[ { "id": "authors:evfd1-z5c19", "collection": "authors", "collection_id": "evfd1-z5c19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170615-083539862", "type": "book_section", "title": "Fivebranes and 4-Manifolds", "book_title": "Arbeitstagung Bonn 2013", "author": [ { "family_name": "Gadde", "given_name": "Abhijit", "clpid": "Gadde-A" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Putrov", "given_name": "Pavel", "clpid": "Putrov-P" } ], "contributor": [ { "family_name": "Ballmann", "given_name": "Werner", "clpid": "Ballmann-W" }, { "family_name": "Blohmann", "given_name": "Christian", "clpid": "Blohmann-C" }, { "family_name": "Faltins", "given_name": "Gerd", "clpid": "Faltins-G" }, { "family_name": "Teichner", "given_name": "Peter", "clpid": "Teichner-P" }, { "family_name": "Zagier", "given_name": "Don", "clpid": "Zagier-D" } ], "abstract": "We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2)N=(0,2) theories, we obtain a number of results, which include new 3d N=2N=2 theories T[M 3] associated with rational homology spheres and new results for Vafa\u2013Witten partition functions on 4-manifolds. In particular, we point out that the gluing measure for the latter is precisely the superconformal index of 2d (0,\u20092) vector multiplet and relate the basic building blocks with coset branching functions. We also offer a new look at the fusion of defect lines/walls, and a physical interpretation of the 4d and 3d Kirby calculus as dualities of 2d N=(0,2)N=(0,2) theories and 3d N=2N=2 theories, respectively.", "doi": "10.1007/978-3-319-43648-7_7", "isbn": "978-3-319-43646-3", "publisher": "Springer", "place_of_publication": "New York, NY", "publication_date": "2016-11-13", "pages": "155-245" }, { "id": "authors:qnq5a-eh607", "collection": "authors", "collection_id": "qnq5a-eh607", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160503-081755071", "type": "book_section", "title": "Lectures on Knot Homology and Quantum Curves", "book_title": "Physics and Mathematics of Link Homology", "author": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Saberi", "given_name": "Ingmar", "clpid": "Saberi-Ingmar-A" } ], "contributor": [ { "family_name": "Gukov", "given_name": "Sergei", "clpid": "Gukov-S" }, { "family_name": "Khovanov", "given_name": "Mikhail", "clpid": "Khovanov-M" }, { "family_name": "Walcher", "given_name": "Johannes", "clpid": "Walcher-J" } ], "abstract": "Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this interpretation allows one to pose questions that would not have been asked otherwise, such as, \"Is there a direct relation between Khovanov homology and the A-polynomial of a knot?\" We will explain that the answer to this question is \"yes,\" and introduce a certain deformation of the planar algebraic curve defined by the zero locus of the A-polynomial. This novel deformation leads to a categorified version of the Generalized Volume Conjecture that completely describes the \"color behavior\" of the colored sl(2) knot homology, and eventually to a similar version for the colored HOMFLY homology. Furthermore, this deformation is strong enough to distinguish mutants, and its most interesting properties include relations to knot contact homology and knot Floer homology.", "doi": "10.1090/conm/680/13700", "isbn": "978-1-4704-1459-7", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2016", "pages": "59-97" }, { "id": "authors:th675-h6y10", "collection": "authors", "collection_id": "th675-h6y10", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150114-205456566", "type": "book_section", "title": "Surface Operators", "book_title": "New Dualities of Supersymmetric Gauge Theories", "author": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "contributor": [ { "family_name": "Teschner", "given_name": "J\u00f6rg", "clpid": "Teschner-J" } ], "abstract": "We give an introduction and a broad survey of surface operators in 4d gauge theories, with a particular emphasis on aspects relevant to AGT correspondence. One of the main goals is to highlight the boundary between what we know and what we don't know about surface operators. To this end, the survey contains many open questions and suggests various directions for future research. Although this article is mostly a review, we did include a number of new results, previously unpublished.", "doi": "10.1007/978-3-319-18769-3_8", "isbn": "978-3-319-18768-6", "publisher": "Springer", "place_of_publication": "Cham, Switzerland", "publication_date": "2015-11-18", "pages": "223-259" }, { "id": "authors:zmh6w-mv829", "collection": "authors", "collection_id": "zmh6w-mv829", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170706-082515229", "type": "book_section", "title": "A-Polynomial, B-Model, and Quantization", "book_title": "Homological Mirror Symmetry and Tropical Geometry", "author": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Su\u0142kowski", "given_name": "Piotr", "orcid": "0000-0002-6176-6240", "clpid": "Su\u0142kowski-Piotr" } ], "contributor": [ { "family_name": "Castano-Bernard", "given_name": "Ricardo", "clpid": "Castano-Bernard-R" }, { "family_name": "Catanese", "given_name": "Fabrizio", "clpid": "Catanese-F" }, { "family_name": "Kontsevich", "given_name": "Maxim", "clpid": "Kontsevich-M" }, { "family_name": "Pantev", "given_name": "Tony", "clpid": "Pantev-T" }, { "family_name": "Soibelman", "given_name": "Yan", "clpid": "Soibelman-Y" }, { "family_name": "Zharkov", "given_name": "Ilia", "clpid": "Zharkov-I" } ], "abstract": "Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly in a suitable semi-classical limit, as \u210f\u21920, and becomes non-commutative or \"quantum\" away from this limit. For a classical curve defined by the zero locus of a polynomial A(x,\u2009y), we provide a construction of its non-commutative counterpart A^(x^,y^) using the technique of the topological recursion. This leads to a powerful and systematic algorithm for computing A^ that, surprisingly, turns out to be much simpler than any of the existent methods. In particular, as a bonus feature of our approach comes a curious observation that, for all curves that \"come from geometry,\" their non-commutative counterparts can be determined just from the first few steps of the topological recursion. We also propose a K-theory criterion for a curve to be \"quantizable,\" and then apply our construction to many examples that come from applications to knots, strings, instantons, and random matrices. The material contained in this chapter was presented at the conference Mirror Symmetry and Tropical Geometry in Cetraro (July 2011) and is based on the work: Gukov and Su\u0142kowski, \"A-polynomial, B-model, and quantization\", JHEP 1202 (2012) 070.", "doi": "10.1007/978-3-319-06514-4_4", "isbn": "978-3-319-06513-7", "publisher": "Springer", "place_of_publication": "Cham, Switzerland", "publication_date": "2014-06-24", "pages": "87-151" }, { "id": "authors:f16jr-nw068", "collection": "authors", "collection_id": "f16jr-nw068", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130212-111358154", "type": "book_section", "title": "Homological algebra of knots and BPS states", "book_title": "String-Math 2011", "author": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" }, { "family_name": "Sto\u0161i\u0107", "given_name": "Marko", "clpid": "Sto\u0161i\u0107-M" } ], "contributor": [ { "family_name": "Block", "given_name": "Jonathan", "clpid": "Block-J" }, { "family_name": "Distler", "given_name": "Jacques", "clpid": "Distler-J" }, { "family_name": "Donagi", "given_name": "Ron", "clpid": "Donagi-R" }, { "family_name": "Sharpe", "given_name": "Eric", "clpid": "Sharpe-E" } ], "abstract": "It is known that knot homologies admit a physical description as spaces of open BPS states. We study operators and algebras acting on these spaces. This leads to a very rich story, which involves wall crossing\nphenomena, algebras of closed BPS states acting on spaces of open BPS states, and deformations of Landau-Ginzburg models.\nOne important application to knot homologies is the existence of \"colored differentials\" that relate homological invariants of knots colored by different\nrepresentations. Based on this structure, we formulate a list of properties of the colored HOMFLY homology that categorifies the colored HOMFLY polynomial. By calculating the colored HOMFLY homology for symmetric and anti-symmetric representations, we find a remarkable \"mirror symmetry\" between these triply-graded theories.", "doi": "10.1090/pspum/085", "isbn": "9780821872956", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2012", "pages": "125-171" }, { "id": "authors:csrxr-6vg78", "collection": "authors", "collection_id": "csrxr-6vg78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110707-110110717", "type": "book_section", "title": "Quantum Field Theory and the Volume Conjecture", "book_title": "Interactions between hyperbolic geometry, quantum topology, and number theory", "author": [ { "family_name": "Dimofte", "given_name": "Tudor", "clpid": "Dimofte-T-D" }, { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "contributor": [ { "family_name": "Champanerkar", "given_name": "A.", "clpid": "Champanerkar-A" }, { "family_name": "Dasbach", "given_name": "O.", "clpid": "Dasbach-O" }, { "family_name": "Kalfagianni", "given_name": "E.", "clpid": "Kalfagianni-E" }, { "family_name": "Kofman", "given_name": "I.", "clpid": "Kofman-I" }, { "family_name": "Neumann", "given_name": "W.", "clpid": "Neumann-W" }, { "family_name": "Stoltzfus", "given_name": "N.", "clpid": "Stoltzfus-N" } ], "abstract": "The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary three-manifolds.", "doi": "10.1090/conm/541/10678", "isbn": "978-0-8218-4960-6", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2011", "pages": "41-67" }, { "id": "authors:z6j4s-kxr53", "collection": "authors", "collection_id": "z6j4s-kxr53", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160505-120947331", "type": "book_section", "title": "Surface Operators and Knot Homologies", "book_title": "New Trends in Mathematical Physics", "author": [ { "family_name": "Gukov", "given_name": "Sergei", "orcid": "0000-0002-9486-1762", "clpid": "Gukov-S" } ], "contributor": [ { "family_name": "Sidoravi\u010dius", "given_name": "V.", "clpid": "Sidoravi\u010dius-V" } ], "abstract": "Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N=2 and N=4 twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant.", "doi": "10.1007/978-90-481-2810-5_22", "isbn": "978-90-481-2809-9", "publisher": "Springer Netherlands", "place_of_publication": "Berlin Heidelberg", "publication_date": "2009", "pages": "313-343" } ]