Download 6d strings and exceptional instantons

6d strings and exceptional instantons
Seok Kim
(Seoul National University)
KEK theory workshop
Dec 9, 2016
Talk based on:
Hee-Cheol Kim (Harvard), SK, Jaemo Park (POSTECH),
“6d strings from new chiral gauge theories” 1608.03919.
Hee-Cheol Kim, Joonho Kim (KIAS), SK, Jaemo Park,
“6d strings and G2 instantons” work in progress.
4d N=2 gauge theories & instantons
•
Simplest: SU(2) without matters [Seiberg, Witten] (1994)
•
Seiberg-Witten effective action in the Coulomb branch:
•
Acquires non-perturbative contributions from Yang-Mills instantons
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Computed microscopically by Nekrasov’s instanton calculus. [Nekrasov] (2002)
•
Coefficients Zk(v) are computed from the following ADHM matrix model
[Atiyah, Drinfeld, Hitchin, Manin] [Christ, Weinberg, Stanton] (1978)
𝑞 𝑘×2 , 𝑞෤ 2×𝑘
SU(2)
𝑎 𝑘×𝑘 , 𝑎෤ 𝑘×𝑘
U(k)
adjoint
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5d N=1 theories
•
R4 x S1: 4d effective action in Coulomb branch. KK tower’s contribution on S1.
•
This is actually where Nekrasov started his studies (then take S1 to be small)
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Zk is a Witten index of k instanton “particles” (& W-bosons in Coulomb branch).
𝑞(𝑡) 𝑘×2 , 𝑞(𝑡)
෤
2×𝑘
𝑎(𝑡) 𝑘×𝑘 , 𝑎(𝑡)
෤
𝑘×𝑘
•
Can use Zk to study higher dimensional quantum field theories (5d/6d SCFTs)
•
Zk computed from the ADHM quantum mechanics
𝑞(𝑡) 𝑘×2 , 𝑞(𝑡)
෤
2×𝑘
SU(2)
U(k)
𝑎(𝑡) 𝑘×𝑘 , 𝑎(𝑡)
෤
𝑘×𝑘
adjoint
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6d N=(1,0) theories… ?
•
R4 x T2: 6d SYM & instanton strings’ partition function…?
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Gauge anomaly cancelation: couple 6d SYM to a tensor multiplet.
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Classical & 1-loop anomalies may cancel via Green-Schwarz mechanism.
-
Without matters, this mechanism works for SU(2), SU(3), G2, SO(8), F4, E6, E7, E8
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Global anomaly free condition. [Bershadsky, Vafa] (1997)
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SU(2) SYM without matters is inconsistent. (Those with matters: Higgsable)
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In a sense, the simplest 6d SYM comes with SU(3) gauge group.
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6d SCFTs and Yang-Mills
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SU(3) pure SYM theory is one of the important building blocks of the 6d SCFTs.
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“atoms” of 6d SCFTs [Morrison,Vafa] [Witten] (1996) [Morrison,Taylor] (2012)
[Heckman, Morrison, Vafa] (2013) [Heckman, Morrison, Rudelius, Vafa] (2015)
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make quivers: glue two CFTs using n=1 SCFT, gauging subgroups of E8.
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“unHiggs” to bigger gauge groups w/ more hypermultiplet matters
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6d SCFTs have self-dual tensor multiplets
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SYM provides a description of the CFT in the tensor branch
VEV
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6d self-dual strings and instanton strings
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6d tensor branch: analogous to Coulomb branch in 4d and 5d
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Coulomb branch: W-bosons (& superpartners), monopoles, dyons …
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6d tensor branch: just self-dual strings
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Examples:
M-strings (strings of 6d (2,0) theory)
M5
M2
E-strings
M5
M2
E8
M9
•
Played important roles in studying 6d SCFTs recently: SUSY partition functions.
•
With 6d gauge symmetry: instanton strings = self-dual strings
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SU(3) strings from instanton strings
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Self-dual strings’ worldsheet QFTs: 2d N=(0,4) SCFTs.
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Their partition functions (elliptic genus) are 6d Nekrasov partition functions.
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2d ADHM gauge theories are UV theories (GLSM) which flow to these SCFTs
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However, SU(3) ADHM has U(k) gauge anomaly. A bad UV description.
SU(3)
U(k)
adjoint
~
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The cure for SU(3)
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Result: can’t make N=(0,4) gauge theory. Should sacrifice some SUSY in UV.
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Add the following N=(0,2) superfields to the anomalous SU(3) ADHM :
-
anomaly: SU(k)
from ADHM ~
from others ~
U(1)
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Can turn on superpotentials to get the correct SU(3) instanton moduli space: but
preserving only N=(0,1) SUSY [H.-C.Kim, SK, J. Park]
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Elliptic genus
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elliptic genus:
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Easy to compute w/ gauge theory
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Our theory: contour integral formula. Residues classified by Young diagrams
[Benini, Eager, Hori, Tachikawa] (2013)
[Flume, Poghossian] [Bruzzo, Fucito, Morales, Tanzini] (2002)
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1d limit agrees with Nekrasov’s SU(3) partition function: alternative “ADHM-like” formalism
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Novel results in 2d: For simplicity, we discuss single string, k=1, in this talk
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Comparison with topological strings
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k=1 (tests also done at k=2,3): computation from topological strings [Haghighat, Klemm, Lockart, Vafa]
black numbers:
computed from
top. strings
red: our prediction
complete agreement
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UnHiggsing to exceptional instantons
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6d SCFTs can be unHiggsed to richer models. Can also generalize 2d QFTs
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unHiggsing to exceptional gauge theories
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G2 & SO(7) instantons w/ 6d matters [Hee-Cheol Kim, Joonho Kim, SK, Jaemo Park] in progress.
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Quiver for SO(7) instantons: only SU(4) ⊂ SO(7) is manifest in UV
Standard SU(4) ADHM
Extra chiral multiplet to make
it a novel “SO(7) ADHM”
Extra 2d field induced
by 6d hypers in 8
-
Can Higgs SO(7) to G2 with one 7. Further Higgsing to our alternative SU(3) ADHM.
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Exceptional instanton partition functions
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Reduce to 1d. ADHM-like descriptions for exceptional instantons
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Nekrasov partition function in the Coulomb branch: e.g. one G2 instanton
One (k=1) G2 instanton partition function
from 1d gauge theory
[Cremonesi, Ferlito, Hanany, Mekareeya] (2014)
•
Can add hypermultiplet matters in 7: We can compute for Nf ≤ 3
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G2 at Nf = 1, and SO(7) at Ns = 2: One can uplift to anomaly-free 2d QFT
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More 2d QFTs for self-dual strings of other 6d atoms
[Hee-Cheol Kim, Joonho Kim, SK, Jaemo Park], work in progress.
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Concluding remarks
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Progress on 6d self-dual strings & their partition functions. Bonus:
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All order genus sums for topological string partition functions
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Novel ADHM-like descriptions of some exceptional instantons.
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Application: Can study S-duality of 4d SCFTs and their 6d uplifts on T2
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6d (2,0) SCFTs: 4d maximal SYM & its 6d uplift
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Nekrasov’s partition functions from self-dual strings [Haghighat, Iqbal, Kozcaz, Lockhart, Vafa]
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High-T asymptotic free energy ~ N3 from S-duality [SK, J. Nahmgoong] work in progress
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N=(1,0) SCFTs: E.g. E-string theory → 4d Minahan-Nemeschansky E8 SCFT.
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Forgets τ in 4d. As one compactifies 6d on S1 to 5d, phase transition to isolated 5d SCFT
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Richer possibilities
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General 6d strings? 5d & 4d compactifications? High T d.o.f. from S-duality?
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