Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Non-Lagrangian theories from brane junctions †L. Bao,∗1 V. Mitev,∗2 E. Pomoni,∗3 M. Taki∗4 and F. Yagi∗5In a seminal article1) , Gaiotto argued that a large
class, called class S, of N = 2 superconformal ﬁeld theories (SCFT) in four dimensions (4D) can be obtainedby a twisted compactiﬁcation of a 6D (2, 0) SCFT on
a Riemann surface of genus g with n punctures. Thebuilding blocks of the class S theories are tubes andpairs of pants that correspond to gauge groups andmatter multiplets, respectively. Subsequently, a relation between the partition functions of the N = 2
SU (N ) gauge theories and the correlation functions of
the 2D AN −1 Toda CFTs was proposed.2) Computation of 2-point and 3-point functions in a CFT would
in principle yield a complete understanding of the npoint functions.
It is important to note that there is a fundamental
diﬀerence between the SU (2) and the SU (N ), N > 2,cases. For the SU (2) quiver gauge theories2) that arerelated to the 2D Liouville CFT, there is only one typeof puncture on the Riemann surface and hence the Liouville CFT has only one class of 2D 3-point functions
to be calculated. On the other hand, the SU (N ) casewith N > 2 has more than one kind of puncture. Sofar, the case with three special SU (N ) punctures TN
has remained elusive, since neither the TN Nekrasov
partition functions nor the Toda three-point correlators are known. The situation is further aggravatedby the fact that the corresponding 4D theories do nothave a Lagrangian description. Even though there isno known Lagrangian description of the 4D TN theories, we are able to obtain the partition functions fortheir 5D uplift3) using topological strings on the dual
geometry of the 5-brane junctions.
In this paper, we compute the Nekrasov partition
functions of the TN junctions as reﬁned topologicalstring partition functions.4) At this point, we make useof the quite recent conjecture of Iqbal and Vafa5) thatsays that the 5D superconformal index, which is thepartition function on S 4 × S 1 , can be obtained fromthe 5D Nekrasov partition function and thus from thetopological string partition function∫∫5D
I 5D = da |ZNek(a)|2 ∝ da |Ztop (a)|2 .
(1)

coincide with those of Kim et al.,6) computed via localization. When parallel external 5-brane legs appear
in the toric web diagram, the corresponding partition
functions contain extra degrees of freedom. In contrast
to the massive spectrum in 5D, which forms a representation of the Wigner little group SU (2)×SU (2), referred to as the full spin content representation, these
extra states do not transform as a correct representation under the Poincaré symmetry. Therefore, we callthem non-full spin content contributions. We interpret this part as the contribution to the extra degrees
of freedom appearing from the parallel 5-branes explained above. It should therefore be removed. Toobtain the superconformal index from the topological string partition function, we have to eliminate all
the non-full spin content from the partition function.
Schematically, the partition function can be expressed
as a sum of Young diagrams assigned to the productof strip geometries as

The E6 superconformal index is obtained from theT3 Nekrasov partition function by using the idea presented in Iqbal and Vafa,5) and we ﬁnd that the results

1) D. Gaiotto: JHEP 1208, 034 (2012) [arXiv:0904.2715[hep-th]].
2) L. F. Alday, D. Gaiotto, and Y. Tachikawa,
Lett.Math.Phys. 91, 167–197 (2010).
3) F. Benini, S. Benvenuti, and Y. Tachikawa, JHEP 0909,052 (2009).
4) A. Iqbal, C. Kozcaz, and C. Vafa, JHEP 10, 069 (2009).
5) A. Iqbal and C. Vafa, arXiv:1210.3605.
6) H.-C. Kim, S.-S. Kim, and K. Lee, JHEP 1210, 142(2012).

†∗1∗2∗3∗4∗5

arXiv:1301.0721
Chalmers University of Technology
Institut für Mathematik und Institut für Physik, HumboldtUniversität zu Berlin
DESY Theory GroupRIKEN Nishina Center
Korea Institute for Advanced Studyr

ZTN =1Znon-full spinN∑∏stripZi(Y ) .
Y(2)i=1

The factor Znon-full spin is the BPS spectrum, which
does not form a representation of the Poincaré symmetry, and Z strip is the partition function of the strip
geometry.
Finally, the 5D version of the AGTW relation, whichsuggests that the 5D Nekrasov partition functions areequal to the conformal block of q-deformed WN Toda,implies the following relation between the superconformal index and the correlation functions of the corresponding q-deformed Toda ﬁeld theory:
∫�
�2
� 5D
�
I 5D (x, y) = [da]�ZNek(a, m, β, 1,2 )�∝ Vα1 (z1 ) · · · Vαn (zn )q-Toda .

(3)This is an important entry in the dictionary of the5D/2D AGTW correspondence. The partition functions of the TN brane junctions predict, up to an overall coeﬃcient, the corresponding DOZZ formula for thethree-point functions.
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