
!
bl
+ bl
topological fugacities.
U(1) U(1) SU(2) SU(2) SU(2).
instantons are also needed. They contribute:
IndexhSiUghe(2r)insStU.1(3)SU(2) = x
+ blqz 2 +
instanton symmetry and span the U(2) group by:
tive part is:
+ p4 + 5p2 +
+14+ f 2 +
(0,1)+(1,1), and their contribution is:
IndexiSnUst0.(2)SU0(4)+2F = x
B4 +1+
+ A+
+ Aa+
This matches the indices to order x3. Further identifying
aB2 =
t3 and Ba2 =
aAf = qtlb 23 z 41 and
completes the matching.
The index can be written in SU(2)3 characters as:
aA
+ aA
aAf + aAf
Two extra nodes
7The last two can again be exchanged by discrete symmetries.
each edge group. These two are the generalizations of (5.1).
SU(2) SU 12 (3) SU 1 (3) SU0(2) 2F + SU1(4) SU1(4) + 2F
2
SU(2) SU 1 (3) SU 12 (3) SU0(2) 1F + SU1(3) SU0(3) SU1(3) + 1F (5.16)
2
SU0(2) SU 1 (3) SU 1 (3) SU0(2) 3F + SU0(5) SU0(3) + 1F .
2 2
describing 3F + SU0(5) SU0(3) + 1F .
1
CS terms 2 . These are distinguished by the orientations of the CS levels and angles
duals and wont be considered here.
only carried it to order x3.
shown in figure 14, we find:
ISU(4)2 = 1 + x2 7 + c2 +
One can see that there appears to be an enhancement of the
instantonicSU(2)4 U(1)3 global symmetry. Indeed the index can be concisely written as:
SU(4)s and the line is the bifundamental.
SU(2)s, the squares the SU(3)s, and the lines are bifundamentals.
the instanton contributions. The perturbative part is:
(0,1,1,0) instantons. The full instanton contribution is:
3 +B3B22 I3 +
B3B22 I3
3 +B3B22 I3 +
+I4B32 +
B3B22 I3
+I4B32 +
3 3
thermore, setting z2 = I2B22 , c2 = B3B22 I3, HdA2 = I4B32 , bHa 2 = I13 and dbH = q I2I3B3
B1
Next we move to the case of SU(3)3.
To order x3, we get contributions of the
higher orders.
bifundamentals.
shown in figure 16, we find:
ISU(3)3 = 1+x
a3F22 p
a1a2F22
a1a2F22
a3F22 p
a2a3p
a1a2F22
a1a2a3p
a2a3p
a1a2a3p
a2a3p
a3p
pa2a3 +
a2a3p
a3p
a1a2F22
SU(3) SU(3) global symmetry.
written as:
1 1 . The index can then be
ISU(3)3 = 1 + x2 (3 + [8, 1] + [1, 8])
F1a13 a23 a33 p 6
F1a13 a23 a33 p 6
the instanton part is different. We find:
(, 12 , 21 ,0) = x2 I13 +
3 =
3a2 3 , I13 =
F123 f
of charge conjugation and group reflection.
in figure 17. For the SU(3) SU(5) theory we find:
I3I4B3
I3I4B3B22
B3B1
I2I3B1
B3B1
B1B3I2I3I4
+ I4B32 +
+ I2B1B22 +
+ B1pB3I2I3I4 +
I2B1B22
ISU(3)SU(5) = 1+x2 7+f d+
F Z d
F Z
I3B3B22
the SU(5), and the line is a bifundamental.
then, SU(3) SU(2)2 U(1)3, and the index can be written as:
this differs from the previous cases only by the instanton part. We find:
I(in0,s21t., 21 ,0) = x2 I1B12 +
+ I3B3B22 +
B1B3I2I3
I1B12 +
I3B3B22
+ I2B1B22 +
I2B1B22
+ B1B3I2I3 +
I3B3B22 = fd2 , I2B31 = df2 ,
SU(2) theory is mapped to charge conjugation in the SU(3) SU(5) theory.
Conclusions
is shown in table 1.
1F + SU(2) SU(2) + 1F
SU(2) SU 12 (3) SU 12 (3) SU0(2)
SU1(3) + 4F
SU0(3) + 4F
1F + SU1(3) SU1(3) + 1F
SU0(2) SU0(4) + 2F
2F + SU1(4) SU1(4) + 2F
SU0(2) SU 12 (3) SU 12 (3) SU0(2)
3F + SU0(5) SU0(3) + 1F
U(1)3 SU(2)2 SU(3)
theory, and the last column specifies the quantum global symmetry.
Global symmetry
U(1) SU(2) SU(4)
U(1)2 SU(4)
U(1)3 SU(2)2
U(1)2 SU(2)3
U(1)3 SU(2)4
U(1)3 SU(3)2
M > 2 also from this perspective.
to test these conjectures by index calculations.
hanced symmetries. It will be interesting to also study these theories.
Acknowledgments
under grant no. 1156-124.7/2011.
branes then determine the rank and level of the theory as shown in the figure.
Determining gauge theory parameters from the web
arguments this is related to the original one by:
Na Nb
where Na(Nb) is the number of flavors integrated from above (below).
out. We will see several examples of this in section 5.
is 2 . Applying the same procedure on (b) shows its CS level is 12
1
. From this we also determine
integrating from below corresponds to a positive mass.
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