Instantons from blow-up
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Springer
Received: September 25, 2019 Accepted: October 26, 2019 Published: November 15, 2019
Joonho Kim,a Sung-Soo Kim,b Ki-Hong Lee,c Kimyeong Leea and Jaewon Songa a
Department of Physics and Astronomy & Center for Theoretical Physics, Seoul National University, 1 Gwanak-ro, Gwanak-gu, Seoul 08826, Korea b School of Physics, University of Electronic Science and Technology of China, No. 4, section 2, North Jianshe Road, Chengdu, Sichuan 610054, China c School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We generalize Nakajima-Yoshioka blowup equations to arbitrary gauge group with hypermultiplets in arbitrary representations. Using our blowup equations, we compute the instanton partition functions for 4d N = 2 and 5d N = 1 gauge theories for arbitrary gauge theory with a large class of matter representations, without knowing explicit construction of the instanton moduli space. Our examples include exceptional gauge theories with fundamentals, SO(N ) gauge theories with spinors, and SU(6) gauge theories with rank-3 antisymmetric hypers. Remarkably, the instanton partition function is completely determined by the perturbative part. Keywords: Brane Dynamics in Gauge Theories, Differential and Algebraic Geometry, Field Theories in Higher Dimensions, Solitons Monopoles and Instantons ArXiv ePrint: 1908.11276
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2019)092
JHEP11(2019)092
Instantons from blow-up
Contents 1 Introduction
1 4 4 9 13
3 Examples 3.1 Theories with known ADHM description 3.2 Theories with spinor hypermultiplets 3.3 Theories with an exceptional gauge group 3.4 SU(6) theory with a rank-3 antisymmetric hypermultiplet
17 18 21 26 28
4 Conclusion
33
A One-instanton partition functions
34
1
Introduction
The Seiberg-Witten prepotential provides a complete description for the low energy dynamics of 4d N = 2 or 5d N = 1 gauge theory in its Coulomb branch [1, 2]. It is a function of the vacuum expectation value (VEV) of the scalar in the vector multiplet that parameterizes the Coulomb branch moduli space. Quantum correction to the prepotential is known to be one-loop exact, while there also exist non-perturbative corrections coming from Yang-Mills instantons. An efficient way to compute the fully quantum corrected prepotential F is to study the Nekrasov partition function Z on Ω-deformed C2 or C2 × S 1 . It can be written as the product of the classical, one-loop, and instanton contributions, Z(~a, m, ~ 1 , 2 , q) = Zclass (~a, 1 , 2 , q) Z1-loop (~a, m, ~ 1 , 2 ) Zinst (~a, m, ~ 1 , 2 , q),
(1.1)
where the instanton piece is the fugacity sum over all multi-instanton contributions: Zinst (~a, m, ~ 1 , 2 , q) = 1 +
∞ X
q n Zn (~a, m, ~ 1 , 2 ).
(1.2)
n=1
Once the Nekrasov partition function is known, one can extract the Seiberg-Witten prepotential via taking the 1 ,
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