Argyres-Douglas matter and S-duality. Part II
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Springer
Received: January 8, 2018 Accepted: March 25, 2018 Published: March 29, 2018
Dan Xiea,b and Ke Yec,d a
Center of Mathematical Sciences and Applications, Harvard University, Cambridge, 02138, U.S.A. b Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. c Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, U.S.A. d Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, U.S.A.
E-mail: [email protected], [email protected] Abstract: We study S-duality of Argyres-Douglas theories obtained by compactification of 6d (2,0) theories of ADE type on a sphere with irregular punctures. The weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points, among which three punctured sphere represents isolated superconformal theories. We also discuss twisted irregular punctures and their S-duality. Keywords: Supersymmetry and Duality, Differential and Algebraic Geometry ArXiv ePrint: 1711.06684
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP03(2018)186
JHEP03(2018)186
Argyres-Douglas matter and S-duality. Part II
Contents 1 Introduction
2 3 4 4 7 7 9 11 14 15
3 Mapping to a punctured Riemann surface 3.1 AD matter and S-duality 3.2 Central charges
17 19 20
4 S-duality for DN theory 4.1 Class (k, 1) 4.1.1 Coulomb branch spectrum 4.1.2 Constraints on coefficient matrices 4.1.3 Generating S-duality frame 4.2 Class (k, b) 4.2.1 Coulomb branch spectrum and degenerating coefficient matrices 4.2.2 Duality frames 4.3 Z2 -twisted theory 4.3.1 Twisted regular punctures 4.3.2 Twisted irregular puncture 4.3.3 S-duality for twisted DN theory of class (k, 1)
21 21 21 22 24 32 32 33 34 35 36 37
5 Comments on S-duality for E-type theories 5.1 Irregular puncture and S-duality for E6 theory 5.1.1 S-duality for E6 theory 5.2 E7 and E8 theory
39 39 40 42
6 Conclusion and discussion
43
A Type IIB construction for AD theories
44
B Grading of Lie algebra from nilpotent orbit
45
C Recover missing Kac diagrams
48
–1–
JHEP03(2018)186
2 SCFTs from M5 branes 2.1 Classification of punctures 2.1.1 Regular punctures 2.2 Irregular puncture 2.2.1 Grading of the Lie algebra 2.2.2 From irregular puncture to parameters in SCFT 2.2.3 Degeneration and graded Coulomb branch dimension 2.2.4 Constraint from conformal invariance 2.3 SW curve and Newton polygon
1
Introduction
• We revisit the classification of irregular singularity of class (k, b) in [5, 8]: Tk
Φ∼ z
2+ kb
+
X −b≤l −b. • We successfully represent our theory by an auxiliary punctured sphere from the data defining our theory from 6d (2,0) SCFT framework, and we then find weakly coupled gauge theory descriptions by studying degeneration limit of new punctured sphere. For instance, we find that for g = DN , b = 1 and large k and all coefficient matrices regular semisimple, one typical duality frame looks like
T1
......
SO(6) T2
SO(2N − 2) TN −2
T3 [1;2i]×(k+1) ,[
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