Paper
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On the curious spectrum of duality invariant higher-derivative gravity
Authors
Olaf Hohm,Usman Naseer
Barton Zwiebach,Elad Salomons,Asya Kudryavtseva,Martin Hergersberg,Elena Vasenina,Ulises Cuore,Elisabeth Dietze,Sara Petty,Andreas Hofmann,Andrey Poliakov,Svetlana Shabalina,Yulia Smirnova,n sandeep chaitanya,Irina Snegireva,CARLA TRIBIOLI,Nadine Rons,Tatiana Davydova,Pei-Ying WU,Татьяна Степанова,Artyom Nikitin,Anatoly Khavkin,Ирина Ездакова,Novikov Yurii,Daria Gognieva,Frieder Schock,Rais Gizatullin,Ilya Leyderman,Елена Меньшикова,VLADIMIR PAVLOV,Marina Petrova,Kirk Jensen,Evgenii Azarov,Анна Спиваковская,Halyna Voznyak,William Dirienzo,OZGUR COGULU,xianqiong zou,Alexey Ryzhenko,Joachim Dengler,Scott Fleming,Joachim Kaiser,Li Rebekah Feng,Keisuke Yano,Lawrence Kennedy,Gary Carver,Luis Emilio Cruz-Barba,Rajesh Dachiraju,Torben Gaedt,Dimitrios Zoakos,Steven Shannon,Simone Giacomelli,Vishnu Ram OV,Bulent Basol,Dagnachew Muluye Fetene,Miha Moškon,Ramnath Subbaraman,Dobrolyubska Yuliya,Vladimir Shats,Steven Burian,Joao Figueiredo,Марина Мазитова,Omar Diego Lopez,Сергей Кичигин,Élodie Brun,Yuji Omura,Nathaniel Butlin,Dimitrios Lykomitros,Engel Roza,Noah Goldberg,Nicholas Demarais,Stephan Endres,Jacqueline Radigan,Anna Frebel,Alexander Szalay,Tamas Budavari,Asiya Safina,Resi Mencacci,Andrew Battisti,Jennifer Sobeck,Kristina Nyland,Remi Leandre,Luciana Bianchi,REIHANEH KARDEHI MOGHADDAM,Marcin Badziak,Alessandra Gnecchi,Rintze Zelle,Sara Böger,Ernest Williams,Tzu-Jung Fang Fang,Сергей Пономарёв,Chia-hung Tsai,G KUMAR,Alexander Polyakov,Przemysław Mańkowski,Iris Kriest,Wojciech Bartoszek,Ze-Jia Chen,Светлана Ивановна Малявская,Miguel Angel Parada,David Newnham,Sandra Clauder-Muenster,Irina Makarova,也铃 周,Зоя Атаманюк,Stefan Vermeulen,Nikolay Demidov,Hanna Telnova,Reiner Lauterbach,maria ofelia moroni,Ta-Chien Chan,Clemens Schannwell,Marsha Speevak,Iain Hrynaszkiewicz,Vera Okorokova,Ivar McDonald,Ирина Чорномидз,Walter Lima TORRES NETO,Xu-Liang Fan,Анатолій Карась,Shi Zhou,Maryna Kuznietsova ,Monica Rodriguez,András Földi,Fanjasoa RAKOTOMANANA,Stefano Menegon,Adriano de Souza Martins,Dmytro Havrilov,CHUNG-JEN CHEN,Igor Okunev,Mariusz Ciesla,Yutaka Hayano,Hiroshi Terada,Gholamreza Naser,Pedro da Costa Junior,Dorota Bors,Dhananjay Suresh,Michele Del Zotto,Andrei Constantin,Orkut GUCLU,Ludmila Popova,Jingwu Duan,Brian J Sudlow,Puma Shen,Jens Strauss,Елена Соловьева,Timo Heimerdinger,Martin Gonzalez-Alonso,Andreu Casero-Ripolles,LI-JEN WENG,Takumi Kuwahara,Junjie Rao,Leslaw Rachwal,Davide Cassani,Elma Uddameri,Javier Camacho-Rodriguez,Mark Neubauer,Shuxia Pan,Maksim Podvalnyi,Jordi Martorell,Dariusz Klejnowski-Różycki,Brendon Bradley,vesna cukic,LI HSUAN CHANG,Wing Kin Puk,Sergei Filippov,Richard Wintle,Nick Halmagyi,Peter Linde,Francesc Prats,nataliia feseniuk,Hüseyin Topan,Mojegan Kowsari,James Jia-Hau Liu,Oscar Chacaltana,N/A ,Mariana Grana,Simone Musso,Billy Connor,Luis Enrique Nores Torres,Oya Topuz,Fakhri Zahedy,Sebastian Rahtz,Pedro Gomes,JORGE ALBERTO Hidalgo-Toledo,Frank Günther,Alessandro Di Cicco,Guido Fantini,V.E. Khoronenko,Ислам Шидаков,Isaac Wait,Luca Giovangiuseppe LIMINTA,Justin Hendrix,IVAN-CHEN CHENG,Aleksandr Mokhirev,Aram Mokarizadeh,Oleg Tarasov,Stefan Zellmann,Elizabeth Wood,María Begoña Sánchez Galán,Andrea Thamm,Gonzalo Miguez,Elias Kaufhold,CHIA-LING WU,Mario Amalfi,Natalia Borysenko,David Martín López,Дрєєв Олександр,Gavin Kenny,Jorge Rezende,Stefan Scheiner,Bryce Lawrence,Bugaychuk ,Suzanna Ward,Maya Velmuradova,Iveta Līce-Zikmane,Оксана Халіман,Alan Ivkovic,Svitlana Ryndiuk,Mary Robledo,Yury Plugatar,Cesar Henriquez-Camacho,Sergey Sheshnitsan,Iryna Mynchynska,Mirela Kozlovsky,Belén Sola,Marie-Luise Kapsch,Anna Shcherbakova,Юрій Дутчак,Adrian Hartley,Татьяна Лабутина,Алексей Белоус,Allison Youngblood,Andrey Teslya,Gloria Titos Vela,JUAN ANTONIO UREÑA SALCEDO,Gena Heidary,Bruna Donato,Oleg Kildyushov,Guzel Nurullina,Margaret Froelich,Ya-Chih Tien,Marc VILLAR,Joaquim Caner,Concepcion Saiz,Miguel Crispim Romão,Drew Davis,Milene Trindade,Dominique Mouazé,Hongliang Zhang,Jianlin Hu,Magda Eldomiaty,amal ezzeldin,Pavel Shcherbinin,Roman Kralik,Samar Khatiwala,Светлана Чупахина,PING-YIN KUAN,Anja Knaupp,Lee Patrick,Bushra Nasir,Alcina Maria de Castro Martins,Viacheslav Zhuravlev,Whei-Lin Pan,Isha Kotecha,Roman Shakhnovich,THOMAS MON-HSIAN HSIEH,Egor Kuzmin,Misko Cubrinovski,Evgeniya Kirakosyan,Evgeniya Shatokhina,Evgeny Grachikov,Chi shing Chen,Ken Mimasu,Firuz Ulmasov,GALO WILFRIDO TOBAR FARIAS,Олеся Васильевна,Paula Leiria Pinto,Suresh Cuddapah,arifin noor sugiharto,Thomas Kahle,Laura Kozell,Renato Zambon,SANTOSH GHIMIRE,Pavle Stamenović,Michael Dukes,Regan Murray,Kai-Ping Huang,Wayne Rankin,Joseph Broach,Takahiro Endo,Ilya Kofiadi,Huei-Lih Hwang,Bora Erdağı,Jens Kleimann,Francesco SCALERA,Nathan McNeil,David Thompson,Louise Sandberg Sørensen,Зураб Хондзия,Zarif Sobirov,Gideon Keren,Vaiva Žeimantienė,Nikolay Lyutorovich,Safiullah Khan Achakzai,Alex McCarthy,Keitaro Satoh,Dhundi Pathak,Pai-Hsing Wu,Yongliang Song,Dirk Hoffmeister,Gail Kinman,Eleni Kasapidou,Muhammad Babar Imran,Marc Le Menn,GEMA CHOCANO,Евгений Игумнов,Renata Cristina Picão,Yelena Likhoshway,Scott B Miles,Harry Broening,Jose Antonio Jimenez-Berni,Jambul Yusupov,Yang Lyu (Lv),Fabian Christandl,Marina NODEL,Michael Crisp,Khrystyne Duddleston,Miguel Trelles De Belaunde,Aurora Meroni,Ricardo Amorim,Marcos Martins,Joel Ariate,Evgeniy Mekhamadiev,Andrew Frampton,Haicheng Zhang,Matthias Jochner,Chris Pridgeon,Sofishchenko Iryna,João Henrique da Silva,Галина Требелева,Silke Schiewer,Lim Jia Qi,Carlos Montemayor,Leandro Silva Pimenta,Alexandra Filipova,Ivan Alekseev,Kehinde David Oyeyemi,André Ary Leonel,Renmin Yuan,Debadin Bose,Raisa Uteeva,Alexander Topham,Rui Mendes,András Salamon,Le Kuai,Hernan PEREZ MOLANO,Chakrit Pongkitivanichkul,Andreas Søgaard,Roberta Jorge Luz,Elbegjargal Nasanbat,Waldemar Andrzej Gorzym-Wilkowski,Janet Hunter,Alex Manetta,Anna Koseniuk,Annie Putman,Edson Silva ,Anjali Tripathi,Guadalupe Ruiz,Putu Aditya Ferdian Ariawantara,Claudio Cabral-Romero,FABIANO CASTRO,Ivan Semenchuk,H. 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arXiv:1607.01784v2 [hep-th] 19 Sep 2016
July 2016
MIT-CTP-4815
On the curious spectrum of
duality invariant higher-derivative gravity
Olaf Hohm,1 Usman Naseer,2 and Barton Zwiebach2
1 Simons Center for Geometry and Physics,
Stony Brook University,
Stony Brook, NY 11794-3636, USA
2 Center for Theoretical Physics,
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
ohohm@scgp.stonybrook.edu, unaseer@mit.edu, zwiebach@mit.edu
Abstract
We analyze the spectrum of the exactly duality and gauge invariant higher-derivative double
field theory. While this theory is based on a chiral CFT and does not correspond to a standard
string theory, our analysis illuminates a number of issues central in string theory. The full
quadratic action is rewritten as a two-derivative theory with additional fields. This allows for a
simple analysis of the spectrum, which contains two massive spin-2 ghosts and massive scalars,
in addition to the massless fields. Moreover, in this formulation, the massless or tensionless
limit α′ → ∞ is non-singular and leads to an enhanced gauge symmetry. We show that the
massive modes can be integrated out exactly at the quadratic level, leading to an infinite series
of higher-derivative corrections. Finally, we present a ghost-free massive extension of linearized
double field theory, which employs a novel mass term for the dilaton and metric.
July 2016
MIT-CTP-4815
On the curious spectrum of
duality invariant higher-derivative gravity
Olaf Hohm,1 Usman Naseer,2 and Barton Zwiebach2
1 Simons Center for Geometry and Physics,
Stony Brook University,
Stony Brook, NY 11794-3636, USA
2 Center for Theoretical Physics,
Massachusetts Institute of Technology
Cambridge, MA 02139, USA
ohohm@scgp.stonybrook.edu, unaseer@mit.edu, zwiebach@mit.edu
Abstract
We analyze the spectrum of the exactly duality and gauge invariant higher-derivative double
field theory. While this theory is based on a chiral CFT and does not correspond to a standard
string theory, our analysis illuminates a number of issues central in string theory. The full
quadratic action is rewritten as a two-derivative theory with additional fields. This allows for a
simple analysis of the spectrum, which contains two massive spin-2 ghosts and massive scalars,
in addition to the massless fields. Moreover, in this formulation, the massless or tensionless
limit α′ → ∞ is non-singular and leads to an enhanced gauge symmetry. We show that the
massive modes can be integrated out exactly at the quadratic level, leading to an infinite series
of higher-derivative corrections. Finally, we present a ghost-free massive extension of linearized
double field theory, which employs a novel mass term for the dilaton and metric.
Contents
1 Introduction 1
2 Full quadratic theory and non-derivative terms 4
2.1 Full quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Full non-derivative Lagrangian and vacua . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Spectrum of the quadratic theory 7
3.1 Spectrum of the two-derivative quadratic theory . . . . . . . . . . . . . . . . . . . . . 8
3.2 Spectrum of the full six-derivative quadratic theory . . . . . . . . . . . . . . . . . . . . 8
4 Massive linearized DFT 11
5 Tensionless limit and degrees of freedom 14
5.1 Tensionless limit, enhanced gauge symmetry, and Bianchi identities . . . . . . . . . . 14
5.2 Higher derivatives that reduce the number of degrees of freedom . . . . . . . . . . . . 15
6 Eliminating the massive fields 17
A Degrees of freedom 21
A.1 Degrees of freedom of two-derivative HSZ theory . . . . . . . . . . . . . . . . . . . . . 22
A.2 Degrees of freedom of full quadratic HSZ theory . . . . . . . . . . . . . . . . . . . . . . 23
A.3 Degrees of freedom of massive DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1 Introduction
Some of the salient characteristics of string theory are the presence of higher-derivative α′ corrections,
massive modes of higher spin, and duality invariance, such as T-duality. In this paper we aim to
illuminate the interplay between these aspects by analyzing the quadratic approximation to the α′-
deformed double field theory (DFT) constructed by Siegel and two of the authors in [1] (generalizing
[2–6] and further investigated in [7–15]). This theory, henceforth called HSZ theory, contains higher-
derivative corrections and is exactly duality1 and gauge invariant and hence well-suited for this purpose.
1In the following we refer to the global O(d, d, R) invariance emerging upon dimensional reduction on a torus, which
is made manifest in DFT, for brevity simply as ‘duality invariance’.
1
1 Introduction 1
2 Full quadratic theory and non-derivative terms 4
2.1 Full quadratic Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Full non-derivative Lagrangian and vacua . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Spectrum of the quadratic theory 7
3.1 Spectrum of the two-derivative quadratic theory . . . . . . . . . . . . . . . . . . . . . 8
3.2 Spectrum of the full six-derivative quadratic theory . . . . . . . . . . . . . . . . . . . . 8
4 Massive linearized DFT 11
5 Tensionless limit and degrees of freedom 14
5.1 Tensionless limit, enhanced gauge symmetry, and Bianchi identities . . . . . . . . . . 14
5.2 Higher derivatives that reduce the number of degrees of freedom . . . . . . . . . . . . 15
6 Eliminating the massive fields 17
A Degrees of freedom 21
A.1 Degrees of freedom of two-derivative HSZ theory . . . . . . . . . . . . . . . . . . . . . 22
A.2 Degrees of freedom of full quadratic HSZ theory . . . . . . . . . . . . . . . . . . . . . . 23
A.3 Degrees of freedom of massive DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1 Introduction
Some of the salient characteristics of string theory are the presence of higher-derivative α′ corrections,
massive modes of higher spin, and duality invariance, such as T-duality. In this paper we aim to
illuminate the interplay between these aspects by analyzing the quadratic approximation to the α′-
deformed double field theory (DFT) constructed by Siegel and two of the authors in [1] (generalizing
[2–6] and further investigated in [7–15]). This theory, henceforth called HSZ theory, contains higher-
derivative corrections and is exactly duality1 and gauge invariant and hence well-suited for this purpose.
1In the following we refer to the global O(d, d, R) invariance emerging upon dimensional reduction on a torus, which
is made manifest in DFT, for brevity simply as ‘duality invariance’.
1
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