Paper
0
F-theoretic vs microscopic description of a conformal N=2 SYM theory
Authors
Marco Billò,Laurent Gallot
Alberto Lerda,igor pesando,Isabel Lousada,José María González Linares,ROBERT ANDRES,Sang Wook Lee,Hazen Russell,Panagiota Michalakakou,MARIA AURORA ARMIENTA,Galinovskiy Andrey,Nina Konovalova,Thomas McCauley,Vladimir Smirnov,Alan Aitken,Mariusz Wojewoda,Oscar Blasco,Natalia Kovaleva,Jorge Guillermo Russo,Helena Van Coller,Stephen Naculich,Ernest ILISCA,Maria Teresa Fernandez-Sanchez,Maurício Pietrocola,Orlando Peres,Rafael Moreno-Vozmediano,Olga Karasik,Trevor Glasbey,Carlos de Carvalho,Joachim Kaiser,Hartmut Gimpel,Alexandre Brea Rodríguez,Kunal Vyas,Diego Regalado,Dragan Bataveljić,Rajesh Dachiraju,James Booth,Dejan Matić,Milivoje Lapčević,Veljko Vlašković,Bojan Urdarević,Zoran Jovanović,Emilija Stanković,Jelena Vučković,Tamara Đurđić Milošević,Srđan Vladetić,Jovan Vujičić,Sveto Puríć,Marko Slavković,Dragica Živojinović, Dj,Dragan Vujisić,Milan Palević,Peter Wolfsteiner,Mirjana Knežević,Sung-Soo Kim,Srdjan Đorđević,Michele Portolan,Garrett Poe,Jelena Nikolić,Laura Ledesma,Jasmina Labudović Stanković,Jelena Eric Nielsen,Katarina Borisavljević,Dejana Zlatanović,Veljko Marinković,Srđan Šapić,Xabier Marcano,Alain Jaccard,Carolyn Dzierba,Ihsan Halifeoglu,Marc Torrens,Benjamin Bellenie,Simone Giacomelli,Suddhasattwa Brahma,Anne Co,Andy Van Brocklin,Song He,Antony Fray,Slađana Savović,Anatoly Dymarsky,Bernard Gagnon,Joseph Fotsing,David Van Wie,Seok Ki Choi,Matti Jarvinen,Vishnu Ram OV,Marie Connett,TE-FANG CHU,Diogo Aurélio,Thorsten Alexander Kern,Bo Feng,Mukund Rangamani,Pierre Vanhove,Ian D. Hogg,Kimmo Tuominen,Nasuf SONMEZ,Valentin V Khoze,kanaka b,Daisuke Kageyama,Anne Smith,Hatem Ltaief,Santiago Folgueras,Prateek Agrawal,Nicoline Schiess,Yasunori Nomura,Ahmad Jabbarzadeh,Gustavo Alberto Burdman,Ana M Ibarra,wang kunjie,Sergio Angel,Yan Zhao,José Ignacio Morales,Brady Lee,Stefan Fredenhagen,Aude Gehrmann-De Ridder,Isabel María del Águila Cano,Alison Murray,Rafael Maestre,Deyan Yordanov,Shlomo Sergei Razamat,pallab basu,Linus Wulff,Ehsan Hatefi,Tatsuhiro Misumi,Jian-You Guo,Chengkang Zhang,Boris Pioline,Adalto Gomes,Jan Pieter van der Schaar,Clare Burrage,Steven Smith,Mª Angeles Egusquiza Balmaseda,Yue-Lin Sming Tsai,Parsa Hossein Ghorbani,Nico Callewaert,Sai Putcha,Riccardo Scateni,Akiyoshi Nakayama,Marinei Lopes Pedralli,Erik Schnetter,Luca Girlanda,Marco CIRELLI,Louis Strigari,Alex Gomes Dias,Daisuke Yamauchi,Amol Upadhye,Poonam Mehta,Yu-Xiao Liu,Michael Strickland,Christoph Englert,Stefano Profumo,Partha Konar,Sanjeev Seahra,Camilo Rindt,Jin Min Yang,David Mitchell,Ronnie Andersson,Paul Heslop,Daniel Thompson,Constantinos Papageorgakis,Alexander Smirnov,Florence Mahuteau-Betzer,Sotaro Sugishita,Tommi Alanne,Daniel O'Keefe,Danning Li,Martin Brodsky,Niko Jokela,Roberto Avendaño,Rosa Maria VITRANO,Gianguido Dall'Agata,Vincenzo Levizzani,James Ferrando,Sinisa Vidovic,Alexander Puzrin,Aleksandr N. Ignatov,Valerie Domcke,Rintze Zelle,amin salehi,Matthias Niessner,Chethan Krishnan,Andre Lessa,Maxim Kozlikin,Jacobo Lopez Pavon,Hoda Ali,Laura Andrianopoli,Antonio Amariti,Mohammad Hossein Namjoo,Alberto Mariotti,Yuichiro TADA,Dmitry Vichuzhanin,JOSE FRANCISCO DE ASSIS DIAS,Piotr Edelman,Filippo Sala,Jindřich Bečvář,Ewa Piwowarska,Zahid Zakir,S. Prem Kumar,robert de mello koch,Gonzalo Torroba,Dionysios Triantafyllopoulos,alberto zaffaroni,Mark Baker,fabio de rensis,Lumi-Pyry Wahlman,Philipp Mertsch,Layne Price,Jessie Shelton,Idoia Rosales,Vladimir Gabrinets,Larisa Stepanova,Ivan Jordanov,Laura Bohn,Syksy Räsänen,Stefan Hohenegger,Adhimar Oliveira,Hiroshi Yokoya,Pawel Caputa,Chirag Bhatt,Christopher McCabe,Max Langbein,Dirk Witthaut,Giuseppe Dibitetto,Deepak Gajanana,KAZUO HOSOMICHI,Arman Esmaili,Jenny Tillotson,Kirill Pavsky,Yoshiharu Takayama,Diego Trancanelli,Robert Stankey,Roberto Quadrini,Marco Billo',Jan Troost,David Pirtskhalava,Xiaoli Lu,Małgorzata Szpilska,Ivana Truccolo,Angel Uranga,David Mateos,Tiago de Jesus Sousa,Maresuke Shiraishi,Iñaki García Etxebarria,Vladimir Popov,Richard Easther,Oleg Evnin,Masako Mouri,Nicola Tamanini,Xun Zhu,Navid Ghavi Hossein-Zadeh,Jingwu Duan,PIERO MALGUZZI,Torsten Bringmann,Bartosz Grzegorz Sołowiej,Kari Enqvist,Seung J. Lee,Eero Vainikko,Takahiro Ueda,Anton de la Fuente,Iason Baldes,Shalender Singh,Gim Seng Ng,Jiaju Zhang,Jose Zurita,Karla McGregor,Troy McMullin,Corrado Boragno,Andrea Doris Kupferschmid,Gherardo Vita,Daniele Pelliccia,Dean Sumic,Lukas Arnold,Gopolang Mohlabeng,Simon Caron-Huot,Kevin Eboigbodin,Alejandro Ruiperez,Ching-Lien Hsiao,Gary Shiu,Jacob Bourjaily,David Gutzler,Ryosuke Sato,Josef Kainz,Andrew Abbo,Tomáš Gřivna,Stefano Di Chiara,Sandra Kvedaraitė,Evgeny Skvortsov,zain saleem,Michele Lucente,Paul Peter Cook,Yusuke Yamada,Kazuki Sakurai,Jack Setford,William Emond,Sibylle Driezen,Motoi Endo,Yi Pang,Kiel Howe,Jorge Norena,Jere Remes,Andrea Puhm,MATTHEW DOLAN,Earl Bardsley,Daniel Blaschke,Norihiro Tanahashi,Hai Siong Tan,Willy Fischler,Eduardo Gomez-Casado,Janusz Kapusniak,Nicolò Petri,Meng Fai Lim,Jean-Pascal Bergé,Antonella Pasquato,Bliss Cunningham,Adar Sharon,Taro Kimura,Dan Seidov,Patricia Dias,Yuan Zhong,Yi Wang,Juan F. Pedraza,Barbara Mathews,Luiz Carlos Kucharski,Sérgio MO Tavares,Amelia Lake,Jérémy Anquetin,Thiago Pereira,Dr. Saleem Iqbal,Christian Byrnes,Antonio Racioppi,Larry Berg,Serap Gultekin,Violetta Jaros,Supriya Chakrabarti,Sonia Garcia,Miguel Figueiredo,Алексей Новиков,Paul Willis,Hongliang Jiang,Sreenath Vijayakumar,Olga Plotnikova,Timothy McKnight,Massimo Taronna,Ольга Горюнова,Francisco José Aranda Pérez,Yashar Akrami,Devon Greyson,Александр Кириллов,Mario Ballardini,Admir Greljo,Tiziana Trombetti,L. Sriramkumar,JOSE ALBERTO RUIZ CEMBRANOS,Otto Monge,Brent Valle,Jason Veatch,N/A ,Fabrizio Nesti,Paolo Ponzio,Darko Prebežac,Alexey Koshelev,Mónica Aguilar-Alba,Otávio Coaracy Brasil Gandolfo,Hooshyar Assadullahi,Diana White,Jolanta Bojarszczuk,Jibitesh Dutta,Jacob Holm,Francesca Calore,Rita de Cassia dos Anjos,Phornphop Naiyanetr,Ge Peng,Eloy Ayón-Beato,ALEKSANDR KHODINOV,Marcin Wodziński,Oliver Piattella,Mattia Manfredonia,Stanislav Chuprakov,Rudranil Basu,Aurelio del Portillo García,Vincenzo Cavaliere,AKHILA MOHAN,Arianna Jakositz,Lorenzo Bordin,Titouan Lazeyras,Liliya Nakashydze,Сергей Кулабухов,Hüseyin Şahin,Protsiv Volodymyr,Raul Mur-Artal,Gizem Şengör,Debika Chowdhury,Stefanie Klenow,Kasey Legaard,Nehal Gamal,Ilya Blinov,Clement Levallois,Jason Pine,Christopher Couzens,Elizabeth Sim,Сергей Ильин,Taylor Turner,CARMEN BONASERA,Hyunjae Daniel Shin,Muhammad Mudasar Saqab,Colin Kruse,UGAITZ GAZTELU ONAINDIA,Lynne Lafave,Dan Řezníček,Edgardo Mejía Roa,Brian Geist,Ashish Kumar Katiyar,Izabela Skomerska-Muchowska,Jingya Zhu,Mohammed Moussa,Mohammad Raqibul Hasan Siddique,Edilson de Carvalho Filho,Naoya Kitajima,Olga Rodrigo Pedrosa,shaojie chen,Juliana Stracieri,Ēriks Treļs,Mikhail Denissenya,Claudia Krull,Vladimir Uspenskiy,Jean-Sebastien Pelletier,Sheila Gutiérrez López,Daniele Marangotto,Paul Lemieux III,Melissa Fernandez,Charif BENBOULAID,Kinjal Banerjee,Elena Ryabova,FREDY ROBERT ROSAS CULCOS,Pierre Bourdin,Igor Areh,Faa-Jeng Lin,Caner Unal,Angela Carter,DOGAN TUTAK,Edith Lara-Carrillo,CÉSAR AUGUSTO RANILLA FALCÓN,Keith Williams,Ying Zhang,Jessielem Soares ribeiro,Cristina Polito,Vladimir Kirilin,Maria Grazia NACCI,Nora Ramtke,Magnus Strandh,Rafael Larrosa Jiménez,Casey Kennington,Eirh-Yu Hsie,Germán Quijano Mena,Fei Wang,Levon Vladimirovich Batiev ,Kateřina Matějková,Valeri Vardanyan,Eric Kuo,Crist´ovão Rodrigues,Fabiana Kneubil,Julio Dutra,Jun-Ping Du,Ilderlane da Silva Lopes,Randy Agra Pratama,Vladimir Vereshagin,Melissa Zweng,Deivys Joel Pérez Chuquihuanca,Elena Gordina,Edwin Herrera,Adnan Adnan,Вячеслав Козоброд,MICHELE LOIZZI,A.D. Degtyareva,Rishi Baral,Alejandra Ramm,Şükrü Arslan,Michael Tanner,Elena Mercedes Perez-Monserrat,FRANCOIS BROQUEDIS,Paolo Martano,VICTOR MANUEL OLMEDO VAZQUEZ,Ashish Narang,Georgi Kadikyanov,Pedrom Jordão,Fernando Moreno,Madawa W Jayawardana,Pedro Augusto Tibúrcio Paulino,Dario Rosa,Dr. Iahtisham-Ul-Haq ,Jasbir Singh,María Jose Legaz,Julian Rode,Rimma Akhmadullina,Kévin Nguyen,Julie Merkle,Martín Eliseo Tamayo Ancona,Martin Aumüller,Davide Gualdi,Ana Carolina Souza,Virginia Romero Plana,Andrzej Rutkiewicz,MARIA PILAR DIEZHANDINO,TATIANA MARIA ANTONIA COSSU,Radek Votrubec,TSUNEHIRO HATO,Mélanie Le Barz,James Alvey,Camila Novaes,Daniel Hepperle,Joaquin Bastias,Adrian Adascalitei,Manuel Velázquez,Manuel Donaire,David Mesterhazy,David Cabrera-García,Agnieszka Berdowska,Kun Wang,João Morais,Nihal Batouty,Ferenc Siklér,Maria da Encarnação Marcelo,Trevor E. Carlson,Juldyz Smagulova,Damião da Cunha,Adriana Antonieta Romero Sandoval,Michael Wagner,NGUENA CHRISTIAN LAMBERT,Francesca Martinelli,Caleb Tse,Taiga Kunishima,Miguel Borja,Alessandro Decarlis,Edgar Neri Castro,Jean-Etienne Fabre,Christophe Rey,Jan Rehor,Kostas Sagonas,Iliana del Carmen Sanmartín-Rojas,shaun tanger,Adil Adatia,Julio C. García-Martínez,SHIPRA GUPTA,Jesús Antonio Camarillo,AHMET AKGÜL,Rui Zhu,Agata Buchwal,R. Jonathan Robitsek,Jose Halim Montes de Oca Yemha,Carolina Gutstein,Daniele De Sensi,Shahid Iqbal,Mustafa Tareq,Mohammad Ebrahim Rezvani,Nils Myszkowski,Raul Sales,HSIANG-KAI DONG,Marta Skiba,Angela DE PALMA,Bernardo Amigo,Serra Topal,Loredana Verdone,Warren Gumbley,Sara María Torres Outón,Roger Davies,Fernando Morenoa,Nicola Martin,Ruyman Reyes,Nathália Gonçalves Zaparolli,Eric Kort,Cezar Gaman,Ayşem Kaya,HASAN BACANLI,ANAI YAH TUZ,Andreea Koreanschi,Gérard Poitras,Filippo Donna,James Xue,Ana Bela Nunes,Odelkis Vázquez González,Gabriel LaPlante,LUIS BRITO-GAONA,Raquel de la Cruz Soriano,Kris Christmann,ISRAEL MORENO-MEJIA,Kirill Semenov-Tian-Shansky,Ann Marie Clark,William Serrato,Stefanie Senger,Scott Reza Jafarian Kerman,aweke shishigu,Markus Didion,Ben Ruijl,Ana María Navarro Casillas,Niels de Hoon,Bruno Guillon,Heitor S. Ramos,Marija Molan,Irina Gasilova,Emerson Izidoro dos Santos,Thaís Cyrino de Mello Forato,Magdalena Wiśniewska-Drewniak,Anna Grygier,Frank Löffler,John de Bono,Carolina Fernanda Gartner Restrepo,Gisleine Coelho de Campos,James Reagan,Huseyin Oz,Nicholas Kiggundu,Fabrizio Caola,Renata Barczynska,Toshiya Takami,Eike Falk Anderson,Raghunayagan Parameswaran,Hernán Alfredo González Leiva,Marwa Elbyaly,Roddy Brett,Amparo Sánchez Cobos,Jennifer Ogilvie,Iskander Yarmakeev,DARIO DELL'OSA,RAJARATNAM RAMESHSHANKER,Amin Safavi,Love Börjeson,Francisco Javier Corbera Peña,dongdong qin,Hodjat Mariji,Ahmet Yildiz,Meegan Kennedy,Juliana Jaramillo-Jaramillo,Laura Prichard,Bahareh Kamranzad,Robert Speck,TAKASHI KUBOTA,Mauricio Loyola,Kelsie Dadd,Matthew Power,Joshua Holden,Jean-Philippe Gourdine,YOSHIHIRO IIJIMA,Rafael Asenjo,Ruben Martinez-Cantin,Gianguido Cossellu,Liliana Maltagliati,Manuel Garcia-Piqueras,Gamaliel Castañeda,Daniel Vitales,GIUSEPPE PAPUZZO,Nikolaj Petrov,Bart Nieuwenhuis,Abdellah Elfeky,Raida Ben Salah,Tien-Chin Tan,Jacek Hulimka,Nuray Voyvoda,DK SINGH,Andrey Fedorov,Haftay Hailu Gebremedhn ,Nedžad Rudonja,Yana Sergeeva,Jeffrey Buyer,Milton Schivani,MARINO NAVAZO,Natasa Honzikova,David Lester,Sarah N. Lim Choi Keung,Cathy Hollis,Katarzyna Skrzypczak,Andreas Galonska,Mario Lillo-Saavedra,André Marques,Rafał Lizut,Samantha McDonnell,Risto Vehmas,Javad Taghizadeh Firouzjaee,Gonzalo Miguez,Guillaume DUFLOS,Angeles G. Navarro,Isabel Cristina Flores Rueda,sada Alyasri,Olumide Ajibola,Maria Elena ROCCIA,Pedro Cruz-Hernandez,Carlos Shahbazi Alonso,Bumsoo Han,Roberto Granados Porras,parviz shahabi,Jérémy Neveu,Alexey Mishonov,Olof Steinthorsdottir,David Hayes,Shokoufeh hajsadeghi,Virginia Karina Rosas Burgos,Mario E. Gomez,Andreas Recknagel,Jesus Garcia-Crespo,masahiro hizume,Alessio Ferrari,Matteo Manzali,Ismeli Alfonso,Carmen Gallego Vega,Walter Yamada,Lisa Sanders,Martin Brazeau,Pere Gironella Gironell,Daniele Pistone,Martin Křížek,Anna Runemark,Mario Renato Cépeda Cáceres,Golam Hossain,Christopher Hollings,Irina Trofimova,Jens Mergenthaler,Jason Otkin,Kaiyou Wang,Danijela Marovic,Fumitake Nishimura,Shanmukhappa Angadi,Sebastian Bysiak,Ran-Zan Wang,Caterina Ripa,Alberto F. Martín,Chris Hughes,Wee Chen Gan,Nuno Teixeira Tavares,Rosario Gonzalez-Ferez,Andre Schneider,Gulińska Hanna,Nicholas Lam,Chi Chung Chan,Billy Makumba,Raisa Ionin,Tetiana Orlova,Roberto Saglia,Chiara Pinto,Pascal martin,Albert Isidro,Maria Mrówczyńska,Yucel Severcan,Nihal Buyukcizmeci,Francois Lanusse,Micaela Caserta,Kevin Davidson,Heli Viiri,Alan Hogg,Arlette Bochud,David M Freire,Jose Ignacio Gomez,Anxela Bugallo-Rodríguez,RENATO BUZIO,Takanori Horii,Miguel Gomez-Heras,Sunil Kulkarni,cuneyt kocas,Jens Ilg,Florian Niebling,Sunny Vagnozzi,Mihkail Milyaev,Nicola Cicero,Jakub Godlewski,Luís Pimentel,Max Chavarría,Yves Carlier,Nikolay Arefiev,Antonio Claret García Martínez,Elsa Gladys Aguilar Ancori,Juan Gómez Luna,Luis Sequeira,Jedrzej Drozdowicz,Karol Wysokiński,Andressa de Souza Pollo,Giuseppe Firpo,Mirosław Tyl,Hector Migallon,Rocco Servidio,Chad Lin,Jose Luis Vazquez-Poletti,Satoshi Takeuchi,karteek guturu,Stefano Andreon,Yutaka Watanuki,Tyanai Masiya,Katerina Nedbalkova,Armando Bernui,ANDREA MARCHETTI,hanna serkowska,Johannes Glodny,jackson cooper,Raphael Proulx,andrea molinari,leonardo Ribeiro Eulalio Cabral,Ehab K. I. Hamad,Bruce Nicoll,Zhien Wang,Luís Nogueira,Laura Gallardo,Bryan Tolson,Beatriz Álvarez-González,Catarina Castro,Peter Grimminger,Natalya Gordina,Igor Camargo Fontana,Efstratios Valakos,Toshiaki Takezawa,Yiannis Giannakopoulos,Leonardo Fabio Martínez Pérez,Daniel Sacristán,Urs Treier,Patrick Martineau,Vadim Menzhulin,Peter Jansson,Jonas Kindberg,Herrera Zendejas Esther,Markus Geimer,Gisela Pujol,Gabriela Besler,Kirsten Hvenegård-Lassen,Rohit Gupta,ANTONIO LOPEZ MAROTO,Gotzon Gangoiti,Hlib Prib,Pawel Kopciewicz,Stephanie Brichau,Ali Unal Sorman,Martin Henk,Eric Jellen,Antonio Bossio,MARIA POOL,Magda Semedo,Rita Martins de Sousa,Christopher Sioris,Xosé A. Armas Castro,MONICA ALVAREZ,Gesine Reinert,Oriane Hidalgo,Genly Leon,Anna Celczyńska,Bruna Sinjari,Jose Luis Solaun-Bustinza,Murat Kazim Ersanli,antonio enea romano,Alexey Boldarev,Abdeslam Mimouni,Maria Regina Menino,Antonello Novelli,Joy Porter,Murray Elder,José António Souto Cabo,Małgorzata Kaczmarska,Amitava Chattopadhyay,Wenran Feng,Laila C. Roisman,José Gracia,Shigeyuki Ishidoya,Eliane Ferreira dos Santos,Roman Czyba,Kamila Myszka,Alfonso Serrano-Maillo,Maria Halabalaki,Marie Mirouze,Ryan Ceddia,Priya Kurian,Ireneusz Stefaniuk,Guy Houlsby,Rui Brás,Renate Egan,Heikki Hirvonen,Héctor Bracho Espinoza,Małgorzata Anna Majcher,Dong Jiang,Ariosto Medina,Dacia Viejo-Rose,Alexandra Alvarsson,Zafeiris Kokkinogenis,kalina burnat,Marek Wójtowicz,María Rebeca Padilla de la Torre,Ulla Susanna Tuomarla,Alexander Nicoletti,Johannes Foufopoulos,Pietro Fecchio,Kelvin Tsun Wai Ng,Vitor Rafael Coluci,Dmitry Serebryakov,Alfred Anwander,Stefano Testa Bappenheim,Irina Garcia-Ispierto,Marc Thomas,Qingyu Ma,Tanglaw Roman,Mariya Gubareva,Guosheng Xu,Anna Lewandowska,COTTONE Antonio,Didier Rémond,Debashish Munshi,Bertrand PEROT,Thomas Eickermann,Karl Meerbergen,Sergey Gilev,Juan Antonio Rodríguez Díaz,Grian A. Cutanda,Cedric Lorce,Juan Rafael de la Haba Rodríguez,Patrick Steinmetz,Fernando Heredia Sánchez,Sharifa Ezat Wan Puteh,Minna Väilä,Valentina Mussi,Denis Lavoie,Gladys Sandi Licapa Redolfo,Yuri Kotelevtsev,Elizabeth Hanna,Yasin Taşpınar,Michal Nedělka,amani taamalli,Philippe Baret,Andrea Sanchini,Mieczyslaw Szalecki,Eliza Freire,Stuart Thomson,David Randall,Jiangbo Yu,Dave Cliff,Vanda Faria dos Santos,Herwig Renner,Raimund Kirner,Anthony Bale,Albert Ellingboe,Nenagh Kemp,Dmitriy Borodin,Allen Good,Jaume Pellicer,José Carlos Quiroga Díaz,Luz Paz-Agras,Ignacio Martin Llorente,Deborah Henderson,Rosa M Badia,Elisa Mele,hongwu liu,Daisuke Takahashi,Sławomir Jabłoński,Sian Sullivan,Mª de la Gloria Espigado Tocino,John Castellani,Eduard Malev,Wim Vanroose,Warrick Dawes,Connie Frank Matthiesen,Paul Gibbon,Petri Kärhä,KOICHI HAMAGUCHI,IRASEMA LUZ VENEGAS AHUMADA,Mariam Bouhmadi Lopez,Massimo Offidani,Matthew Leybourne,SEVINÇ ÜÇGÜL,Isamu Morino,Srinivas Gadipelli,Mireia Morgades,Shuichi Kurabayashi,Miloš Vujisić,Antonio García Hernández,Tsiu-Kwen Lee,Rafael Fort,Erling Andersen,Isabel Paula,Vasilis Strogilos,wenjun ma,Tommy Ohlsson,Teresa Garnatje,Sandra Martins Pereira,Juergen Stolzenberg,Felix Weiss,Hanno Friedrich,Ingrid Tonhajzerova,Ксения Шурда,VANESSA OLIVEIRA OSSOLA DA CRUZ,Kristof Roomp,Elixabete imaz,Abdol Ahad Shadparvar,Juan María Bilbao Ubillos,军帅 栗,Katarzyna Śliżewska,Zachary Hunter,James Muir,Vladislav Belavin,Siegfried Benkner,Jan Klein,Johana Arévalo Cortes,JHON JAIRO CALDERON LEYTON,Kwok Fai Geoffrey TSO,Erik Svensson
+993 authors
,Sonia Regina PAULINOJournal
Published
Aug 31, 2010
Show more
Save
Tip0
TipSave
Preprint typeset in JHEP style. - PAPER VERSION DFTT/7/2010
LAPTH 023/10
NSF-KITP-10-107
F-theoretic vs microscopic description of a conformal
N = 2 SYM theory
Marco Bill`o1,2, Laurent Gallot3, Alberto Lerda4 and Igor Pesando1
1Dipartimento di Fisica Teorica, Universit`a di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
2Kavli Institute for Theoretical Physics, University of California, Santa Barbara,
CA 93106-4030, USA
3LAPTH, Universit´e de Savoie, CNRS
9, Chemin de Bellevue
74941 Annecy le Vieux Cedex, France
4Dipartimento di Scienze e Tecnologie Avanzate, Universit`a del Piemonte Orientale
and I.N.F.N. - Gruppo Collegato di Alessandria - sezione di Torino
Viale T. Michel 11, I-15121 Alessandria, Italy
billo,lerda,ipesando@to.infn.it; laurent.gallot@lapp.in2p3.fr
Abstract: The F-theory background of four D7 branes in a type I′ orientifold was con-
jectured to be described by the Seiberg-Witten curve for the superconformal SU(2) gauge
theory with four flavors. This relation was explained by considering in this background
a probe D3 brane, which supports this theory with SU(2) realized as Sp(1). Here we ex-
plicitly compute the non-perturbative corrections to the D7/D3 system in type I′ due to
D-instantons. This computation provides both the quartic effective action on the D7 branes
and the quadratic effective action on the D3 brane; the latter agrees with the F-theoretic
prediction. The action obtained in this way is related to the one derived from the usual
instanton calculus `a la Nekrasov (or from its AGT realization in terms of Liouville confor-
mal blocks) by means of a non-perturbative redefinition of the coupling constant. We also
point out an intriguing relation between the four-dimensional theory on the probe D3 brane
with SO(8) flavor symmetry and the eight-dimensional dynamics on the D7 branes. On
the latter, SO(8) represents a gauge group and the flavor masses correspond to the vacuum
expectation values of an adjoint scalar field m: what we find is that the exact effective
coupling in four dimensions is obtained from its perturbative part by taking into account
in its mass dependence the full quantum dynamics of the field m in eight dimensions.
Keywords: Superstrings, D-branes, Gauge Theories, Instantons, F-theory.
arXiv:1008.5240v2 [hep-th] 15 Nov 2010
LAPTH 023/10
NSF-KITP-10-107
F-theoretic vs microscopic description of a conformal
N = 2 SYM theory
Marco Bill`o1,2, Laurent Gallot3, Alberto Lerda4 and Igor Pesando1
1Dipartimento di Fisica Teorica, Universit`a di Torino
and I.N.F.N. - sezione di Torino
Via P. Giuria 1, I-10125 Torino, Italy
2Kavli Institute for Theoretical Physics, University of California, Santa Barbara,
CA 93106-4030, USA
3LAPTH, Universit´e de Savoie, CNRS
9, Chemin de Bellevue
74941 Annecy le Vieux Cedex, France
4Dipartimento di Scienze e Tecnologie Avanzate, Universit`a del Piemonte Orientale
and I.N.F.N. - Gruppo Collegato di Alessandria - sezione di Torino
Viale T. Michel 11, I-15121 Alessandria, Italy
billo,lerda,ipesando@to.infn.it; laurent.gallot@lapp.in2p3.fr
Abstract: The F-theory background of four D7 branes in a type I′ orientifold was con-
jectured to be described by the Seiberg-Witten curve for the superconformal SU(2) gauge
theory with four flavors. This relation was explained by considering in this background
a probe D3 brane, which supports this theory with SU(2) realized as Sp(1). Here we ex-
plicitly compute the non-perturbative corrections to the D7/D3 system in type I′ due to
D-instantons. This computation provides both the quartic effective action on the D7 branes
and the quadratic effective action on the D3 brane; the latter agrees with the F-theoretic
prediction. The action obtained in this way is related to the one derived from the usual
instanton calculus `a la Nekrasov (or from its AGT realization in terms of Liouville confor-
mal blocks) by means of a non-perturbative redefinition of the coupling constant. We also
point out an intriguing relation between the four-dimensional theory on the probe D3 brane
with SO(8) flavor symmetry and the eight-dimensional dynamics on the D7 branes. On
the latter, SO(8) represents a gauge group and the flavor masses correspond to the vacuum
expectation values of an adjoint scalar field m: what we find is that the exact effective
coupling in four dimensions is obtained from its perturbative part by taking into account
in its mass dependence the full quantum dynamics of the field m in eight dimensions.
Keywords: Superstrings, D-branes, Gauge Theories, Instantons, F-theory.
arXiv:1008.5240v2 [hep-th] 15 Nov 2010
Contents
1. Introduction and motivations 2
2. F-theory and the D7/D3 system in type I′ 4
2.1 The local case 5
2.2 F-theoretic description and D3 brane probes 6
2.2.1 The SW curve for the D3 gauge theory 8
3. D-instanton corrections 11
3.1 Instanton moduli spectrum 11
3.2 BRST structure of moduli space 13
3.3 Moduli integration via localization and instanton partition function 16
3.4 Non-perturbative prepotentials 18
4. Eight-dimensional effective prepotential and chiral ring 20
5. Four-dimensional effective prepotential 23
6. An intriguing relation 25
7. Comparison with the SU(2) instanton calculus 26
7.1 Instanton calculus 27
7.2 The Nekrasov prepotential from the AGT realization 28
8. Summary and conclusions 30
A. Notations and theta-function conventions 32
B. Flavor invariants 33
B.1 Triality extension of SL(2, Z) 34
C. Details on the D-instanton computation 34
D. Details on the AGT correspondence 36
E. Decoupling limits to Nf = 3, 2, 1, 0. 38
1
1. Introduction and motivations 2
2. F-theory and the D7/D3 system in type I′ 4
2.1 The local case 5
2.2 F-theoretic description and D3 brane probes 6
2.2.1 The SW curve for the D3 gauge theory 8
3. D-instanton corrections 11
3.1 Instanton moduli spectrum 11
3.2 BRST structure of moduli space 13
3.3 Moduli integration via localization and instanton partition function 16
3.4 Non-perturbative prepotentials 18
4. Eight-dimensional effective prepotential and chiral ring 20
5. Four-dimensional effective prepotential 23
6. An intriguing relation 25
7. Comparison with the SU(2) instanton calculus 26
7.1 Instanton calculus 27
7.2 The Nekrasov prepotential from the AGT realization 28
8. Summary and conclusions 30
A. Notations and theta-function conventions 32
B. Flavor invariants 33
B.1 Triality extension of SL(2, Z) 34
C. Details on the D-instanton computation 34
D. Details on the AGT correspondence 36
E. Decoupling limits to Nf = 3, 2, 1, 0. 38
1
1. Introduction and motivations
Phenomenologically viable string models based on consistent D-brane configurations have
attracted a lot of attention in the last years [1]–[3]. In such constructions it is necessary to
take into account possible non-perturbative corrections due D-instantons and (wrapped)
euclidean branes; for a review see, for example, [4]. Some of these instantonic branes
reproduce gauge instantons [5]–[8], other provide inherently stringy (or “exotic”) instanton
effects; these latter can be responsible of important terms in the effective action which would
be perturbatively forbidden [9]–[11]. Recently there has been much progress in the explicit
computation of such contributions, both ordinary and exotic, at least in supersymmetric
cases (again, see [4] and references therein).
Another framework where phenomenological models with highly desirable features can
be set up, and where in particular Grand Unified Theories occur naturally and consistently
[12]–[14], is represented by F-theory compactifications [15] (for reviews see, for instance,
[16, 17]). F-theory gives a non-perturbative geometric description of type IIB backgrounds
containing D7 branes and orientifold planes; it somehow resums the non-perturbative cor-
rections arising from certain instantonic branes. Understanding in detail how this resumma-
tion takes place would improve our knowledge of the relation between the two descriptions.
This could be useful for a better comprehension of further non-perturbative effects in F-
theory through a lift of their type IIB counterparts, a subject that is recently1 receiving
quite some attention [19]–[24].
In this paper we work out a simple, yet non-trivial, example where we are able to
compute the D-instanton effects in the IIB description and show that they reconstruct
the F-theory curve. This example was considered by A. Sen in [25], and is given by the
compactification of F-theory on the orbifold limit of an elliptically fibered K3 surface, for
which the complex structure modulus τ of the fiber is constant. This background was
shown to correspond to the so-called type I′ theory, which is T-dual to type I theory
compactified on a 2-torus T2, and thus possesses one O7 plane at each of the four fixed
points of T2 with four D7 branes on top of it. Focusing on the vicinity of one orientifold
fixed-plane, and allowing the four D7 branes to move out of it, Sen conjectured that the
corresponding F-theory background should be described by the Seiberg-Witten (SW) curve
[26] for the N = 2 superconformal Yang-Mills theory with gauge group SU(2) and Nf = 4
flavors [27]. This relation was later explained by T. Banks, M. Douglas and N. Seiberg
[28] by considering a D3 brane in this background, which indeed supports such a four-
dimensional gauge theory on its world-volume, with SU(2) realized as Sp(1). The SO(8)
flavor symmetry of the Nf = 4 theory is nothing else but the gauge group on the D7 branes.
Here we explicitly compute the non-perturbative corrections to the D7/D3 system in
type I′ due to D-instantons. This requires to identify the spectrum of moduli, i.e. of
excitations of the strings with at least one end-point on the D-instantons, and the moduli
action that arises from disks with at least part of their boundary on a D-instanton, which
was already discussed in [29]. To obtain the non-perturbative effects it is necessary to
integrate over the moduli; we explicitly perform this integration by applying the by-now
1For an earlier discussion in an N = 2 context see [18].
2
Phenomenologically viable string models based on consistent D-brane configurations have
attracted a lot of attention in the last years [1]–[3]. In such constructions it is necessary to
take into account possible non-perturbative corrections due D-instantons and (wrapped)
euclidean branes; for a review see, for example, [4]. Some of these instantonic branes
reproduce gauge instantons [5]–[8], other provide inherently stringy (or “exotic”) instanton
effects; these latter can be responsible of important terms in the effective action which would
be perturbatively forbidden [9]–[11]. Recently there has been much progress in the explicit
computation of such contributions, both ordinary and exotic, at least in supersymmetric
cases (again, see [4] and references therein).
Another framework where phenomenological models with highly desirable features can
be set up, and where in particular Grand Unified Theories occur naturally and consistently
[12]–[14], is represented by F-theory compactifications [15] (for reviews see, for instance,
[16, 17]). F-theory gives a non-perturbative geometric description of type IIB backgrounds
containing D7 branes and orientifold planes; it somehow resums the non-perturbative cor-
rections arising from certain instantonic branes. Understanding in detail how this resumma-
tion takes place would improve our knowledge of the relation between the two descriptions.
This could be useful for a better comprehension of further non-perturbative effects in F-
theory through a lift of their type IIB counterparts, a subject that is recently1 receiving
quite some attention [19]–[24].
In this paper we work out a simple, yet non-trivial, example where we are able to
compute the D-instanton effects in the IIB description and show that they reconstruct
the F-theory curve. This example was considered by A. Sen in [25], and is given by the
compactification of F-theory on the orbifold limit of an elliptically fibered K3 surface, for
which the complex structure modulus τ of the fiber is constant. This background was
shown to correspond to the so-called type I′ theory, which is T-dual to type I theory
compactified on a 2-torus T2, and thus possesses one O7 plane at each of the four fixed
points of T2 with four D7 branes on top of it. Focusing on the vicinity of one orientifold
fixed-plane, and allowing the four D7 branes to move out of it, Sen conjectured that the
corresponding F-theory background should be described by the Seiberg-Witten (SW) curve
[26] for the N = 2 superconformal Yang-Mills theory with gauge group SU(2) and Nf = 4
flavors [27]. This relation was later explained by T. Banks, M. Douglas and N. Seiberg
[28] by considering a D3 brane in this background, which indeed supports such a four-
dimensional gauge theory on its world-volume, with SU(2) realized as Sp(1). The SO(8)
flavor symmetry of the Nf = 4 theory is nothing else but the gauge group on the D7 branes.
Here we explicitly compute the non-perturbative corrections to the D7/D3 system in
type I′ due to D-instantons. This requires to identify the spectrum of moduli, i.e. of
excitations of the strings with at least one end-point on the D-instantons, and the moduli
action that arises from disks with at least part of their boundary on a D-instanton, which
was already discussed in [29]. To obtain the non-perturbative effects it is necessary to
integrate over the moduli; we explicitly perform this integration by applying the by-now
1For an earlier discussion in an N = 2 context see [18].
2
standard techniques based on the BRST structure of the moduli spectrum and action and
its deformation by means of suitable RR backgrounds [30]–[34]. This induces a complete
localization of the integral, similarly to what happens in supersymmetric instanton calculus
in field theory [35]–[40]. These techniques have been recently applied to similar brane set-
ups, such as the D7 system in type I′ [31, 33], and the D7/D3 system on T 4/Z2 [34].
D-instanton effects induce corrections to both the quartic effective action on the D7
branes and the quadratic effective action on the D3 brane. We use the prescription proposed
in [34] to disentangle the two contributions. The non-perturbative action on the D7’s turns
out to be exactly the same of the D7 system in type I′ theory considered in [31]. The non-
perturbative effective coupling on the D3 brane agrees with the one extracted from the SW
curve, that is with the F-theoretic prediction.
One interesting question is the precise relation between the eight-dimensional quartic
effective action on the D7 branes and the F-theory curve. This problem was already
addressed in the past using the duality of certain F-theory compactifications, including the
one corresponding to type I′ theory, to heterotic models. Despite some interesting results
[41, 42], this relation is not yet totally clear. Since in our case the F-theory curve is nothing
else but the SW curve encoding the effective theory on the D3 probe, another way to state
the above question is: what is the relation between the non-perturbative effective actions
on the D7 branes and on the D3 brane? More generally, how are the quantum dynamics
on the D7’s and that on the D3 related to each other?
We do not have a full answer to this question, but we uncover an intriguing relation
that goes as follows. On the D7’s, the SO(8) flavor symmetry represents a gauge group
and the flavor masses mi correspond to the vacuum expectation values of an adjoint scalar
field m(X): we find that the exact effective four-dimensional coupling is obtained from
its perturbative part by taking into account in its mass dependence the eight-dimensional
quantum dynamics of the field m, and in particular the so-called “chiral ring” formed by
the correlators 〈Tr m2l〉. In this way the F-theory geometry is related explicitly to these
eight-dimensional quantities. It would be very interesting to investigate whether such sort
of perturbative propagation of the full quantum dynamics on a brane stack (in our case,
the D7’s) to another one (in our case the D3 probe) takes place also in other systems2.
The theory that, in our example, lives on the D3 brane is a four-dimensional conformal
N = 2 theory. This class of theories have recently attracted much attention in relation to
the so-called AGT conjecture put forward by L. F. Alday, D. Gaiotto and Y. Tachikawa
[44]. This conjecture relates the effective actions obtained from usual instanton calculus `a
la Nekrasov [35] to suitable correlators of the Liouville theory in two dimensions [45] (see
also [46]–[48]). In particular, the non-perturbative action for the SU(2) theory with Nf = 4
can be extracted from the 4-point functions on the sphere. It is interesting to compare these
results to what we get in the D7/D3 system in type I′, where the conformal theory is realized
as an Sp(1) gauge theory. It turns out that the effective action derived from the Nekrasov’s
prescription for the SU(2), Nf = 4 case or, more efficiently, from its AGT realization in
2Recently, D3 probes in F-theory have been considered in [20, 43]. In particular in [43] it is pointed out
that the interplay between the eight-dimensional theory on the D7 and the four-dimensional theory on the
probe, and the interpretation of parameters in the latter as adjoint fields on the former, plays a crucial rˆole.
3
its deformation by means of suitable RR backgrounds [30]–[34]. This induces a complete
localization of the integral, similarly to what happens in supersymmetric instanton calculus
in field theory [35]–[40]. These techniques have been recently applied to similar brane set-
ups, such as the D7 system in type I′ [31, 33], and the D7/D3 system on T 4/Z2 [34].
D-instanton effects induce corrections to both the quartic effective action on the D7
branes and the quadratic effective action on the D3 brane. We use the prescription proposed
in [34] to disentangle the two contributions. The non-perturbative action on the D7’s turns
out to be exactly the same of the D7 system in type I′ theory considered in [31]. The non-
perturbative effective coupling on the D3 brane agrees with the one extracted from the SW
curve, that is with the F-theoretic prediction.
One interesting question is the precise relation between the eight-dimensional quartic
effective action on the D7 branes and the F-theory curve. This problem was already
addressed in the past using the duality of certain F-theory compactifications, including the
one corresponding to type I′ theory, to heterotic models. Despite some interesting results
[41, 42], this relation is not yet totally clear. Since in our case the F-theory curve is nothing
else but the SW curve encoding the effective theory on the D3 probe, another way to state
the above question is: what is the relation between the non-perturbative effective actions
on the D7 branes and on the D3 brane? More generally, how are the quantum dynamics
on the D7’s and that on the D3 related to each other?
We do not have a full answer to this question, but we uncover an intriguing relation
that goes as follows. On the D7’s, the SO(8) flavor symmetry represents a gauge group
and the flavor masses mi correspond to the vacuum expectation values of an adjoint scalar
field m(X): we find that the exact effective four-dimensional coupling is obtained from
its perturbative part by taking into account in its mass dependence the eight-dimensional
quantum dynamics of the field m, and in particular the so-called “chiral ring” formed by
the correlators 〈Tr m2l〉. In this way the F-theory geometry is related explicitly to these
eight-dimensional quantities. It would be very interesting to investigate whether such sort
of perturbative propagation of the full quantum dynamics on a brane stack (in our case,
the D7’s) to another one (in our case the D3 probe) takes place also in other systems2.
The theory that, in our example, lives on the D3 brane is a four-dimensional conformal
N = 2 theory. This class of theories have recently attracted much attention in relation to
the so-called AGT conjecture put forward by L. F. Alday, D. Gaiotto and Y. Tachikawa
[44]. This conjecture relates the effective actions obtained from usual instanton calculus `a
la Nekrasov [35] to suitable correlators of the Liouville theory in two dimensions [45] (see
also [46]–[48]). In particular, the non-perturbative action for the SU(2) theory with Nf = 4
can be extracted from the 4-point functions on the sphere. It is interesting to compare these
results to what we get in the D7/D3 system in type I′, where the conformal theory is realized
as an Sp(1) gauge theory. It turns out that the effective action derived from the Nekrasov’s
prescription for the SU(2), Nf = 4 case or, more efficiently, from its AGT realization in
2Recently, D3 probes in F-theory have been considered in [20, 43]. In particular in [43] it is pointed out
that the interplay between the eight-dimensional theory on the D7 and the four-dimensional theory on the
probe, and the interpretation of parameters in the latter as adjoint fields on the former, plays a crucial rˆole.
3
terms of Liouville conformal blocks does not coincide, at first sight, with our results, nor
with the SW curve proposed in [27]. There is a discrepancy already at the massless level: in
this case from the SW curve (and from our microscopic computation) we see that the tree-
level coupling receives no corrections. When computed following Nekrasov’s prescription
for SU(2), or by the AGT method, instead, the coupling gets non-perturbatively modified3.
This suggests that a redefinition of the coupling constant is needed in order to compare
the two approaches; after such a redefinition is performed, remarkably the two methods
are reconciled and the two results agree completely also in the massive case.
The paper is subdivided into several sections as we now describe. In Section 2 we
introduce the model and its F-theory description through the SW curve; from this curve
we extract the instanton expansion of the effective coupling. In Section 3 we describe
the microscopic computation of D-instanton contributions in our model. The resulting
effective action on the D7 branes is discussed in Section 4, while in Section 5 we write the
D-instanton induced effective action on the D3 brane, which is in full agreement with the F-
theoretic description. In Section 6 we put forward our conjecture about the exact effective
coupling on the D3 being determined by its perturbative part plus the eight-dimensional
dynamics of the mass parameters. Finally, in Section 7 we discuss how our results compare
to Nekrasov instanton calculus (or its AGT realization) for the SU(2) theory with Nf = 4,
and in Section 8 we present our conclusions. In the appendices some technical details and
the extension of some results to asymptotically free cases with Nf < 4 are given.
2. F-theory and the D7/D3 system in type I′
In F-theory compactifications over an elliptically fibered manifold [15], the complex struc-
ture modulus τ of the fiber corresponds to the varying axio-dilaton profile of a suitable type
IIB compactification on the base manifold. In [25] A. Sen studied F-theory on an ellipti-
cally fibered K3 surface, which is conjectured to be dual to heterotic string compactified
on a two-dimensional torus. He considered the particular case in which the K3 is at the
orbifold limit in moduli space where it is described by the following curve in Weierstrass
form
y2 = x3 − 1
4 G2(z) x − 1
4 G3(z) . (2.1)
Here z is the coordinate on the base of the fibration and
G2(z) ∝ Q2(z) , G3(z) ∝ Q3(z) , Q(z) =
4∏
I=1
(z − fI ) , (2.2)
with fI constants. The absolute modular invariant of this curve
J = G3
2
G3
2 − 27 G2
3
(2.3)
3A discrepancy in this sense was already noticed at the two-instanton level in [49, 50], where the direct
integration over the moduli was performed without resorting to localization techniques.
4
with the SW curve proposed in [27]. There is a discrepancy already at the massless level: in
this case from the SW curve (and from our microscopic computation) we see that the tree-
level coupling receives no corrections. When computed following Nekrasov’s prescription
for SU(2), or by the AGT method, instead, the coupling gets non-perturbatively modified3.
This suggests that a redefinition of the coupling constant is needed in order to compare
the two approaches; after such a redefinition is performed, remarkably the two methods
are reconciled and the two results agree completely also in the massive case.
The paper is subdivided into several sections as we now describe. In Section 2 we
introduce the model and its F-theory description through the SW curve; from this curve
we extract the instanton expansion of the effective coupling. In Section 3 we describe
the microscopic computation of D-instanton contributions in our model. The resulting
effective action on the D7 branes is discussed in Section 4, while in Section 5 we write the
D-instanton induced effective action on the D3 brane, which is in full agreement with the F-
theoretic description. In Section 6 we put forward our conjecture about the exact effective
coupling on the D3 being determined by its perturbative part plus the eight-dimensional
dynamics of the mass parameters. Finally, in Section 7 we discuss how our results compare
to Nekrasov instanton calculus (or its AGT realization) for the SU(2) theory with Nf = 4,
and in Section 8 we present our conclusions. In the appendices some technical details and
the extension of some results to asymptotically free cases with Nf < 4 are given.
2. F-theory and the D7/D3 system in type I′
In F-theory compactifications over an elliptically fibered manifold [15], the complex struc-
ture modulus τ of the fiber corresponds to the varying axio-dilaton profile of a suitable type
IIB compactification on the base manifold. In [25] A. Sen studied F-theory on an ellipti-
cally fibered K3 surface, which is conjectured to be dual to heterotic string compactified
on a two-dimensional torus. He considered the particular case in which the K3 is at the
orbifold limit in moduli space where it is described by the following curve in Weierstrass
form
y2 = x3 − 1
4 G2(z) x − 1
4 G3(z) . (2.1)
Here z is the coordinate on the base of the fibration and
G2(z) ∝ Q2(z) , G3(z) ∝ Q3(z) , Q(z) =
4∏
I=1
(z − fI ) , (2.2)
with fI constants. The absolute modular invariant of this curve
J = G3
2
G3
2 − 27 G2
3
(2.3)
3A discrepancy in this sense was already noticed at the two-instanton level in [49, 50], where the direct
integration over the moduli was performed without resorting to localization techniques.
4
is z-independent, and so is its complex structure modulus τ which can be determined from
J by inverting the relation
J =
( ϑ8
2(τ ) + ϑ8
3(τ ) + ϑ8
4(τ )
24 η8(τ )
)3
(2.4)
where the ϑa’s are the Jacobi theta-functions and η is the Dedekind function. By studying
the metric on the base space, it can be seen that the latter has the geometry of a torus
orbifold of the type T2/Z2, with Z2 acting as parity reflection along T2, in which the fI ’s
appearing in (2.2) correspond to the points of T2 that are fixed under the Z2 action.
This specific F-theory background can be identified with the so-called type I′ theory,
namely type IIB compactified on a torus T2 and modded out by
Ω = ω (−1)FL I2 , (2.5)
where ω is the world-sheet parity reversal, FL is the left-moving space-time fermion number
and I2 the inversion on T2. This is the T-dual version of type I theory compactified on T2,
and possesses four O7 orientifold planes located at the points of T2 that are fixed under
I2 (see Fig. 1a). Each orientifold plane carries (−4) units of 7-brane charge, which need
to be neutralized by putting 16 D7 branes transverse to T2. If we place them in groups of
4 over each orientifold plane, the tadpole cancellation becomes local and the axio-dilaton
is constant over T2. From now on we take a local perspective and focus on one of the
orientifold fixed planes (say, the one at z = f1) and its associated stack of 4 D7 branes.
2.1 The local case
The action of the orientifold projection Ω is such that each group of 4 D7 branes supports
an eight-dimensional theory with gauge group SO(8). Indeed, the massless degrees of
freedom of the 7/7 open strings build up an eight-dimensional chiral superfield in the
adjoint representation of SO(8), whose first few terms are
M (X, Θ) = m(X) + √2 Θ σ(X) + 1
2 ΘγM N Θ fM N (X) + . . . , (2.6)
where fM N is the field-strength, σ is the gaugino, m is a complex scalar, and (X, Θ) are
the eight-dimensional super-coordinates.
If the D7 branes are moved away from the orientifold plane, i.e. if we give a diagonal
vacuum expectation value to the scalar field m, the charges no longer cancel locally, and
correspondingly the solution of the equation of motion for τ displays logarithmic singu-
larities at the orientifold and D7 brane locations. Let us parametrize the region near the
orientifold fixed point with a coordinate4 w and let the D7 branes and their images be
located at w = ±mi/√2 with i = 1, . . . , 4 (see Fig. 1b). This corresponds to choose the
following vacuum expectation values5
〈m〉 = diag (m1/√2, . . . , m4/√2, −m1/√2, . . . − m4/√2 ) . (2.7)
4With respect to the global coordinate z used above, w ∝ (z − f1).
5Notice that m is a complex field in the adjoint of SO(8), i.e. it is a complex antisymmetric matrix. If
m were real, its eigenvalues mi would be imaginary.
5
J by inverting the relation
J =
( ϑ8
2(τ ) + ϑ8
3(τ ) + ϑ8
4(τ )
24 η8(τ )
)3
(2.4)
where the ϑa’s are the Jacobi theta-functions and η is the Dedekind function. By studying
the metric on the base space, it can be seen that the latter has the geometry of a torus
orbifold of the type T2/Z2, with Z2 acting as parity reflection along T2, in which the fI ’s
appearing in (2.2) correspond to the points of T2 that are fixed under the Z2 action.
This specific F-theory background can be identified with the so-called type I′ theory,
namely type IIB compactified on a torus T2 and modded out by
Ω = ω (−1)FL I2 , (2.5)
where ω is the world-sheet parity reversal, FL is the left-moving space-time fermion number
and I2 the inversion on T2. This is the T-dual version of type I theory compactified on T2,
and possesses four O7 orientifold planes located at the points of T2 that are fixed under
I2 (see Fig. 1a). Each orientifold plane carries (−4) units of 7-brane charge, which need
to be neutralized by putting 16 D7 branes transverse to T2. If we place them in groups of
4 over each orientifold plane, the tadpole cancellation becomes local and the axio-dilaton
is constant over T2. From now on we take a local perspective and focus on one of the
orientifold fixed planes (say, the one at z = f1) and its associated stack of 4 D7 branes.
2.1 The local case
The action of the orientifold projection Ω is such that each group of 4 D7 branes supports
an eight-dimensional theory with gauge group SO(8). Indeed, the massless degrees of
freedom of the 7/7 open strings build up an eight-dimensional chiral superfield in the
adjoint representation of SO(8), whose first few terms are
M (X, Θ) = m(X) + √2 Θ σ(X) + 1
2 ΘγM N Θ fM N (X) + . . . , (2.6)
where fM N is the field-strength, σ is the gaugino, m is a complex scalar, and (X, Θ) are
the eight-dimensional super-coordinates.
If the D7 branes are moved away from the orientifold plane, i.e. if we give a diagonal
vacuum expectation value to the scalar field m, the charges no longer cancel locally, and
correspondingly the solution of the equation of motion for τ displays logarithmic singu-
larities at the orientifold and D7 brane locations. Let us parametrize the region near the
orientifold fixed point with a coordinate4 w and let the D7 branes and their images be
located at w = ±mi/√2 with i = 1, . . . , 4 (see Fig. 1b). This corresponds to choose the
following vacuum expectation values5
〈m〉 = diag (m1/√2, . . . , m4/√2, −m1/√2, . . . − m4/√2 ) . (2.7)
4With respect to the global coordinate z used above, w ∝ (z − f1).
5Notice that m is a complex field in the adjoint of SO(8), i.e. it is a complex antisymmetric matrix. If
m were real, its eigenvalues mi would be imaginary.
5
100%
