Topological string theory, modularity and non-perturbative physics [Elektronische Ressource] / Marco Rauch

Marco Rauch

Sujets

• Physik

Informations

• Publié par: rheinische_friedrich-wilhelms-universitat_bonn
• Publié le: 01 janvier 2011
• Nombre de lectures: 58
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Topological string theory, modularity
and non-perturbative physics
Dissertation
zur
Erlangung des Doktorgrades (Dr. rer. nat.)
der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der
Rheinischen Friedrich-Wilhelms-Universit¨at Bonn
vorgelegt von
Marco Rauch
aus
Memmingen
Bonn 2011Angefertigt mit Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨at
der Rheinischen Friedrich-Wilhelms-Universita¨t Bonn.
1. Gutachter: Prof. Dr. Albrecht Klemm
2. Gutachter: PD Dr. Stefan F¨orste
Tag der Promotion: 19. September 2011
Erscheinungsjahr: 2011ABSTRACT
In this thesis the holomorphic anomaly of correlators in topological string theory, matrix
models and supersymmetric gauge theories is investigated. In the first part it is shown how
the techniques of direct integration known from topological string theory can be used to solve
the closed amplitudes of Hermitian multi-cut matrix models with polynomial potentials. In
the case of the cubic matrix model, explicit expressions for the ring of non-holomorphic
modular forms that are needed to express all closed matrix model amplitudes are given. This
allows to integrate the holomorphic anomaly equation up to holomorphic modular terms that
are fixed by the gap condition up to genus four. There is an one-dimensional submanifold of
the moduli space in which the spectral curve becomes the Seiberg–Witten curve and the ring
reduces to the non-holomorphic modular ring of the group Γ(2). On that submanifold, the gap
conditions completely fix the holomorphic ambiguity and the model can be solved explicitly
to very high genus. Using these results it is possible to make precision tests of the connection
between the large order behavior of the 1/N expansion and non-perturbative effects due to
instantons. Finally, it is argued that a full understanding of the large genus asymptotics in
the multi-cut case requires a new class of non-perturbative sectors in the matrix model. In the
second part a holomorphic anomaly equation for the modified elliptic genus of two M5-branes
wrapping a rigid divisor inside a Calabi-Yau manifold is derived using wall-crossing formulae
and the theory of mock modular forms. The anomaly originates from restoring modularity of
an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-
D2-D0 brane systems. The compatibility of this equation with anomaly equations previously
2observed in the context ofN = 4 topological Yang-Mills theory onP and E-strings obtained
from wrapping M5-branes on a del Pezzo surface which in turn is related to topological string
theory is shown. The non-holomorphic part is related to the contribution originating from
bound-states of singly wrapped M5-branes on the divisor. In examples it is shown that the
information provided by the anomaly is enough to compute the BPS degeneracies for certain
charges.
iiiACKNOWLEDGEMENTS
I am indebted to my “Doktorvater” Prof. Dr. Albrecht Klemm for giving me the opportunity
to work, learn and study with him on this exciting subject. I am grateful for his support and
that he accompanied me over the last couple of years.
Not the least less importantly I have to thank my other collaborators Prof. Dr. Marcos
Marin˜o, Dr. Murad Alim, Dr. Babak Haghighat, Michael Hecht and Thomas Wotschke for
sharing their ideas with me and for countless brilliant discussions. It was challenging, but
still fun to work with and learn from them.
Further, I would like to thank the other members of our group Dr. Denis Klevers, Daniel
Viera Lopes, Maximilian Poretschkin, Jose Miguel Zapata Rolon and Marc Schiereck as well
as the former group members Dr. Thomas Grimm, Dr. Tae-Won Ha and Dr. Piotr Su lkowski
for patiently answering numerous questions and for a joyful time and atmosphere in the offices
of room 104.
Last, this work was partially supported by the Bonn-Cologne Graduate School of Physics
and Astronomy.
iiiivContents
1 Introduction 1
2 Topological string theory, matrix models and BPS states 9
2.1 Topological string theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 N = (2, 2) superconformal field theory . . . . . . . . . . . . . . . . . . 9
2.1.2 Twisting and topological field theory . . . . . . . . . . . . . . . . . . . 12
2.1.3 Coupling to gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.4 Mirror symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.5 The holomorphic anomaly equations . . . . . . . . . . . . . . . . . . . 16
2.1.6 Background independence . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Basics of matrix models and saddle-point analysis . . . . . . . . . . . 20
2.2.2 Matrix models, supersymmetric gauge theory and topological strings . 22
2.2.3 Loop equations, holomorphic anomaly and their solution . . . . . . . . 24
2.2.4 A digression on non-perturbative effects and large order behavior . . . 26
2.3 BPS states and wall-crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 BPS black holes, the attractor mechanism and BPS indices . . . . . . 29
2.3.2 The wall-crossing phenomenon . . . . . . . . . . . . . . . . . . . . . . 30
3 Direct integration and non-perturbative effects in matrix models 35
3.1 Introduction and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Direct integration of the holomorphic anomaly equation . . . . . . . . . . . . 37
3.2.1 Direct integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.2 Modular covariant formulation . . . . . . . . . . . . . . . . . . . . . . 39
3.3 The two-cut cubic matrix model . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 The geometrical setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.2 Direct integration, boundary conditions and integrability . . . . . . . 44
3.3.3 Modular covariant formulation . . . . . . . . . . . . . . . . . . . . . . 45
3.4 The cubic model on a slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 The geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.2 Direct integration and higher genus amplitudes . . . . . . . . . . . . . 51
3.4.3 Boundary conditions and integrability . . . . . . . . . . . . . . . . . . 53
3.5 Non-perturbative aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5.1 Non-perturbative effects in the one-cut matrix model . . . . . . . . . . 54
3.5.2 Non-perturbative effects in the cubic matrix model . . . . . . . . . . . 55
3.5.3 Asymptotics and non-perturbative sectors . . . . . . . . . . . . . . . . 58
4 Wall-crossing, mock modularity and multiple M5-branes 63
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Effective descriptions of wrapped M5-branes . . . . . . . . . . . . . . . . . . . 65
4.2.1 The elliptic genus and D4-D2-D0 branes . . . . . . . . . . . . . . . . . 66
4.2.2 N = 4 SYM, E-strings and bound-states . . . . . . . . . . . . . . . . . 69
vvi CONTENTS
14.2.3 Holomorphic anomaly via mirror symmetry for K3 . . . . . . . . . . 72
2
4.2.4 Generating functions from wall-crossing . . . . . . . . . . . . . . . . . 76
4.3 Wall-crossing and mock modularity . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.1 D-branes and sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.3.2 Relation of KS to Go¨ttsche’s wall-crossing formula . . . . . . . . . . . 81
4.3.3 Holomorphic anomaly at rank two . . . . . . . . . . . . . . . . . . . . 82
4.4 Application and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.1 Blow-up formulae and vanishing chambers . . . . . . . . . . . . . . . . 85
+4.4.2 Applications to surfaces with b = 1 . . . . . . . . . . . . . . . . . . . 852
4.4.3 Extensions to higher rank and speculations . . . . . . . . . . . . . . . 89
5 Conclusions 93
Appendices 97
A Modularity 97
A.1 Elliptic modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.2 Mock modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A.3 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
B Calabi-Yau spaces and its divisors 101
B.1 Rigid divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2 An elliptically fibered Calabi-Yau space . . . . . . . . . . . . . . . . . . . . . 102
C Results 105
C.1 Data of the two-cut example . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.1.1 Large Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.1.2 Conifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.2 Modular properties of the elliptic genus . . . . . . . . . . . . . . . . . . . . . 107
1C.3 Elliptic genera of K3 and K3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2
Bibliography 111