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5d/6d DE instantons from trivalent gluing of web diagrams
Received: March
Published for SISSA by Springer
Hirotaka Hayashi 0 1 2 5 6 7
Kantaro Ohmori 0 1 2 3 4 6 7
0 7-3-1 Hongo , Bunkyo-ku, Tokyo 113-0033 , Japan
1 1 Einstein Drive , Princeton, NJ 08540 , U.S.A
2 4-1-1 Kitakaname , Hiratsuka-shi, Kanagawa 259-1292 , Japan
3 School of Natural Sciences, Institute for Advanced Study
4 Department of Physics, Faculty of Science, The University of Tokyo
5 Department of Physics, School of Science, Tokai University
6 Open Access , c The Authors
7 sions, Topological Strings , F-Theory
We propose a new prescription for computing the Nekrasov partition functions of ve-dimensional theories with eight supercharges realized by gauging non-perturbative avor symmetries of three ve-dimensional superconformal eld theories. The topological vertex formalism gives a way to compute the partition functions of the matter theories with avor instanton backgrounds, and the gauging is achieved by summing over Young diagrams. We apply the prescription to calculate the Nekrasov partition functions of various ve-dimensional gauge theories such as SO(2N ) gauge theories with or without hypermultiplets in the vector representation and also pure E6; E7; E8 gauge theories. Furthermore, the technique can be applied to computations of the Nekrasov partition functions of vedimensional theories which arise from circle compacti cations of six-dimensional minimal superconformal eld theories characterized by the gauge groups SU(3); SO(8); E6; E7; E8. We exemplify our method by comparing some of the obtained partition functions with known results and nd perfect agreement. We also present a prescription of extending the gluing rule to the re ned topological vertex.
Conformal Field Models in String Theory; Field Theories in Higher Dimen-
-
DE instantons from
trivalent gluing of web
1 Introduction
2 A dual description of 5d gauge theory with D; E-type gauge group 2.1 5d SO(2N + 4) gauge theory
2.2 5d pure E6; E7; E8 gauge theories
3 Gluing rule and 5d SO(2N + 4) gauge theory
3.2 5d pure SO(2N + 4) gauge theory
Example: 5d pure SO(8) gauge theory
3.3 Adding avors
Example: 5d SO(8) gauge theory with four avors
4 5d gauge theory with E-type gauge group
4.1 5d pure E6 gauge theory
4.2 5d pure E7 gauge theory
4.3 5d pure E8 gauge theory
5 A 5d description of non-Higgsable clusters 5.1 5.2 5.3
O( n) model with n = 6; 8; 12
Another non-Higgsable cluster
6 Re nement
7 Conclusion
A 5d SO(2N + 3) gauge theory
B Some formulae for computation
B.1 Re ned topological vertex
B.2 Nekrasov partition function
B.3 Schur functions
Re ned partition function of Db2(SU(2)) matter from
6.2 Examples: 5d pure SO(8) gauge theory and O( 4) model
The (re ned) topological vertex is a powerful tool to compute the all genus topological
string amplitudes for toric Calabi-Yau threefolds [1{4]. One can compute the full
topological string partition function like a Feynman diagram-like method and it can yield the full
list of the Gromov-Witten invariants and the Gopakumar-Vafa invariants of a toric
CalabiYau threefold in principle. The topological string partition function also has a physical
interpretation through string theory or M-theory. When we consider M-theory on a
noncompact Calabi-Yau threefold with a compact base that is contractible, the low energy
e ective eld theory gives rise to a
ve-dimensional (5d) theory with eight supercharges
which has a ultraviolet (UV) completion [5{8]. Then M2-branes wrapping various
holomorphic curves in the Calabi-Yau threefold yield BPS particles in the 5d theory. Therefore,
the curve counting for a non-compact Calabi-Yau threefold is equivalent to the counting of
BPS particles of the 5d theory and this implies that the topological string partition
function is equal to the Nekrasov partition function up to some extra factors. Indeed several
checks of the equality have been done for example in [9{13] for 5d SU(N ) gauge theories
with avors by utilizing the method of the topological vertex.
Recently, the topological vertex formalism has been extended for computing the
topological string partition functions of certain non-toric Calabi-Yau threefolds [14{16].1 The
new method makes use of a Higgs prescription of the superconformal index in [19, 20].2
In fact, some non-toric Calabi-Yau threefold can be obtained from a topology changing
transition or a Higgsing from a toric Calabi-Yau threefold. Then applying the Higgsing
prescription for the topological string partition function of the \UV" Calabi-Yau
threefold gives rise to the topological string partition function of the \infrared" (IR) non-toric
Calabi-Yau threefold. This new technique enables us to compute the Nekrasov partition
functions of the 5d rank one E7; E8 theories [14, 15], the 5d SU(N ) gauge theory with
a hypermultiplet in the antisymmetric representation and also the 5d Sp(N ) gauge
theory [25]. Furthermore, it has been also applied to the calculation of the Nekrasov partition
functions of 5d theories which has a six-d (...truncated)