Black Holes, Branes, and Knots in String Theory by Kevin Alan Schaeffer A dissertation submitted ...

228 Pages·2013·2.6 MB·English

Black Holes, Branes, and Knots in String Theory
by
Kevin Alan Schaeffer
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Physics
in the
Graduate Division
of the
University of California, Berkeley
Committee in charge:
Professor Mina Aganagic, Chair
Professor Ori Ganor
Professor Nicolai Reshetikhin
Spring 2013
Black Holes, Branes, and Knots in String Theory
Copyright 2013
by
Kevin Alan Schaeffer
1
Abstract
Black Holes, Branes, and Knots in String Theory
by
Kevin Alan Schaeffer
Doctor of Philosophy in Physics
University of California, Berkeley
Professor Mina Aganagic, Chair
String theory has proven to be fertile ground for interactions between physical and
mathematical ideas. This dissertation develops several new points of contact where
emerging mathematical ideas can be applied to stringy physics, and conversely where
stringy insights suggest new mathematical structures. In the first half, I explain how
the formula of Kontsevich and Soibelman (KS) can be used to compute the spectrum
of stable BPS particles in Calabi-Yau compactifications and I show that dimer models
can be used to prove the KS formula for walls of the second kind in toric Calabi-Yau
manifolds. I also explain how dimer models give a way of associating integrable
systems to Calabi-Yau threefolds and show that this map agrees with an existing
gauge-theoretic map. The second half of this dissertation focuses on a background in
M-theory that defines the refined topological string. I explain how orientifolds can be
introduced into this background, leading to new integrality conditions on amplitudes
and to new invariants for torus knots. Finally, I introduce a new duality that relates
the refined counting of supersymmetric balck hole entropy and refined topological
string theory.
i
Contents
1 Introduction 1
2 Wall Crossing, Quivers, and Crystals 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Walls of the Second Kind and Seiberg Dualities . . . . . . . . 5
2.1.2 Dimer models, Seiberg Dualities, and BPS Degeneracies . . . . 6
2.1.3 Calabi-Yau Crystals, Quivers and Topological Strings . . . . . 7
2.2 Walls of the Second Kind and Seiberg Duality . . . . . . . . . . . . . 8
2.2.1 KS conjecture and Seiberg duality . . . . . . . . . . . . . . . . 10
2.2.2 A simple example . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Quivers from Calabi-Yau Threefolds . . . . . . . . . . . . . . . . . . 13
2.3.1 The mirror of P1 ×P1 example . . . . . . . . . . . . . . . . . 16
2.3.2 Toric Quivers and Dimers on a Torus . . . . . . . . . . . . . 18
2.4 BPS Degeneracies and Wall Crossing from Crystals and Dimers . . . 19
2.4.1 BPS states of Quivers and Melting Crystals . . . . . . . . . . 20
2.4.2 Wall Crossing and Crystals . . . . . . . . . . . . . . . . . . . 23
2.4.3 Dimers and Wall Crossing . . . . . . . . . . . . . . . . . . . . 25
2.4.4 The proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Mirror Symmetry and Quivers . . . . . . . . . . . . . . . . . . . . . 31
2.5.1 Mirror Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 Mirror Symmetry, D-branes and Quivers . . . . . . . . . . . . 32
2.5.3 P1 ×P1 example . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Monodromy and B-field . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.1 The P1 ×P1 example . . . . . . . . . . . . . . . . . . . . . . 37
2.7 Geometry of the D6 brane bound states . . . . . . . . . . . . . . . . 42
2.7.1 Geometry of D6 branes . . . . . . . . . . . . . . . . . . . . . . 43
2.7.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7.3 Geometric interpretation of the D6 brane bound states . . . . 49
2.7.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7.5 The large B-field limit and DT/GW correspondence . . . . . . 51
ii
3 Dimer Models and Integrable Systems 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Some Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Integrable Systems from Dimers, 5d and 4d . . . . . . . . . . . . . . . 58
3.4 The Periodic Toda Chain . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4.1 Yp,0 Integrable Systems . . . . . . . . . . . . . . . . . . . . . 60
3.4.2 Kasteleyn matrix . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.3 More Relativistic Generalizations of Toda from Yp,q . . . . . . 66
3.4.4 Yp,p Integrable Systems . . . . . . . . . . . . . . . . . . . . . 67
3.4.5 Yp,q Integrable Systems as Spin Chains . . . . . . . . . . . . . 69
3.5 Generating New Integrable Systems via Partial Resolution . . . . . . 70
3.5.1 Examples: Partial Resolutions of Y4,0 . . . . . . . . . . . . . . 72
3.6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Orientifolds and the Refined Topological String 80
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2 M-Theory and Refined Topological Strings . . . . . . . . . . . . . . . 82
4.3 Orientifolds and M-Theory . . . . . . . . . . . . . . . . . . . . . . . 86
4.4 A New Integrality Conjecture from M-Theory . . . . . . . . . . . . . 92
4.5 Open Strings from Refined Chern-Simons Theory . . . . . . . . . . . 96
4.5.1 Refined Chern-Simons as a Topological Field Theory . . . . . 97
4.6 Refined Kauffman Invariants . . . . . . . . . . . . . . . . . . . . . . 102
4.6.1 Relation to Knot Homology . . . . . . . . . . . . . . . . . . . 104
4.6.2 Example: The Hopf Link . . . . . . . . . . . . . . . . . . . . . 105
4.6.3 Example: The Trefoil Knot . . . . . . . . . . . . . . . . . . . 106
4.6.4 Example: The General T(2,2m+1) Torus Knot . . . . . . . . 107
4.6.5 Example: The T(3,4) Knot . . . . . . . . . . . . . . . . . . . 107
4.7 The Large N Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5 Refined Black Hole Ensembles and Topological Strings 111
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.2 The OSV Conjecture: Unrefined and Refined . . . . . . . . . . . . . 114
5.2.1 Refining the Conjecture . . . . . . . . . . . . . . . . . . . . . 116
5.3 Motivating the Refined Conjecture . . . . . . . . . . . . . . . . . . . 119
5.3.1 Refined OSV and The Wave Function on the Moduli Space . . 121
5.4 Refined Topological String on L ⊕L → Σ . . . . . . . . . . . . . . 124
1 1
5.4.1 A TQFT for the Refined Topological String . . . . . . . . . . 125
5.4.2 Branes in the Fiber . . . . . . . . . . . . . . . . . . . . . . . 130
5.4.3 Refined Topological Strings on L ⊕ L → Σ and 5d U(1)
1 2 g
Gauge Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.5 Refined Black Hole Entropy . . . . . . . . . . . . . . . . . . . . . . . 134
5.5.1 D4 branes on L ⊕L → Σ . . . . . . . . . . . . . . . . . . . 137
1 2
iii
5.5.2 From 4d N = 4 Yang-Mills to 2d (q,t)-deformed Yang-Mills . 138
5.5.3 Refined Chern-Simons Theory and a (q,t)-deformed Yang-Mills 143
5.5.4 A Path Integral for (q,t)-deformed Yang-Mills . . . . . . . . . 144
5.5.5 Example: O(−1) → P1 . . . . . . . . . . . . . . . . . . . . . 148
5.6 Large N Factorization and the Refined OSV Conjecture . . . . . . . 150
5.6.1 Genus g ≥ 1 Case . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.6.2 Genus g = 0 Case . . . . . . . . . . . . . . . . . . . . . . . . 156
5.7 Black Hole Entropy and Refined Wall Crossing . . . . . . . . . . . . 157
Bibliography 162
A The Conifold 181
B The local P2 example 183
C The (2,0) Theory on Seifert Three-Manifolds 187
C.1 Computing the refined index . . . . . . . . . . . . . . . . . . . . . . . 192
D Background on Macdonald polynomials 195
E Facts about SO(2N) 199
F Refined Indices 201
F.1 Five Dimensional Indices . . . . . . . . . . . . . . . . . . . . . . . . . 201
F.2 Three Dimensional Indices . . . . . . . . . . . . . . . . . . . . . . . . 202
G Identities for Macdonald Polynomials 205
G.1 SU(N) Macdonald Polynomials . . . . . . . . . . . . . . . . . . . . . 206
G.2 SU(∞) Macdonald Polynomials . . . . . . . . . . . . . . . . . . . . . 207
H Gopakumar-Vafa Invariants 209
I Factorization of the (q,t)-Dimension and Metric 212
J Refined S-Duality 219
iv
Acknowledgments
This dissertation would not have been possible without the support and encourage-
ment of many people. First of all, thanks go to Mina Aganagic for advising this thesis
and to Vincent Bouchard, Richard Eager, Sebastian Franco, and Kolya Reshetikhin
for being enthusiastic and insightful collaborators. The members of the BCTP, espe-
cially Ori Ganor, Petr Horava, and Raphael Bousso, also deserve special thanks for
patiently answering my questions and explaining their ideas over the years. I am also
grateful to my fellow graduate students and postdocs, especially Chris Beem, Tudor
Dimofte, Abhijit Gadde, Stefan Leichenauer, and Michael Zaletel for many inspiring
and wonderful physics conversations. I also want to thank Anne Takizawa whose
dedication and concern for graduate students has improved my time at Berkeley in
many ways. Finally, I owe debts of gratitude to Ed Chiang, who first introduced me
to the beauty of physics, to my parents for their continual encouragement throughout
this journey, and to Erika for her unwavering love and support.
1
Chapter. 1
Introduction
String theory is an ambitious enterprise – not only does it give a consistent way
of combining quantum mechanics and general relativity, it also gives a unified frame-
work for explaining the origin of all fundamental forces and matter. Historically, it
was first studied as a fundamental theory by perturbatively quantizing the classical
string, leading to the requirement of ten microscopic spacetime dimensions. This per-
turbative approach was useful for understanding the most basic spectrum of stringy
excitations and for obtaining a coarse picture of the theory, but non-perturbative
effects were poorly understood. This changed dramatically during the second string
revolution in the ’90s, leading to remarkable non-perturbative dualities and break-
throughs in our understanding of deep questions in quantum gravity such as black
hole entropy [261] and holography [146,204,263,281]. Although these developments
greatly enriched our perspective on string theory, they also raised questions about
whether strings should really be viewed as the fundamental degrees of freedom in
“string” theory.
Muchofthesubsequentresearchinstringtheoryhasfocusedonobtainingabetter
understanding of stringy dualities in order to find the deep structures that underpin
the theory. Remarkably, ideas from mathematics have emerged as the structures con-
trolling many of the dualities that appear in string theory – this has given convincing
evidence that the mathematical languages of geometry, topology, and combinatorics
are the correct ones for understanding many phenomena. At the same time, math-
ematics has also emerged as a useful testing ground for string theory – although
experimental tests of string theory are still far away, string theorists can use dualities
to make many nontrivial mathematical predictions. The verification of these predic-
tions by mathematicians has given powerful evidence for the consistency and power
of string theory.
Inthisdissertation,Iwilldiscussseveralmathematicalstructuresthathaveemerged
from studying supersymmetric string backgrounds. This work is centered around
string theory on geometries of the form R3,1×X and M-theory on R4,1×X, where X
is a Calabi-Yau threefold. Originally, such backgrounds were studied as a way of pre-
2
serving supersymmetry while reducing the macroscopic dimensionality of space-time
from10to4, inagreementwitheverydayobservations. However, ithasalsobeenreal-
ized that such compactifications are extremely rich mathematically – for example, the
topological string is critical when its target space is a Calabi-Yau threefold (cˆ= 3). It
is an open question whether this particular connection between mathematical beauty
and physical relevance is simply coincidental or if it gives some explanation for the
macroscopic dimensionality of our universe.
In the Chapter 2, I focus on IIA string theory on R3,1 ×X in the case where X
is a noncompact Calabi-Yau threefold. Such backgrounds preserve N = 2 supersym-
metry in four dimensions (8 supercharges). It is particularly interesting to study the
spectrum of stable BPS branes wrapping the cycles of X. A remarkable fact is that
as the moduli of X are varied, certain branes can become unstable and the spectrum
can jump. These jumps occur along real codimension-one “walls” in the moduli space
and a fundamental problem is to predict the form of these jumps under such “wall
crossing.”
This wall crossing phenomenon was first pointed out by Seiberg and Witten in
their solution of pure N = 2 Yang-Mills – they realized that for their theory to be
consistent, infinitely many dyons that are stable at large VEVs would have to decay
in the interior of the moduli space [250]. Progress in understanding wall crossing from
a physical perspective has been made in [12,27,28,58,73,84,95,97,124–126,175,236].
However, the greatest leap forward was made by Kontsevich and Soibelman (KS)
[188], who proposed a mathematical formula that determines the jump in spectrum
in complete generality. In this chapter, I explain how the KS formula can be tested
by studying the intricate spectrum of BPS branes in toric Calabi-Yaus with compact
four-cycles. For these geometries, I show that the KS formula for crossing walls of
the second kind can actually be proven using the tool of dimer models which can be
used to count BPS states. I also discuss the connections between crossing walls of
the second kind and Seiberg duality in quiver quantum mechanics, and conclude by
explaining how topological string theory emerges in the limit of large B-field. This
work was done in collaboration with Mina Aganagic and appeared previously in [16]
In Chapter 3, I focus on Calabi-Yau compactifications in M-theory which engineer
five-dimensional supersymmetric gauge theories. Starting with the work of [142,222],
it has been known that four- and five-dimensional gauge theories with 8 super-
charges are intimately related to algebraic integrable systems. The connection is
made through the low energy solution of these gauge theories, which naturally has
the structure of a torus-fibration over the Coulomb branch. Since these gauge the-
ories are geometrically engineered by compactifying M-theory or string theory on a
Calabi-Yau, this gives a way of assigning integrable systems to threefolds.
Recent work by Goncharov and Kenyon (GK) gave a mathematically precise map
from dimer models to integrable systems [137]. As explained in Chapter 2, there is
also a very useful assignment of dimer models to Calabi-Yau manifolds. Combining
these two observations gives a method for obtaining explict integrable systems from
3
Calabi-Yauthreefolds. Thisraisesthenaturalquestion: dothetwomethodsoffinding
integrable systems (the gauge-theoretic method and the mathematical GK method)
give the same results? In this chapter, I explicitly show that this is true for Yp,0
Calabi-Yaus which are known to engineer pure SU(p) supersymmetric gauge theories.
This work was carried out in collaboration with Richard Eager and Sebastian Franco,
and previously appeared in [103].
Chapters4and5focusonaone-parametergeneralizationofthetopologicalstring,
known as “refined topological string theory.” To put this generalization into context,
it helps to recall the origin of topological string theory and its connection with string
duality. In [276], it was discovered that two-dimensional N = (2,2) sigma models
could be topologically twisted, making them independent of the worldsheet metric.
Depending on the kind of twist (A or B), the topological theory is also independent
of the complex structure of the target space (A twist) or independent of the Ka¨hler
structre of the target (B twist). It was further explained in [276] that the twisted
sigma model can be coupled to worldsheet gravity in a way analogous to the bosonic
string, leading to a topological string theory.
Althoughoriginallydefinedontheworldsheet, itwasquicklyrealizedthattopolog-
icalstringtheorycanalsobedefinedascomputingF-termsinlow-energysupergravity
after Type II string theory has been compactified on a Calabi-Yau threefold, X [45]
Equivalently, the topological string partition function can be computed in M-theory
by counting M2-brane bound states wrapping cycles in X [91,138,139]. It is this
last perspective that is especially valuable for generalization – these M2-branes can
be counted in a more refined way, giving an implicit definition of refined topological
string theory [162,171,225]. Ideally, there would also be an explicit worldsheet defi-
nition of the theory, but such a definition is currently unknown1 and is not necessary
to perform many computations.
Taking this definition as a starting point, much progress has been made in com-
puting the partition function of the refined topological string in both the closed and
open cases [2,18,20,21,60,90,91,95,172]. In Chapter 4, I explain how to introduce ori-
entifolds into both refined and ordinary topological string computations – this leads
to new integrality conditions for the partition functions in both cases. It also gives a
way of defining open topological string theory, and therefore, a refinement of SO(N)
Chern-Simons theory. This can then be used to compute a refined generalization of
the Kaufmann polynomial for torus knots, and further leads to a new instance of large
N duality. All of these computations are made possible by the intricate mathematical
structure of D-type Macdonald Polynomials. This work was done in collaboration
with Mina Aganagic and appeared previously in [15].
Finally, in Chapter 5 I discuss how refined topological string theory is related to
black holes. This work generalizes the seminar discovery of Ooguri, Strominger, and
Vafa (OSV) that the mixed partition function of BPS black holes is computed by the
1For some recent progress in this direction see [29,221]