Paper
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Anomaly matching on the Higgs branch
Authors
Hiroyuki Shimizu,Yuji Tachikawa
Gabi Zafrir,Paulo S. Hemsi,Juan F. Escobar-Ibáñez,Maria Marusina,Gayane Tovmasyan,Анна Харитонова,Yi-Tsung Chiu,Mikhail Akselrov,Oleh Kabat,Krzysztof Jakubiak,Lynne Bianchi,Kirill Kolosov,Alyona Mironova,Olesya Dolmatova,Oleg Apolikhin,Keti Chukhrov,Evgenii Morozov,Александра Шахаева,CHUDAMANI PRANESACHAR ANIL KUMAR,Dmitry Akhaladze,MOSA AHer,Alexandra Lunina,CHING-HUI HSU,Ekaterina A. Shibaeva,Ирина Голобородова,Olga Tsybulko,Oksana Hrytsiuta,Guilherme Fioramonte,Karen Guihou,Natalia Uskova,Vladimir Oleshko,Александр Гордиенко,Niyaz Abdikeev,Adel Enikeev,Светлана Рыжкова,Aleksandr Naboka,Sergey Soloviov,Jeffrey Gerber,Максим Романов,Boney Kuriakose,Svitlana Stavytska,Olga Protasenko,Chien-Chou Hou,ChianWen Wang,Andrey Sivkov,Halyna Radchuk,Shih-An Liu,Nailya Urazayeva,Simone Antonio De Rose,German Герман Borlikov Борликов,Sharon Maresse,Robert Sparkes,Héctor Arce,Qian-Sheng Zhang,Olena Stashchuk,Zagira Askarova,Alisa Zubova,Manuela Kulaxizi,Juan Carlos Rodríguez-Cordero,Kunal Vyas,Silvia Nagy,Aziza Ulugova,Simone Giacomelli,Joseph Fotsing,Aaron Balog,Gregory Mendez,ENRICO SPACONE,NATALIIA VOLKOVA,Artem Lyamin,Majeti Narasimha Vara Prasad Prasad,Thierry Marchant,Byron Wicks,Swati Modi,Kazuyuki Furuuchi,Sergey Cherkis,José Antonio Santiago Lastra,Tatyana Cherkashina,Derek Lamb,Jacqueline Radigan,Xiang-Dong Li,Allan Schmitt,Светлана Мерзлякова,Ji Wang,Jorrit Leenaarts,Zoltan Balog,Alexis Finoguenov,Masateru Ishiguro,Bart De Pontieu,Anna Cibinel,Hiroyuki Takahashi,Julian Krolik,Roberto Sanchis-Ojeda,A A Alazba,Youjun LU,Leslie Looney,Nelson Caldwell,Igor Prisukha,Chi-Shu Ko,Евгений Зибарев,Vitalii Serdiuk,Arun Thalapillil,Nadezhda Pripuzova,Nadia Hamdy,Stephen Schauer,Alessandra Gnecchi,Evgeniya Zhuravleva,Nataliya Velts,Aleksandr Danchenkov,Orren Schneider,Nina Polyakova,Alexey Krasnov,Anna Shchetinina,Sergey Perminov,Anna Eremkina,Anna Sugak,Evgeniya Glukhova,Adrian Slater,Аліна Бурляй,Dr Siddarth Rai,Susanna Sologova,RAJIV MOHAN DAVID,Ksenia Dobryakova,Halyna Matiko,Zalina Batyrova,Santino Di Berardino,Alexander Kulkov,Tomonari Mizoguchi,Юлия Мамырко,Сергей Подлесный,Gabriele Franciolini,Davide Racco,Li Zeng,ANNE HOULLIER,CHIU-KUEI NIEN,Nadiia Pysarenko,Valentyn Khokhlov,Andrei Sergeev,Saglara Mirzaeva,CHUN-HUNG LIN,Hanna Tereshchenko,Mykola Veselov,Halil ARI,Marco Silva,Robert Simpson,CHIA-CHIEN CHANG,Petr Balling,Kseniia Yutilova,Adrian Dawson,Priyotosh Bandyoapdhyay,Ragab Mousa,Tatyana Bukatina,Joko Siswanto,Fupeng Zhang,YAO-CHIEN Alex CHANG,Katarzyna Czajkowska,Alexander Andreev,Николай Первышин,Gaspar Bakos,George Kosar,Vladislav Schukin,Thomas Mak,Samuel Grunblatt,Yegor ,Irina Erofeeva,Kenneth Park,ralph bohlin,Pavel Bakhvalov,Philip Waite,Jeppe Andersen,Yurii Poskrypko,Thomas Woodcock,Luís Cerqueira,Julius Ibenthal,Cheng-Hung Tsai,Ta-Peng Wu,Julian Avery,David Turton,Marina Dubyago,Alexander Teplov,Viacheslav Zaitsev,Ewa Wójcik,Anna Gusakova,Александр Френкель,Anna Markova,Satoshi Okuzumi,KUEI-LAN TSAI,Konstantin Rukomojnikov,Антон Сурков,Anna Vitiuk,宜穎 詹,Марианна Хобейш,Emilia Herman,Andreas Maier,Santiago Cabrera,James Edwards,Gonzalo Pizarro,Yang Zhou,Denis Suslov,Georgina Tsolidis,Yunfeng Jiang,Rina Štrajn,Andrii Iudin,Helen Andersson,Emmanuele Battista,Tony Honorato,Carola Westermeier,Carmen Giunta,Marina Klimova,ANNA CORRAL FULLA,Irina Ponomareva,Yana Sizova,Ion Preda,Azam Imomov,Timothy Birdsall,Babalola Ogunsua,Дмитро, Dmytro Якимчук, Yakymchuk,Andreas Mölder,Ігор Калімбет,Leonid Tsepov,Дзюндзя, Dzyundzya Оксана, Oksana,Fernando Plata Parga,Viliam Pavlík,Graeme Smith,Alexandra Volosova,Геннадий Орехов,Chih Hsin Wang,Алексей Кисляков,Евгений Журбин,Steven Lanham,Polina Kuznetsova,Irina Goroshinskaya,Antonio GESUALDO,Alexander Kazakov,N/A ,Dana Krauß,Marina ,Anton Dronov,Timur Radbil,Chernukhin Roman,Vadim Timofeev,Aleksey Khoreshok,Aleksandra Malʼchik,Buyalich Konstantin,Roman Nekrasov,Alexandra Zakharova,Darja Kaļužnaja,Elena Shandarova,Konstantin Cherevyk,Hossein Motamedi,Michalis Vousdoukas,Ольга Александрова,Alexey Kuzmenkov,Valery Poturay,Oleg Shvetsov,Ivan Trushin,sarhad namiq,Mike Blake,Elena Svechnikova,Antonio Costantini,Leonid Fituni,Jennifer Graham,Soraya Paz,Daniela Neuhofer,Laylson da Silva Borges,Alexey Zelepugin,Zahra Lari,Elena Danilova,Mahmoud Safari,Alexander Tchoudinov,Maxim Kharin,Mohit P. 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arXiv:1703.01013v1 [hep-th] 3 Mar 2017
IPMU-17-0032
Anomaly matching on the Higgs branch
Hiroyuki Shimizu, Yuji Tachikawa and Gabi Zafrir
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
We point out that we can almost always determine by the anomaly matching the full anomaly
polynomial of a supersymmetric theory in 2d, 4d or 6d if we assume that its Higgs branch is the
one-instanton moduli space of some group G. This method not only provides by far the simplest
method to compute the central charges of known theories of this class, e.g. 4d E6,7,8 theories of
Minahan and Nemeschansky or the 6d E-string theory, but also gives us new pieces of information
about unknown theories of this class.
IPMU-17-0032
Anomaly matching on the Higgs branch
Hiroyuki Shimizu, Yuji Tachikawa and Gabi Zafrir
Kavli Institute for the Physics and Mathematics of the Universe,
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
We point out that we can almost always determine by the anomaly matching the full anomaly
polynomial of a supersymmetric theory in 2d, 4d or 6d if we assume that its Higgs branch is the
one-instanton moduli space of some group G. This method not only provides by far the simplest
method to compute the central charges of known theories of this class, e.g. 4d E6,7,8 theories of
Minahan and Nemeschansky or the 6d E-string theory, but also gives us new pieces of information
about unknown theories of this class.
1 Introduction
Instantons of classical groups can be described in terms of the ADHM construction [1], which
can in turn be realized as the Higgs branch of supersymmetric gauge theories [2, 3]. These gauge
theories arise as the worldvolume theories on perturbative p-branes probing perturbative (p +
4)-branes, and the motion into the Higgs branch corresponds to the process where p-branes get
absorbed as instantons of the gauge fields on (p + 4)-branes.
In string/M/F theory, there are also non-perturbative branes that host exceptional gauge groups,
and if we probe them by lower-dimensional branes, we get supersymmetric theories whose Higgs
branch equals to the instanton moduli spaces of exceptional groups. Among them we can count
the 4d theories of Minahan and Nemeschansky [4, 5] for E6,7,8 instantons and the 6d E-string
theory [6, 7].
The theories obtained this way do not usually have any conventional Lagrangian descriptions,
and were therefore rather difficult to study. Even their anomaly polynomials, or equivalently the
conformal central charges assuming that they become superconformal in the infrared, needed to
be computed first with stringy techniques [8, 9] and then with rather lengthy field theoretical
arguments on the Coulomb branch in 4d or on the tensor branch in 6d [10, 11, 12].
In this paper, we point out that the anomaly matching on the Higgs branch almost always
allows us to determine the full anomaly polynomial, when the theory is 6d N = (1, 0), 4d N = 2,
or 2d N = (4, 0), and when the Higgs branch is assumed to be the one-instanton moduli space of
some group G. This is because on the generic point of the Higgs branch the theory becomes free
and the unbroken symmetry still knows the SU(2)R symmetry at the origin.
This method provides the simplest way to compute the anomaly polynomials of 4d theories of
Minahan and Nemeschansky and the 6d E-string theory. But more importantly, this method gives
us new pieces of information about a theory whose Higgs branch is the one-instanton moduli space
of the group G, even when no string/M/F theory construction is known. For example, in [13], the
conformal bootstrap method was used to determine the conformal central charges of the 4d theory
whose Higgs branch is the one-instanton moduli space of G2 or F4. Our method reproduces the
values they obtained, and not only that, we find a strong indication that the F4 theory does not
exist because of a field theoretical inconsistency. Similarly, we will see that there cannot be any
6d E6,7 theory.
The rest of the paper is organized as follows. In Section 2, we describe in more detail how
the anomaly matching on the Higgs branch works if the Higgs branch is the one-instanton moduli
space of some group G. In Section 3, we summarize the results which we obtained in this paper.
Then in Section 4, 5, 6, we study the 6d N = (1, 0) theories, the 4d N = 2 theories, and the 2d
N = (4, 0) theories in turn. In Appendix A, we collect the formulas for characteristic classes used
throughout in this paper.
1
Instantons of classical groups can be described in terms of the ADHM construction [1], which
can in turn be realized as the Higgs branch of supersymmetric gauge theories [2, 3]. These gauge
theories arise as the worldvolume theories on perturbative p-branes probing perturbative (p +
4)-branes, and the motion into the Higgs branch corresponds to the process where p-branes get
absorbed as instantons of the gauge fields on (p + 4)-branes.
In string/M/F theory, there are also non-perturbative branes that host exceptional gauge groups,
and if we probe them by lower-dimensional branes, we get supersymmetric theories whose Higgs
branch equals to the instanton moduli spaces of exceptional groups. Among them we can count
the 4d theories of Minahan and Nemeschansky [4, 5] for E6,7,8 instantons and the 6d E-string
theory [6, 7].
The theories obtained this way do not usually have any conventional Lagrangian descriptions,
and were therefore rather difficult to study. Even their anomaly polynomials, or equivalently the
conformal central charges assuming that they become superconformal in the infrared, needed to
be computed first with stringy techniques [8, 9] and then with rather lengthy field theoretical
arguments on the Coulomb branch in 4d or on the tensor branch in 6d [10, 11, 12].
In this paper, we point out that the anomaly matching on the Higgs branch almost always
allows us to determine the full anomaly polynomial, when the theory is 6d N = (1, 0), 4d N = 2,
or 2d N = (4, 0), and when the Higgs branch is assumed to be the one-instanton moduli space of
some group G. This is because on the generic point of the Higgs branch the theory becomes free
and the unbroken symmetry still knows the SU(2)R symmetry at the origin.
This method provides the simplest way to compute the anomaly polynomials of 4d theories of
Minahan and Nemeschansky and the 6d E-string theory. But more importantly, this method gives
us new pieces of information about a theory whose Higgs branch is the one-instanton moduli space
of the group G, even when no string/M/F theory construction is known. For example, in [13], the
conformal bootstrap method was used to determine the conformal central charges of the 4d theory
whose Higgs branch is the one-instanton moduli space of G2 or F4. Our method reproduces the
values they obtained, and not only that, we find a strong indication that the F4 theory does not
exist because of a field theoretical inconsistency. Similarly, we will see that there cannot be any
6d E6,7 theory.
The rest of the paper is organized as follows. In Section 2, we describe in more detail how
the anomaly matching on the Higgs branch works if the Higgs branch is the one-instanton moduli
space of some group G. In Section 3, we summarize the results which we obtained in this paper.
Then in Section 4, 5, 6, we study the 6d N = (1, 0) theories, the 4d N = 2 theories, and the 2d
N = (4, 0) theories in turn. In Appendix A, we collect the formulas for characteristic classes used
throughout in this paper.
1
G h∨ G′ R′ short comment
SU(n) n U(1)F × SU(n − 2) (n − 2)−n ⊕ (n − 2)+n
SO(n) n − 2 SU(2)F × SO(n − 4) 2F ⊗ (n − 4)
Sp(n) n + 1 Sp(n − 1) 2n − 2
E6 12 SU(6) 20 3-index antisym.
E7 18 SO(12) 32 chiral spinor.
E8 30 E7 56
F4 9 Sp(3) 14′ 3-index antisym. traceless.
G2 4 SU(2)F 4 3-index sym.
Table 1: The data. For SU(n), U(1)F is normalized so that n splits as (n − 2)−2 and 2n−2. For
SO(n), n is assumed to be ≥ 5.
2 Basic idea
We consider a theory with 6d N = (1, 0) or 4d N = 2 or 2d N = (4, 0) supersymmetry has a Higgs
branch given by the one-instanton moduli space MG of a group G.
Geometric data: Let us first recall some basic information on MG, whose detail can be found
e.g. in [14] and the references therein. The quaternionic dimension of MG is h∨(G) − 1. We note
that for G = Sp(n), the one-instanton moduli space is simply Hn/Z2, where H is the space of
quaternions.
Furthermore, the moduli space is smooth on a generic point, and the symmetry SU(2)R × G
acting on MG is broken to SU(2)D × G′, where SU(2)X × G′ ⊂ G is a particular subgroup
described in more detail below and SU(2)D is the diagonal subgroup of SU(2)R and SU(2)X .
The subgroup SU(2)X is the SU(2) subgroup associated to the highest root of G and G′ is its
commutant within G. The tangent space of MG at a generic point transforms under SU(2)X × G′
as a neutral hypermultiplet and a charged half-hypermultiplet in a representation R, with the rule
g = g′ ⊕ su(2) ⊕ R. (2.1)
Here R is always of the form of the doublet of SU(2)X tensored with a representation R′ of G′.
The subgroup G′ and the representation R′ are given in the table 1.
Strategy of the matching: Now let us explain how the anomaly matching on the Higgs branch
works. At the origin, the theory has the symmetry SU(2)R × G where SU(2)R is (part of) the
R-symmetry.
2
SU(n) n U(1)F × SU(n − 2) (n − 2)−n ⊕ (n − 2)+n
SO(n) n − 2 SU(2)F × SO(n − 4) 2F ⊗ (n − 4)
Sp(n) n + 1 Sp(n − 1) 2n − 2
E6 12 SU(6) 20 3-index antisym.
E7 18 SO(12) 32 chiral spinor.
E8 30 E7 56
F4 9 Sp(3) 14′ 3-index antisym. traceless.
G2 4 SU(2)F 4 3-index sym.
Table 1: The data. For SU(n), U(1)F is normalized so that n splits as (n − 2)−2 and 2n−2. For
SO(n), n is assumed to be ≥ 5.
2 Basic idea
We consider a theory with 6d N = (1, 0) or 4d N = 2 or 2d N = (4, 0) supersymmetry has a Higgs
branch given by the one-instanton moduli space MG of a group G.
Geometric data: Let us first recall some basic information on MG, whose detail can be found
e.g. in [14] and the references therein. The quaternionic dimension of MG is h∨(G) − 1. We note
that for G = Sp(n), the one-instanton moduli space is simply Hn/Z2, where H is the space of
quaternions.
Furthermore, the moduli space is smooth on a generic point, and the symmetry SU(2)R × G
acting on MG is broken to SU(2)D × G′, where SU(2)X × G′ ⊂ G is a particular subgroup
described in more detail below and SU(2)D is the diagonal subgroup of SU(2)R and SU(2)X .
The subgroup SU(2)X is the SU(2) subgroup associated to the highest root of G and G′ is its
commutant within G. The tangent space of MG at a generic point transforms under SU(2)X × G′
as a neutral hypermultiplet and a charged half-hypermultiplet in a representation R, with the rule
g = g′ ⊕ su(2) ⊕ R. (2.1)
Here R is always of the form of the doublet of SU(2)X tensored with a representation R′ of G′.
The subgroup G′ and the representation R′ are given in the table 1.
Strategy of the matching: Now let us explain how the anomaly matching on the Higgs branch
works. At the origin, the theory has the symmetry SU(2)R × G where SU(2)R is (part of) the
R-symmetry.
2
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