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:16730", "collection": "thesis", "collection_id": "16730", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09162024-065933174", "type": "thesis", "title": "Spin Geometry and Quantum Diffusion", "author": [ { "family_name": "Gakkhar", "given_name": "Sitanshu", "orcid": "0000-0002-2895-9921", "clpid": "Gakkhar-Sitanshu" } ], "thesis_advisor": [ { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "\u0106a\u0107i\u0107", "given_name": "Branimir Josip", "orcid": "0000-0002-4721-778X", "clpid": "\u0106a\u0107i\u0107-Branimir-Josip" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis studies diffusion processes on spinor endomorphism algebras. The spinor and connection laplacian generated heat semigroups are shown to quantum dynamical semigroups, and after spectral truncation the existence of Evans-Hudson flows is established. The vacuum state expectation of the process is related to the spectral action principle in noncommutative geometry. Examples where the flow is proven to exist for untruncated laplacians are given. Convergence of finite dimensional approximations, through discretization and truncation, to spectral triples encoding Riemannian geometry and their statespaces as quantum metric spaces is also considered.", "doi": "10.7907/ded0-hn95", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "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 differential cohomology groups of a smooth manifold are discretized with respect to a triangulation. The realization of differential cohomology used is Deligne cohomology. A discretized version of the smooth Deligne double complex is constructed from cochain groups defined on simplices of the triangulation. The total cohomology of this double complex is studied and shown to satisfy exact sequences analogous to the standard structural sequences satisfied by differential cohomology. In the degree corresponding to line bundles with connection, our cohomology classes are shown to correspond to isomorphism classes of an existing notion of discrete line bundles with connection. Explicit examples of these discrete line bundles with connection are constructed. A ring structure is defined on the discrete Deligne cohomology groups; it is graded-commutative and non-associative (however, associativity is recovered in the continuum limit). The ring structure allows one to define a more general discrete Chern-Simons action than has previously appeared in the literature.
", "doi": "10.7907/aae8-re02", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:14659", "collection": "thesis", "collection_id": "14659", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012022-020454667", "type": "thesis", "title": "Harmonic Maps of Riemann Surfaces and Applications in Geometry", "author": [ { "family_name": "Sagman", "given_name": "Nathaniel Levi", "orcid": "0000-0002-8485-7073", "clpid": "Sagman-Nathaniel-Levi" } ], "thesis_advisor": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "thesis_committee": [ { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" }, { "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": "Chen", "given_name": "Lei", "orcid": "0000-0002-5941-7914", "clpid": "Chen-Lei" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Harmonic maps are fundamental objects in differential geometry. They play an important role in studying deformations of geometric structures and in various rigidity problems. In this thesis, we present three projects, all of which involve harmonic mappings of Riemann surfaces.
\r\n\r\nIn the first project, we study infinite energy harmonic maps and spacelike maximal surfaces in pseudo-Riemannian manifolds, and give applications to domination for surface group representations and anti-de Sitter geometry. The culminating result is the existence of a new class of anti-de Sitter 3-manifolds and a parametrization of their deformation space.
\r\n\r\nThe second project concerns moduli spaces of harmonic surfaces inside higher dimensional Riemannian manifolds. First, we prove a factorization theorem for harmonic maps. We then use infinite-dimensional transversality theory to prove results about the distribution of certain families of harmonic surfaces inside our moduli spaces.
\r\n\r\nThe final project is motivated by the Labourie conjecture from Higher Teichmueller theory. We prove non-uniqueness results for minimal surfaces in products of hyperbolic surfaces and products of \u211d-trees, and we make a connection to classical minimal surfaces.
", "doi": "10.7907/14dw-zh58", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:14149", "collection": "thesis", "collection_id": "14149", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05162021-224519644", "type": "thesis", "title": "Annular Links with sl\u2082-Irreducible Annular Khovanov Homology", "author": [ { "family_name": "Kim", "given_name": "Juhyun", "orcid": "0000-0002-8447-3758", "clpid": "Kim-Juhyun" } ], "thesis_advisor": [ { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "thesis_committee": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" }, { "family_name": "Wang", "given_name": "Lu", "clpid": "Wang-Lu" }, { "family_name": "Chen", "given_name": "Lei", "orcid": "0000-0002-5941-7914", "clpid": "Chen-Lei" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We prove that the rank of annular Khovanov homology of a braid in its next-to-top annular grading is always greater than 1, and as an immediate consequence prove that annular Khovanov homology of an annular link as a representation over the Lie algebra sl\u2082 is irreducible if and only if the annular link is isotopic to the core of the annulus. We also conjecture an analogue of Fox's trapezoid conjecture in the context of annular Khovanov homology with a computer-assisted supporting evidence.", "doi": "10.7907/rwqc-q126", "publication_date": "2021", "thesis_type": "phd", "thesis_year": "2021" }, { "id": "thesis:11576", "collection": "thesis", "collection_id": "11576", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05302019-200300602", "type": "thesis", "title": "Light and Heat: Nonlocal Aspects in Conformal Field Theories", "author": [ { "family_name": "Kolo\u011flu", "given_name": "Murat", "orcid": "0000-0002-5082-8434", "clpid": "Kolo\u011flu-Murat" } ], "thesis_advisor": [ { "family_name": "Simmons-Duffin", "given_name": "David", "orcid": "0000-0002-2937-9515", "clpid": "Simmons-Duffin-D" } ], "thesis_committee": [ { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Simmons-Duffin", "given_name": "David", "orcid": "0000-0002-2937-9515", "clpid": "Simmons-Duffin-D" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Spiropulu", "given_name": "Maria", "orcid": "0000-0001-8172-7081", "clpid": "Spiropulu-M" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis is dedicated to certain nonlocal aspects of conformal quantum field theories (CFTs). Two main directions are the study of CFTs on a particular globally-nontrivial spacetime corresponding to finite temperature, and the study of particular nonlocal CFT observables localized on light-rays. Specifically, we introduce bootstrap techniques for determining finite-temperature data of CFTs, and make novel predictions for the 2+1-dimensional Ising model. We propose the \u201cstringy equivalence principle\u201d, stating that coincident gravitational shocks commute, as a generalization of the strong equivalence principle of Einstein\u2019s General Relativity that must hold in all consistent theories of gravity. We prove it in Anti-de Sitter (AdS) spacetimes by studying light-ray operators in the holographically dual CFT. We also derive an operator product expansion (OPE) for light-ray operators in CFT, by which two light-ray operators on the same light-sheet can be expanded as a sum of single light-ray operators. Light-ray operators model detectors \u2014 such as calorimeters. We use the light-ray OPE to compute energy event shape observables suitable for conformal collider physics.
\r\n\r\nAn additional part of this thesis determines the low-energy vacua of two-dimensional maximal super-Yang-Mills theory, which describes the dynamics of stacks of D-strings in Type IIB string theory. By computing an invariant of the renormalization group (RG) flow from high to low energy \u2014 a modified thermal partition function named the refined elliptic genus \u2014 we prove the existence of multiple vacua, and identify the superconformal field theories capturing their dynamics. The vacua correspond to bound states of (p,q)-strings in Type IIB string theory. Our computation serves as a check of the strong-weak S-duality of the Type IIB string.
", "doi": "10.7907/DJ2W-AF45", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:11507", "collection": "thesis", "collection_id": "11507", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05102019-160621419", "primary_object_url": { "basename": "Okounkova_Thesis_2019.pdf", "content": "final", "filesize": 6682017, "license": "other", "mime_type": "application/pdf", "url": "/11507/1/Okounkova_Thesis_2019.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Numerical Relativity Beyond General Relativity", "author": [ { "family_name": "Okounkova", "given_name": "Maria", "orcid": "0000-0001-7869-5496", "clpid": "Okounkova-Maria" } ], "thesis_advisor": [ { "family_name": "Teukolsky", "given_name": "Saul A.", "orcid": "0000-0001-9765-4526", "clpid": "Teukolsky-S-A" } ], "thesis_committee": [ { "family_name": "Chen", "given_name": "Yanbei", "orcid": "0000-0002-9730-9463", "clpid": "Chen-Yanbei" }, { "family_name": "Scheel", "given_name": "Mark", "orcid": "0000-0001-6656-9134", "clpid": "Scheel-M-A" }, { "family_name": "Weinstein", "given_name": "Alan Jay", "orcid": "0000-0002-0928-6784", "clpid": "Weinstein-Alan-J-Physics" }, { "family_name": "Adhikari", "given_name": "Rana", "orcid": "0000-0002-5731-5076", "clpid": "Adhikari-R" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Teukolsky", "given_name": "Saul A.", "orcid": "0000-0001-9765-4526", "clpid": "Teukolsky-S-A" } ], "local_group": [ { "literal": "TAPIR" }, { "literal": "Astronomy Department" }, { "literal": "div_pma" } ], "abstract": "Einstein\u2019s theory of general relativity has passed all precision tests to date. At some length scale, however, general relativity (GR) must break down and be reconciled with quantum mechanics in a quantum theory of gravity (a beyond-GR theory). Binary black hole mergers probe the non-linear, highly dynamical regime of gravity, and gravitational waves from these systems may contain signatures of such a theory. In this thesis, we seek to make gravitational wave predictions for binary black hole mergers in a beyond-GR theory. These predictions can then be used to perform model-dependent tests of GR with gravitational wave detections.
\r\n\r\nWe make predictions using numerical relativity, the practice of precisely numerically solving the equations governing spacetime. This allows us to probe the behavior of a binary black hole system through full inspiral, merger, and ringdown. We choose to work in dynamical Chern-Simons gravity (dCS), a higher-curvature beyond-GR effective field theory that couples spacetime curvature to a scalar field, and has motivations in string theory and loop quantum gravity. In order to obtain a well-posed initial value formalism, we perturb this theory around GR. We compute the leading-order behavior of the dCS scalar field in a binary black hole merger, as well as the leading-order dCS correction to the spacetime metric and hence gravitational radiation. We produce the first numerical relativity beyond-GR waveforms in a higher-curvature theory of gravity.
\r\n\r\nThis thesis contains additional results, all of which harness the power of numerical relativity to test GR. We compute black hole shadows in dCS gravity, numerically prove the leading-order stability of rotating black holes in dCS gravity, and lay out a formalism for determining the start time of binary black hole ringdown using information from the strong-field region of a binary black hole simulation.
\r\n", "doi": "10.7907/WPAT-PS25", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:11723", "collection": "thesis", "collection_id": "11723", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06072019-210423914", "type": "thesis", "title": "Quantum Groups and Integralities in Chern-Simons Theory", "author": [ { "family_name": "Chun", "given_name": "Sungbong", "orcid": "0000-0003-3594-2690", "clpid": "Chun-Sungbong" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Wise", "given_name": "Mark B.", "orcid": "0000-0002-9125-801X", "clpid": "Wise-M-B" }, { "family_name": "Ooguri", "given_name": "Hirosi", "orcid": "0000-0001-6021-3778", "clpid": "Ooguri-H" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this dissertation, we investigate integralities in Chern-Simons theory. The integralities of interest arise from non-local observables (Wilson lines) in Chern-Simons theory and the partition function itself. In the associated supersymmetric gauge theories (via 3d-3d correspondence), they encode certain BPS spectrum, which are often identified with homological invariants of links and three-manifolds. In this dissertation, we observe that all of them are equipped with non-trivial algebraic structures, such as quantum group actions, modularity, and logarithmic vertex algebras. In the first half of this dissertation, we identify quantum group representations with the dynamics of line operators and their lift to surface operators. In the second half, Chern-Simons partition functions on Seifert manifolds are studied in detail, and its ``hidden'' integralities are identified with quantum modular forms and the characters of logarithmic vertex operator algebra. From the latter, we also observe that quantum group actions control the ``dynamics'' of characters.", "doi": "10.7907/2CW7-5J29", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" }, { "id": "thesis:10844", "collection": "thesis", "collection_id": "10844", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05032018-153255038", "primary_object_url": { "basename": "Ye_Ke_2018_thesis.pdf", "content": "final", "filesize": 5040684, "license": "other", "mime_type": "application/pdf", "url": "/10844/1/Ye_Ke_2018_thesis.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "A Symphony of Supersymmetry and Geometry: Invariants, Dualities and Chiral Rings", "author": [ { "family_name": "Ye", "given_name": "Ke", "orcid": "0000-0002-2978-2013", "clpid": "Ye-Ke" } ], "thesis_advisor": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "thesis_committee": [ { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "family_name": "Wise", "given_name": "Mark B.", "orcid": "0000-0002-9125-801X", "clpid": "Wise-M-B" }, { "family_name": "Intriligator", "given_name": "Kenneth A.", "orcid": "0000-0002-8045-1869", "clpid": "Intriligator-Kenneth-A" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Nakayama", "given_name": "Yu", "orcid": "0000-0002-1747-5147", "clpid": "Nakayama-Yu" } ], "local_group": [ { "literal": "Walter Burke Institute for Theoretical Physics" }, { "literal": "div_pma" } ], "abstract": "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:8059", "collection": "thesis", "collection_id": "8059", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:01242014-104001899", "primary_object_url": { "basename": "chan_y_park_2013_thesis.pdf", "content": "final", "filesize": 13395553, "license": "other", "mime_type": "application/pdf", "url": "/8059/1/chan_y_park_2013_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Branes and Supersymmetric Quantum Field Theories", "author": [ { "family_name": "Park", "given_name": "Chan Youn", "clpid": "Park-Chan-Youn" } ], "thesis_advisor": [ { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" } ], "thesis_committee": [ { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" }, { "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": "Porter", "given_name": "Frank C.", "clpid": "Porter-F-C" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.
\r\n\r\nOne example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don\u2019t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.
\r\n\r\nFirst, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.
", "doi": "10.7907/Y1VH-E821", "publication_date": "2014", "thesis_type": "phd", "thesis_year": "2014" }, { "id": "thesis:7208", "collection": "thesis", "collection_id": "7208", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:09192012-150137728", "primary_object_url": { "basename": "Setter_thesis.pdf", "content": "final", "filesize": 1327269, "license": "other", "mime_type": "application/pdf", "url": "/7208/1/Setter_thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Topological Quantum Field Theory and the Geometric Langlands Correspondence", "author": [ { "family_name": "Setter", "given_name": "Kevin Luke", "clpid": "Setter-Kevin-Luke" } ], "thesis_advisor": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" } ], "thesis_committee": [ { "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" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Wise", "given_name": "Mark B.", "orcid": "0000-0002-9125-801X", "clpid": "Wise-M-B" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "In the pioneering work of A. Kapustin and E. Witten, the geometric Langlands program of number theory was shown to be intimately related to duality of GL-twisted N=4 super Yang-Mills theory compactified on a Riemann surface. In this thesis, we generalize Kapustin-Witten by investigating compactification of the GL-twisted theory to three dimensions on a circle (for various values of the twisting parameter t). By considering boundary conditions in the three-dimensional description, we classify codimension-two surface operators of the GL-twisted theory, generalizing those surface operators studied by S. Gukov and E. Witten. For t=i, we propose a complete description of the 2-category of surface operators in terms of module categories, and, in addition, we determine the monoidal category of line operators which includes Wilson lines as special objects. For t=1 and t=0, we discuss surface and line operators in the abelian case.
\r\n\r\nWe generalize Kapustin-Witten also by analyzing a separate twisted version of N=4, the Vafa-Witten theory. After introducing a new four-dimensional topological gauge theory, the gauged 4d A-model, we locate the Vafa-Witten theory as a special case. Compactification of the Vafa-Witten theory on a circle and on a Riemann surface is discussed. Several novel two- and three-dimensional topological gauge theories are studied throughout the thesis and in the appendices.
\r\n\r\nIn work unrelated to the main thread of the thesis, we conclude by classifying codimension-one topological defects in two-dimensional sigma models with various amounts of supersymmetry.
", "doi": "10.7907/RK2P-2H81", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:6387", "collection": "thesis", "collection_id": "6387", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05122011-105136289", "primary_object_url": { "basename": "main_doc.pdf", "content": "final", "filesize": 2175352, "license": "other", "mime_type": "application/pdf", "url": "/6387/1/main_doc.pdf", "version": "v5.0.0" }, "type": "thesis", "title": "Integrability of N = 6 Chern-Simons theory", "author": [ { "family_name": "Lipstein", "given_name": "Arthur Elias", "clpid": "Lipstein-Arthur-Elias" } ], "thesis_advisor": [ { "family_name": "Schwarz", "given_name": "John H.", "clpid": "Schwarz-J-H" } ], "thesis_committee": [ { "family_name": "Schwarz", "given_name": "John H.", "clpid": "Schwarz-J-H" }, { "family_name": "Gukov", "given_name": "Sergei", "clpid": "Gukov-S" }, { "family_name": "Porter", "given_name": "Frank C.", "clpid": "Porter-F-C" }, { "family_name": "Wise", "given_name": "Mark B.", "clpid": "Wise-M-B" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "In 2008, Aharony, Bergman, Jafferis, and Maldacena (ABJM) discovered a three-dimensional Chern-Simons theory with N = 6 supersymmetry and conjectured that in a certain limit, this theory is dual to type IIA string theory on AdS4xCP3. Since then, a great deal of evidence has been accumulated which suggests that the ABJM theory is integrable in the planar limit. Integrability is a very useful property that allows many physical observables, such as anomalous dimensions and scattering amplitudes, to be computed efficiently. In the first half of this thesis, we will explain how to use integrabilty to compute the anomalous dimensions of long, single-trace operators in the ABJM theory. In particular, we will describe how to compute them at weak coupling using a Bethe Ansatz, and how to compute them at strong coupling using string theory. The latter approach involves using algebraic curve and world-sheet techniques to compute the energies of string states dual to gauge theory operators. In the second half of this thesis, we will discuss integrability from the point of view of on-shell scattering amplitudes in the ABJM theory. In particular, we will describe how to parameterize the amplitudes in terms of supertwistors and how to relate higher-point tree-level amplitudes to lower-point tree-level amplitudes using a recursion relation. We will also explain how this recursion relation can be used to show that all tree-level amplitudes of the ABJM theory are invariant under dual superconformal symmetry. This symmetry is hidden from the point of the action and implies that the theory has Yangian symmetry, which is a key feature of integrability. This thesis is mainly based on the material in [94], [76], and [77].", "doi": "10.7907/QDSM-8448", "publication_date": "2011", "thesis_type": "phd", "thesis_year": "2011" }, { "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" }, { "id": "thesis:2495", "collection": "thesis", "collection_id": "2495", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06072006-174745", "primary_object_url": { "basename": "thesis.pdf", "content": "final", "filesize": 838808, "license": "other", "mime_type": "application/pdf", "url": "/2495/1/thesis.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "On Quantum Interacting Embedded Geometrical Objects of Various Dimensions", "author": [ { "family_name": "Evnin", "given_name": "Oleg", "orcid": "0000-0001-5999-778X", "clpid": "Evnin-Oleg" } ], "thesis_advisor": [ { "family_name": "Gubser", "given_name": "Steven Scott", "orcid": "0000-0001-5214-1786", "clpid": "Gubser-S-S" }, { "family_name": "Schwarz", "given_name": "John H.", "orcid": "0000-0001-9861-7559", "clpid": "Schwarz-J-H" } ], "thesis_committee": [ { "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": "Preskill", "given_name": "John P.", "orcid": "0000-0002-2421-4762", "clpid": "Preskill-J" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "local_group": [ { "literal": "Caltech Theory" }, { "literal": "div_pma" } ], "abstract": "Modern string theory naturally gives rise to an assortment of dynamical geometrical objects of various dimensions (collectively referred to as \"branes\") embedded into spacetime. The aim of this thesis is to present a series of results (of varying novelty and rigor) pertinent to dynamics of the low-dimensional geometrical objects of this kind. The processes considered are the D0-brane recoil and annihilation, \"local recoil\" of D1-branes (which is a peculiar effect manifested by one-dimensional topological defects in response to an impact, and closely related to soliton recoil), and D- and F-string loop mixing. Apart from the practical relevance within the formalism of string theory, such considerations are worthwhile in that the quantum dynamics of the geometrical objects involved is complex enough to be interesting, yet simple enough to be tractable. Furthermore, some of the results derived here within the string theory formalism may give valuable insights into the dynamics of low-dimensional field-theoretical topological defects.", "doi": "10.7907/1VN2-VZ71", "publication_date": "2006", "thesis_type": "phd", "thesis_year": "2006" } ]