This thesis studies the \u1e90-invariants of three-manifolds, q-series valued invariants which are half-indices of 3d N=2 theories T[M\u2083] obtained by compactifying the 6d (2,0) theory on a three-manifold. These invariants sit at the intersection of supersymmetric field theory, low-dimensional topology, and number theory. We develop their structure along four directions. A unifying theme is that the natural structures associated with \u1e90 are organized by enlargements of the usual moduli spaces appearing in Chern-Simons theory.
\r\n\r\nFirst, we construct the \u1e90-TQFT, a generalized topological quantum field theory that computes \u1e90. Unlike Atiyah-type TQFTs, this framework involves infinite-dimensional Hilbert spaces and depends on Spin\u1d9c-structures. A central ingredient is a Q-extended quantization of SL(2C) Chern-Simons theory on the torus in which the space of flat connections (C/Z)\u00b2 is replaced by C\u00b2. Second, we propose a combinatorial categorification of WRT invariants. The resulting homologies have \u1e90 as their graded Euler characteristic, whose weighted sum over Spin\u1d9c-structures recovers the WRT invariant. These homologies are computable for a large class of three-manifolds and provide a prototype for the physically defined but largely inaccessible BPS cohomologies of T[M\u2083]. Third, we study the quantum modularity of \u1e90-invariants with line defects. The half-indices with different line defects organize into a module P over the ring of polynomials in q, and we uncover a connection between P and a module of certain meromorphic functions that elucidates the algebraic structure of P. Finally, we introduce ceff, an effective central charge measuring BPS degrees of freedom in 3d N=2 theories. When applied to T[M\u2083], this quantity yields a new topological invariant that provides a particular integer lift of the Chern-Simons invariant. We use ceff as a probe to test competing proposals for extending \u1e90 beyond the class of weakly negative-definite plumbed three-manifolds, finding that the resurgence-based and regularized-surgery approaches yield different results.
", "doi": "10.7907/7756-7774", "publication_date": "2026", "thesis_type": "phd", "thesis_year": "2026" }, { "id": "thesis:15256", "collection": "thesis", "collection_id": "15256", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012023-043437790", "type": "thesis", "title": "Regularities, Resurgence and R-Matrices in Chern Simons Theory", "author": [ { "family_name": "Gruen", "given_name": "Angus Fred Wilkinson", "orcid": "0000-0003-0284-009X", "clpid": "Gruen-Angus-Fred-Wilkinson" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ekholm", "given_name": "Tobias", "orcid": "0000-0003-2060-8462", "clpid": "Ekholm-Tobias" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis aims to address two related but distinct problems in Chern Simons theory:
\r\n\r\n1. In 2019, Gukov and Manolescu observed that for fixed a knot K, the family of coloured Jones polynomials Jk(K; q) display regularity in colour k and conjectured that this could be captured by a 2 variable series FK(x, q). Over the subsequent few years, Park proved that, for a large family of knots, FK(x, q) could be computed using the R-matrix for a particular Verma module.
\r\n\r\nWe will show that it is possible to extend the work of Park to compute the 2 variable series FNK(x, q) associated to other lie groups, slN, which capture a similar regularity in the quantum invariants PNk(K; q). Following on from this we will further show that in many cases these series FNK(x, q) themselves display a regularity in N, reminiscent of the HOMFLY-PT polynomial, allowing the construction of a 3 variable series FK(x, a, q) interpolating FNK(x, q) for all N.
\r\n\r\n2. Complex Chern Simons theory is a rare example of Quantum field theory with both interesting non-perturbative behaviour and whose perturbative expansion can be computed to high order. For a nice class of 3-manifolds, namely surgeries on knot complements, we will show how to predict aspects of the non-perturbative behaviour first semi-classically and then, using resurgence, through studying just the perturbative expansion around the trivial flat connection. Finally, we show that contrary to expectation, these families of 3-manifolds display regularity in the surgery coefficient.
", "doi": "10.7907/dr5q-3074", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:14606", "collection": "thesis", "collection_id": "14606", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05252022-010117961", "primary_object_url": { "basename": "PhD Thesis.pdf", "content": "final", "filesize": 1167098, "license": "other", "mime_type": "application/pdf", "url": "/14606/1/PhD Thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "3-Manifolds, Q-Series, and Topological Strings", "author": [ { "family_name": "Park", "given_name": "Sunghyuk", "orcid": "0000-0002-6132-0871", "clpid": "Park-Sunghyuk" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Manolescu", "given_name": "Ciprian", "orcid": "0000-0003-0600-8751", "clpid": "Manolescu-Ciprian" }, { "family_name": "Ekholm", "given_name": "Tobias", "orcid": "0000-0003-2060-8462", "clpid": "Ekholm-Tobias" }, { "family_name": "Hutchcroft", "given_name": "Thomas", "orcid": "0000-0003-0061-593X", "clpid": "Hutchcroft-Tom" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "\u1e91 is a 3d TQFT whose existence was predicted by S. Gukov, D. Pei, P. Putrov, and C. Vafa in 2017. To each 3-manifold equipped with a spinc structure, \u1e91 is supposed to assign a q-series with integer coefficients that is categorifiable and provides an analytic continuation of the Witten-Reshetikhin-Turaev invariants. In 2019, S. Gukov and C. Manolescu initiated a program to mathematically construct \u1e91 via Dehn surgery, and as part of that they conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series FK(x,q), which is \u1e91 for the knot complement. Following those developments, in this thesis we develop further and generalize the theory of \u1e91. Some of the main results are:
\r\n1. Proof of Gukov-Manolescu conjecture for a big class of links, including all homogeneous braid links, which gives a mathematical definition of \u1e91 for the complements of those links;
\r\n2. Generalization of Gukov-Pei-Putrov-Vafa formula for \u1e91 for negative-definite plumbed 3-manifolds to general Lie algebra;
\r\n3. Various conjectures coming out of the interpretation of FK(x,q) in terms of topological strings, such as the HOMFLY-PT analogue (i.e., a-deformation) of FK(x,q) and the holomorphic Lagrangian generalizing the A-polynomial.
The present dissertation discusses aspects of supersymmetric quantum field theory, whose main themes are two-folded. First, we explore connections between superconformal theories in various dimensions and geometric invariants. Such correspondence arises from compactification of string theory or M-theory, which encodes geometric quantities into physical observables. Second, we study in detail the chiral rings and their quantum corrections in certain supersymmetric gauge theory. The goal is to shed some light on the hitherto mysterious electric-magnetic dualities.
\r\n\r\nWe first consider M5 brane on the product manifold L(k, 1) \u00d7 M3, where M3 = L(p, 1). Compactification on L(p, 1) gives rise to three dimensional theory T[L(p, 1)] whose partition function, according to 3d-3d correspondence, is equivalent to Chern-Simons invariants with complex gauge group on L(p, 1). We test the statement in Chapter 2 by taking k = 0 and calculating the supersymmetric index. We find a full agreement between two seemingly distinct quantities. In particular, when p = 1, we see the familiar S3 partition function of Chern-Simons theory arises from the index of a free theory.
\r\n\r\nWe then move on in Chapter 3 to consider M3 = S1 \u00d7 \u03a3, and twisted compactification on general Riemann surface \u03a3 with tame punctures. The twisted partition function of lens space theory T[L(k, 1)] on S1 \u00d7 \u03a3 computes the graded dimension of the Hilbert space after geometrically quantizing Hitchin moduli space MH, dubbed as \"tame Hitchin characters\" or \"equivariant Verlinde formula\". We show that this quantity can be computed from the \"Coulomb branch index\" of the class S theory T[\u03a3] on L(k, 1) \u00d7 S1. The gauge groups on two sides of the equivalence are naturally G and the Langlands dual group LG. We check explicitly the relation for G = SU(2) or SO(3). We also consider more general case where G is SU(N) or PSU(N) and show that the SU(N) equivariant Verlinde formula can be derived using field theory via (generalized) Argyres-Seiberg duality.
\r\n\r\nAs a further application, in Chapter 4 we use Coulomb branch indices of Argyres-Douglas theories on S1 \u00d7 L(k, 1) to quantize moduli spaces MH of wild/irregular Hitchin systems. We obtain the \"wild Hitchin characters\", and observe that the characters can always be written as a sum over fixed points in MH under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the vii modular transform STkS in certain vertex operator algebras. The appearance of vertex operator algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.
\r\n\r\nThe BPS spectrum of superconformal theories probe the geometry of Hitchin moduli space. Conversely, physical data of superconformal theories can be read off from Hitchin moduli space as well. We study this dictionary in Chapter 5 for general Argyres-Douglas theories and obtain a refined classification. We also discuss the S-duality of these theories, and find that the weakly coupled descriptions are given by the degeneration limit of auxiliary Riemann sphere with marked points.
\r\n\r\nFinally, in Chapter 6, we analyze classical and quantum chiral ring relations of four dimensional N = 1 adjoint SQCD with superpotential turned on for the adjoint field. In particular, for the mass deformed theory we obtain the complete on-shell vacuum expectation value for various gauge invariant chiral operators and find non trivial gaugino condensations. We argue that the solution of the chiral ring is in one-to-one correspondence with supersymmetric vacua, provided that an additional Konishi anomaly equation is included.
", "doi": "10.7907/K2K2-9790", "publication_date": "2018", "thesis_type": "phd", "thesis_year": "2018" }, { "id": "thesis:9896", "collection": "thesis", "collection_id": "9896", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:07282016-144651775", "primary_object_url": { "basename": "Saberi_Ingmar_2016.pdf", "content": "final", "filesize": 1992686, "license": "other", "mime_type": "application/pdf", "url": "/9896/1/Saberi_Ingmar_2016.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Knots, Trees, and Fields: Common Ground Between Physics and Mathematics", "author": [ { "family_name": "Saberi", "given_name": "Ingmar Akira", "clpid": "Saberi-Ingmar-Akira" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Spiropulu", "given_name": "Maria", "orcid": "0000-0001-8172-7081", "clpid": "Spiropulu-M" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "One main theme of this thesis is a connection between mathematical physics (in particular, the three-dimensional topological quantum field theory known as Chern-Simons theory) and three-dimensional topology. This connection arises because the partition function of Chern-Simons theory provides an invariant of three-manifolds, and the Wilson-loop observables in the theory define invariants of knots. In the first chapter, we review this connection, as well as more recent work that studies the classical limit of quantum Chern-Simons theory, leading to relations to another knot invariant known as the -polynomial. (Roughly speaking, this invariant can be thought of as the moduli space of flat connections on the knot complement.) In fact, the connection can be deepened: through an embedding into string theory, categorifications of polynomial knot invariants can be understood as spaces of BPS states.
\r\n\r\nWe go on to study these homological knot invariants, and interpret spectral sequences that relate them to one another in terms of perturbations of supersymmetric theories. Our point is more general than the application to knots; in general, when one perturbs any modulus of a supersymmetric theory and breaks a symmetry, one should expect a spectral sequence to relate the BPS states of the unperturbed and perturbed theories. We consider several diverse instances of this general lesson. In another chapter, we consider connections between supersymmetric quantum mechanics and the de Rham version of homotopy theory developed by Sullivan; this leads to a new interpretation of Sullivan's minimal models, and of Massey products as vacuum states which are entangled between different degrees of freedom in these models.
\r\n\r\nWe then turn to consider a discrete model of holography: a Gaussian lattice model defined on an infinite tree of uniform valence. Despite being discrete, the matching of bulk isometries and boundary conformal symmetries takes place as usual; the relevant group is , and all of the formulas developed for holography in the context of scalar fields on fixed backgrounds have natural analogues in this setting. The key observation underlying this generalization is that the geometry underlying AdS3/CFT2 can be understood algebraically, and the base field can therefore be changed while maintaining much of the structure. Finally, we give some analysis of -polynomials under change of base (to finite fields), bringing things full circle.
", "doi": "10.7907/Z9VX0DHZ", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:9813", "collection": "thesis", "collection_id": "9813", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05282016-191214961", "type": "thesis", "title": "3d-3d Correspondence for Seifert Manifolds", "author": [ { "family_name": "Pei", "given_name": "Du", "orcid": "0000-0001-5587-6905", "clpid": "Pei-Du" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "In this dissertation, we investigate the 3d-3d correspondence for Seifert manifolds. This correspondence, originating from string theory and M-theory, relates the dynamics of three-dimensional quantum field theories with the geometry of three-manifolds.
\r\n\r\nWe first start in Chapter II with the simplest cases and demonstrate the extremely rich interplay between geometry and physics even when the manifold is just a direct product. In this particular case, by examining the problem from various vantage points, we generalize the celebrated relations between 1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3) quantization of Chern-Simons theory and 4) the index theory of the moduli space of flat connections to a completely new set of relations between 1) the \"equivariant Verlinde algebra\" for a complex group, 2) the equivariant quantum K-theory of the vortex moduli space, 3) quantization of complex Chern-Simons theory and 4) the equivariant index theory of the moduli space of Higgs bundles.
\r\n\r\nIn Chapter III we move one step up in complexity by looking at the next simplest three-manifolds---lens spaces. We test the 3d-3d correspondence for theories that are labeled by lens spaces, reaching a full agreement between the index of the 3d N=2 \"lens space theory\" and the partition function of complex Chern-Simons theory on the lens space.
\r\n\r\nThe two different types of manifolds studied in the previous two chapters also have interesting interactions. We show in Chapter IV the equivalence between two seemingly distinct 2d TQFTs: one comes from the \"Coulomb branch index\" of the class S theory on a lens space, the other is the \"equivariant Verlinde formula\". We check this relation explicitly for SU(2) and demonstrate that the SU(N) equivariant Verlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg dualities.
\r\n\r\nIn the last chapter, we directly jump to the most general situation, giving a proposal for the 3d-3d correspondence for an arbitrary Seifert manifold. We remark on the huge class of novel dualities relating different descriptions of the \"Seifert theory\" associated with the same Seifert manifold and suggest ways that our proposal could be tested.
", "doi": "10.7907/Z9X34VF9", "publication_date": "2016", "thesis_type": "phd", "thesis_year": "2016" }, { "id": "thesis:8920", "collection": "thesis", "collection_id": "8920", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292015-095300444", "primary_object_url": { "basename": "Thesis.pdf", "content": "", "filesize": 1360403, "license": "other", "mime_type": "application/pdf", "url": "/8920/1/Thesis.pdf", "version": "v1.0.0" }, "type": "thesis", "title": "Topological Strings, Double Affine Hecke Algebras, and Exceptional Knot Homology", "author": [ { "family_name": "Elliot", "given_name": "Ross Filip", "clpid": "Elliot-Ross-Filip" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Graber", "given_name": "Thomas B.", "clpid": "Graber-T-B" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we consider two main subjects: refined, composite invariants and exceptional knot homologies of torus knots. The main technical tools are double affine Hecke algebras (\"DAHA\") and various insights from topological string theory.
\r\n\r\nIn particular, we define and study the composite DAHA-superpolynomials of torus knots, which depend on pairs of Young diagrams and generalize the composite HOMFLY-PT polynomials from the full HOMFLY-PT skein of the annulus. We also describe a rich structure of differentials that act on homological knot invariants for exceptional groups. These follow from the physics of BPS states and the adjacencies/spectra of singularities associated with Landau-Ginzburg potentials. At the end, we construct two DAHA-hyperpolynomials which are closely related to the Deligne-Gross exceptional series of root systems.
\r\n\r\nIn addition to these main themes, we also provide new results connecting DAHA-Jones polynomials to quantum torus knot invariants for Cartan types A and D, as well as the first appearance of quantum E6 knot invariants in the literature.
", "doi": "10.7907/Z9ST7MRK", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8496", "collection": "thesis", "collection_id": "8496", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06062014-152022421", "primary_object_url": { "basename": "Thesis_final version_Hee Joong Chung.pdf", "content": "final", "filesize": 883873, "license": "other", "mime_type": "application/pdf", "url": "/8496/1/Thesis_final version_Hee Joong Chung.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Three-Dimensional Superconformal Field Theory, Chern-Simons Theorv, and Their Correspondence", "author": [ { "family_name": "Chung", "given_name": "Hee Joong", "clpid": "Chung-Hee-Joong" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" }, { "family_name": "Porter", "given_name": "Frank C.", "orcid": "0000-0003-1948-8889", "clpid": "Porter-F-C" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "In this thesis, we discuss 3d-3d correspondence between Chern-Simons theory and three-dimensional N = 2 superconformal field theory. In the 3d-3d correspondence proposed by Dimofte-Gaiotto-Gukov information of abelian flat connection in Chern-Simons theory was not captured. However, considering M-theory configuration giving the 3d-3d correspondence and also other several developments, the abelian flat connection should be taken into account in 3d-3d correspondence. With help of the homological knot invariants, we construct 3d N = 2 theories on knot complement in 3-sphere for several simple knots. Previous theories obtained by Dimofte-Gaiotto-Gukov can be obtained by Higgsing of the full theories. We also discuss the importance of all flat connections in the 3d-3d correspondence by considering boundary conditions in 3d N = 2 theories and 3-manifold.", "doi": "10.7907/7CA7-9C79", "publication_date": "2014", "thesis_type": "phd", "thesis_year": "2014" }, { "id": "thesis:5808", "collection": "thesis", "collection_id": "5808", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05142010-131147918", "primary_object_url": { "basename": "DimofteTDofficial.pdf", "content": "final", "filesize": 1810018, "license": "other", "mime_type": "application/pdf", "url": "/5808/1/DimofteTDofficial.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "Refined BPS Invariants, Chern-Simons Theory, and the Quantum Dilogarithm", "author": [ { "family_name": "Dimofte", "given_name": "Tudor Dan", "clpid": "Dimofte-Tudor-Dan" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "In this thesis, we consider two main subjects: the refined BPS invariants of Calabi-Yau threefolds, and three-dimensional Chern-Simons theory with complex gauge group. We study the wall-crossing behavior of refined BPS invariants using a variety of techniques, including a four-dimensional supergravity analysis, statistical-mechanical melting crystal models, and relations to new mathematical invariants. We conjecture an equivalence between refined invariants and the motivic Donaldson-Thomas invariants of Kontsevich and Soibelman. We then consider perturbative Chern-Simons theory with complex gauge group, combining traditional and novel approaches to the theory (including a new state integral model) to obtain exact results for perturbative partition functions. We thus obtain a new class of topological invariants, which are not of finite type, defined in the background of genuinely nonabelian flat connections. The two main topics, BPS invariants and Chern-Simons theory, are connected at both a formal and (we believe) deeper conceptual level by the striking central role that the quantum dilogarithm function plays in each.", "doi": "10.7907/Q6WF-D678", "publication_date": "2010-06-11", "thesis_type": "phd", "thesis_year": "2010" } ]