Paper
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Irregular conformal block, spectral curve and flow equations
Authors
Sang Kwan Choi,Chaiho Rim
Hong Zhang,Dunja Peric,Chimit Ochirov,Pyrieva Ekaterina,Magomed Akhmedov,Alexei Dmitriev,MITCHELL BEGELMAN,Andrej Stopar,Chung-Chih Hung,Александр Довженок,Krzysztof Jakubiak,Agnieszka Judkowiak,Andreas Hofmann,Alyona Mironova,Krystyna Tarasova,Aleksandra Kotlova,Anton Salmin,Irina Zubina,Светлана Денисова,Irina Snegireva,Tatyana Sergeeva,Юрий Мизерницкий,Yulia Ostankova,IRINA NIKITINA,Дмитрий Макаркин,Elena Kornetova,Tatiana Rovinskaya,Alexander Kornetov,Olga Fadeeva,Aidar Ishmukhametov,Tamara Duchyminska,Julian Hopkins,Miroslav Kubelskiy,Владимир Сизонов,Niyaz Abdikeev,Elena Bolgova,Aleksandr Naboka,Galina Ivanova,Tong Liu,Nikita Pavlov,Elena Aleksandrovna Kirova,DIGAMBER PAWAR,Ana Tagliari,Alessandro Pini,Celia Souza,Petr Shumilov,Trevor Glasbey,Sarah Morrison,Joachim Kaiser,Will Best,Jean Young,Giovanni Grilli di Cortona,Rajesh Dachiraju,Mikhail Trifonov,Laura Ledesma,Stefanos Katmadas,Carolyn Dzierba,Reghard Brits,Aaron Vincent,Luis Garcés-Erice,Валентина Вікторівна Виговська ,Lutz S. Goehring,NATALIIA VOLKOVA,Elena Meskina,Nathaniel Butlin,Byron Wicks,Romulo Rougemont,Gregory Carven,Stepan Jenicek,Carlos Alberto Vaquera-Araujo,Victor Shnirelman,Svetlana Prokhorova,Olga Ivanishcheva,Martha Haynes,Basilio Ruiz Cobo,Ryun Young Kwon,Heather Flewelling,Lauranne Lanz,Claes Fransson,Ji-Ming Shi,Eric E. Martinez-Garcia,Светлана Петрова,Wakana Iwakami,Pu Du,Joseph Barranco,Amin Abou Ibrahim,Elvira Grinberg,Tsuyoshi Sawada,Ronnie Andersson,Ronnie Machielsen,Wolfgang Doerr,Natalia Alexeeva,Alex May,Agnese Bissi,Yulia Potekhina,Таисия Скоробогатова,Tatiana Berezovskaia,Andrea Billi,Andrea Messori,Evgeniya Glukhova,Monika Płużyczka,VILLEVO ADANHOUNME,Ze-Jia Chen,Dmitry Kudryavtsev,Chen-Yen Lai,Valeriya Novikova,Sinisa Mesarovic,Константин Григоричев,Василий Акимкин,Pablo Poggi,Philip Yuzlikeev,Caciana Costa,KUO-MING LIN,Oleg Kosolapov,Marcin Brodecki,Sachi Horibata,Sandra Márcia Tietz Marques,Tatiana Gavruseva,Nina Skitina,Nataliia Poliakova,Lysyuk Vladimir,Ірина Топалова,Cristiano de Sousa Zanetti,Andrea Mandarino,Сергей, Serhii Павлов, Pavlov,Tatyana Bukatina,Oleh Romaniv,MARIA TERESA MORIES,Evgeny Schastnyi,Erhan ISCAN,saffet erdoğan,Jan Golembiewski,Stanislav Andriainen,Andrey Chukreyev,Alphia Karaseva,Esperanza Guarneros-Reyes,Ryo Suzuki,David Sarkisyan,Mukul Agnihotri,Rudolph Kalveks,Ryan Ubelhor,Joelle Maslak,Roland Grappin,Gary Coleman,Olga Kravtsova,د. محمد خالد الرهاوي,Leonardo Trombetta,RONG-LIN WANG,Bi Benié Yves BOLI,Anna Gusakova,Matias Leoni-Olivera,Александр Френкель,Liudmyla Shevchuk,Peter van Hoof,Nanyao Lu,Satyanarayana Kalidindi,Антон Сурков,Md. Aminul Islam,Daniel Katukov,Saroj Bembalkar,Ekaterina Makarenko,Shoichi YAMADA,xiaowen song,YUH-LONG YANG,Wan-Zhe Feng,Hiroki Sakamoto,Emre Gurkanli,Karapet Mkrtchyan,Stefano Catani,N/A ,David Outomuro,Carlos Scarinci,Madhusudhan Raman,Max Zoller,Francesco Benini,Pratik Rath,Bette Otto-Bliesner,Anton Bredikhin,Olga Galkina,Vitaly Korchagin,Barbara Sionek,Саидбек Шамилев,Zofia Sokołowicz,Michi Miura,Артем Баталов,Igor Pronin,Mariam Voskanyan,Manoshkina Elena ,Casimir Ehrenborg,Sambit Kumar Dash,Feng Lin,ELena Sapir,Filip Radovanović,Jericó Bello Bello,Vanessa Schein,Cisco de Bruijn,Shamil Kashaf,Benrachou Djamel Eddine,Scott Winroth,Badri Ranganathan,Alexander Kazakov,Mikhail Ryazantsev,YI-LIN CHEN,Nathalia Padilla,Stefan Prohazka,Yuriy Prysyazhnyuk,Сергей Воеводин,Nikolai Tolstov,Shamima Akhter,Jun-qiang Liu,Igor Shevchenko,Kunal Jani,Наталья Бегункова,Elena Balashova,Darja Kaļužnaja,michael griffith,Valeriia Rostovtseva,Nuria Martín,Olivier Gabriel,Hernando Vargas-Herrera,Ольга Лесняк,Jougi Yamashita,Svetlana Ivanova,Александр Щемелев,XOLILE NOKULUNGA MASHWAMA,Thanh Binh Do,Svetlana Bratchenko,Владислав Тєтєрєв,Rokhani Hasbullah,Alexander Lukiyanchenko,Nadezhda Morozova,Michał Górecki,Rafał Smulski,Miguel Ángel Esparza Torres,Tuguldur Sukhbold,Pecheritsa Elena,Gerard (Gerry) Strange,Olena Sakovska,Laura Flamand,Mohamad Shalaby,Nogan Badmaeva,aleksandr rozin,Olga Roshchina,Christoph Pfrommer,Tatiana Anisimova,José Marques Basílio,Javier Matulich,Marco Hufnagel,Safarbi Shkhagapsoev,新逸 徐,Brian Sheil,Jongchul Chae,Michaela Schier,Zdenek Ruzicka,Renat Bekkin,Владислав Панченко,Natalya Zhorzh,Pei Shan Lee,Sergey Ermashkevich,Daniel Reichelt,Diana Arutyunova,Сергій Приймак,Dmitry Neshumaev,Emily Sandford,Алексей Лебедев,Anastasia Tanchuk,Natalya Prokopeva,Andrey Andreev,Fengyan Wang,Ilya Lapovok,Iraida Nekhaeva,Milton Shintaku,Vitaly Kablukov,Jakob Salzer,Haris Kremo,María del Pilar Ramirez Salazar,Søren Kudsk-Iversen,Ihor Kulyniak,Rodolfo José Bueno Rogerio,Naeem Akram,Yulia Gilyarova,Sergey Durmanov,Oksana Medvedeva,Alina Zhuk,Aurélio Ferreira Borges,Li-Yu ,Melese Abate Reta,James Yang,Евгения Каширина,Bruno Salvatore MESSINA,Petri Räisänen,Maria Marta Amorim,Maltseva Ludmila A.,Thomas Walsh,Wojciech Górecki,Владимир Федоров,Ігор Діордіца,Blessing Onoriode Boloje,Anna Augustyńska-Prejsnar,Rhys Fitzgerald,Oskar Englund,Dmytro Sugonyako,Han-Chih Chang,Bradley Markle,Ivan Shcherbatov,Michael C Liu,Zarko Mijajlovic,Astghik Sukiasyan,Gisela Susanne Bahr,John Airey,Mariya Darmostukova,Wei Li,Rose Hsiu-li Juan,David Schroeder,Olga Beketova,aline albi-feldzer,Natalia Pomytkina,Dmitry Fomichev,Chung-I Chen,Alina Mikhnova,Valentyna Hrebenova,Shen-Haw Ju,Laurent Gizon,Maria Klara Wolters,Bruno Lafetá,Iuliia Kobzieva,Catarina Lemos,Anatolii Rubanenko,Luca Santoni,Tünde Pusztahelyi,Оксана Макушкина,Irina Chernova,Руслан Костюкевич,Olga Krivosheina,Alexander Pchelintsev,Zhi-Xiang Zhang,Anton Agapovichev,Meta Sienkiewicz,Сергей Спицын,Ainura Satarkulova,Nadezhda Ryumshina,Виктория Власова,Anna Kornietskaya,Rachel Riemann,Inessa Shevtsova,Matheus Rodrigues Lopes,Ольга Валентинова,Renata Zdařilová,Alexey Zimin,Herman Fesenko,Dionysios Pantazatos,John Levine,David Chappell,Arnold Reinhold,James W. 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arXiv:1510.09060v2 [hep-th] 4 Mar 2016
Irregular conformal block, spectral curve and flow
equations
Sang Kwan Choi1, Chaiho Rim2 and Hong Zhang3
Department of Physics and Center for Quantum Spacetime (CQUeST)
Sogang University, Seoul 121-742, Korea
Abstract
Irregular conformal block is motivated by the Argyres-Douglas type of N=2 super
conformal gauge theory. We investigate the classical/NS limit of irregular conformal block
using the spectral curve on a Riemann surface with irregular punctures, which is equivalent
to the loop equation of irregular matrix model. The spectral curve is reduced to the second
order (Virasoro symmetry, SU (2) for the gauge theory) and third order (W3 symmetry,
SU (3)) differential equations of a polynomial with finite degree. The conformal and W
symmetry generate the flow equations in the spectral curve and determine the irregular
conformal block, hence the partition function of the Argyres-Douglas theory ala AGT
conjecture.
1 Introduction
Irregular conformal block (ICB) is closely related with Argyres-Douglas type (AD) of N=2
super conformal gauge theory in four dimensions [1]. AD type has the non-trivial infrared
fixed point on the Coulomb branch and does not allow marginal deformation. Therefore, AD
type of gauge theory is considered as a special class of super conformal gauge theory.
According to AGT [2], the Nekrasov partition function [3, 4, 5] of the gauge theory is
equivalent to the conformal block of Liouville vertex operators in two dimensions. This con-
nection is understood using the twisted compactification of the six dimensional N = (2, 0)
theory on a punctured Riemann surface [6, 7, 8]. In this context, the Seiberg-Witten curve of
the four-dimensional theory is identified with the spectral curve of Hitchin system on the Rie-
mann surface with regular punctures. The Hitchin system has simple poles and the residues
are associated with the mass parameter of the gauge theory [9, 10]. On the other hand, the
AD type theory is characterized in terms of irregular punctures, poles of higher order [11, 12].
Therefore, the irregular puncture is the key point to understand AD type theory.
It is noted that irregular punctures appear when the regular conformal block has the collid-
ing limit [13, 14]. The colliding limit is a fusion of vertex operators so that multiple moments
of Liouville charges of the vertex operators are present. It is like the collection of charges dis-
tributed over a small region whose collection is viewed as an idealized system of total charge,
dipole, quadrupole and multi-poles. As a consequence, the irregular punctures maintain the
conformal symmetry but change the conformal state. Note that a regular puncture on the
1email:hermit1231@sogang.ac.kr
2email:rimpine@sogang.ac.kr
3email:kilar@sogang.ac.kr
1
Irregular conformal block, spectral curve and flow
equations
Sang Kwan Choi1, Chaiho Rim2 and Hong Zhang3
Department of Physics and Center for Quantum Spacetime (CQUeST)
Sogang University, Seoul 121-742, Korea
Abstract
Irregular conformal block is motivated by the Argyres-Douglas type of N=2 super
conformal gauge theory. We investigate the classical/NS limit of irregular conformal block
using the spectral curve on a Riemann surface with irregular punctures, which is equivalent
to the loop equation of irregular matrix model. The spectral curve is reduced to the second
order (Virasoro symmetry, SU (2) for the gauge theory) and third order (W3 symmetry,
SU (3)) differential equations of a polynomial with finite degree. The conformal and W
symmetry generate the flow equations in the spectral curve and determine the irregular
conformal block, hence the partition function of the Argyres-Douglas theory ala AGT
conjecture.
1 Introduction
Irregular conformal block (ICB) is closely related with Argyres-Douglas type (AD) of N=2
super conformal gauge theory in four dimensions [1]. AD type has the non-trivial infrared
fixed point on the Coulomb branch and does not allow marginal deformation. Therefore, AD
type of gauge theory is considered as a special class of super conformal gauge theory.
According to AGT [2], the Nekrasov partition function [3, 4, 5] of the gauge theory is
equivalent to the conformal block of Liouville vertex operators in two dimensions. This con-
nection is understood using the twisted compactification of the six dimensional N = (2, 0)
theory on a punctured Riemann surface [6, 7, 8]. In this context, the Seiberg-Witten curve of
the four-dimensional theory is identified with the spectral curve of Hitchin system on the Rie-
mann surface with regular punctures. The Hitchin system has simple poles and the residues
are associated with the mass parameter of the gauge theory [9, 10]. On the other hand, the
AD type theory is characterized in terms of irregular punctures, poles of higher order [11, 12].
Therefore, the irregular puncture is the key point to understand AD type theory.
It is noted that irregular punctures appear when the regular conformal block has the collid-
ing limit [13, 14]. The colliding limit is a fusion of vertex operators so that multiple moments
of Liouville charges of the vertex operators are present. It is like the collection of charges dis-
tributed over a small region whose collection is viewed as an idealized system of total charge,
dipole, quadrupole and multi-poles. As a consequence, the irregular punctures maintain the
conformal symmetry but change the conformal state. Note that a regular puncture on the
1email:hermit1231@sogang.ac.kr
2email:rimpine@sogang.ac.kr
3email:kilar@sogang.ac.kr
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