Categorification in geometry, topology, and physics
The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorification is a powerful tool for relating va...
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| Hlavní autori: | , |
|---|---|
| Médium: | E-kniha Kniha |
| Jazyk: | English |
| Vydavateľské údaje: |
Providence, R.I
American Mathematical Society
2017
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| Vydanie: | 1 |
| Edícia: | Contemporary mathematics |
| Predmet: | |
| ISBN: | 1470428210, 9781470428211 |
| On-line prístup: | Získať plný text |
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- Cover -- Title page -- Contents -- Preface -- Geometry and categorification -- 1. Introduction -- 2. K-theory -- 3. The function-sheaf correspondence -- 4. Symplectic resolutions -- References -- A geometric realization of modified quantum algebras -- 1. Introduction -- 2. Preliminary, I -- 3. Preliminary, II -- 4. Convolution product -- 5. Defining relation -- 6. Algebra \KK_{ } -- 7. Relation with the work [Zh08] -- 8. BLM case -- References -- The cube and the Burnside category -- 1. Introduction -- 2. The cube -- 3. The Burnside category -- 4. Functors from the cube to B -- 5. Properties of such functors -- 6. The Khovanov functor -- 7. Spaces -- 8. Some questions -- Acknowledgments -- References -- Junctions of surface operators and categorification of quantum groups -- 1. Introduction -- 2. Junctions of Wilson lines and quantum groups -- 2.1. Junctions of line operators -- 2.2. Web relations -- 2.3. Skew Howe duality and the quantum group -- 2.4. Why "categorification = surface operators" -- 3. Junctions of surface operators -- 3.1. Junctions in 4d \CN=4 theory -- 3.2. Line-changing operators in class \CS and network cobordisms -- 3.3. Junctions in 4d \CN=2 theory -- 3.4. Junctions in 4d \CN=1 theory -- 3.5. OPE of surface operators and the Horn problem -- 3.6. OPE and Schubert calculus -- Domain walls in 4d \CN=2 SQCD -- 4. Categorification and the Landau-Ginzburg perspective -- 4.1. Physics perspectives on categorification -- 4.2. LG theory on "time × knot" -- 4.3. Junctions and LG interfaces -- 4.4. Junctions and matrix factorizations -- 4.5. Junctions and categorification of quantum groups -- 5. What's next? -- Acknowledgments -- Appendix A. Wilson lines and categories Web -- Appendix B. Domain walls, junctions and Grassmannians -- Appendix C. LG Interfaces and the cohomology of Grassmannians
- 9.2. Colored/iterated examples -- 9.3. Generalized twisting -- 9.4. Some examples -- 9.5. Toward the Skein -- Appendix A. Links and splice diagrams -- A.1. Links, cables and splices -- A.2. Splice diagrams -- A.3. Operations on links -- A.4. Equivalent diagrams -- A.5. Connection with DAHA -- References -- Back Cover
- Appendix D. LG Interfaces and 2-categories Foam -- D.1. Derivation of the bubble relation -- D.2. Quantum group relations -- References -- Khovanov-Rozansky homology and 2-braid groups -- 1. Introduction -- 2. Notations -- 3. 2-braid groups -- 4. Hochschild cohomology and traces -- 5. Proofs -- References -- DAHA approach to iterated torus links -- 0. Introduction -- 0.1. Overview -- 0.2. Brief history -- 1. Double Hecke algebras -- 1.1. Affine root systems -- 1.2. Definition of DAHA -- 1.3. The automorphisms -- 1.4. Macdonald polynomials -- 1.5. Evaluation formula -- 2. Integral forms -- 2.1. A preliminary version -- 2.2. The integrality theorem -- 2.3. Using the duality -- 2.4. Concluding the proof -- 2.5. Affine exponents -- 2.6. -polynomials in type -- 3. Topological vertex -- 3.1. Theta-functions -- 3.2. Mehta-Macdonald identities -- 3.3. Norm-formulas -- 3.4. The case of _{ } -- 3.5. High-level 3j-symbols -- 3.6. The coinvariant approach -- 4. DAHA-Jones theory -- 4.1. Iterated torus knots -- 4.2. From knots to links -- 4.3. Splice diagrams -- 4.4. DAHA-Jones polynomials -- 4.5. The polynomiality -- 4.6. Major symmetries -- 5. DAHA-superpolynomials -- 5.1. Main theorem -- 5.2. Sketch of the proof -- 5.3. Super-vertex -- 5.4. HOMFLY-PT polynomials -- 5.5. Connection Conjecture -- 6. Multiple torus knots -- 6.1. Preliminary remarks -- 6.2. Uncolored 2-fold trefoil -- 6.3. Similar links -- 6.4. Uncolored 2-fold (4,3) -- 6.5. Uncolored 2-fold (5,3) -- 6.6. Uncolored 3-links -- 7. Hopf links -- 7.1. Basic constructions -- 7.2. Colored Hopf 3-links -- 7.3. Hopf 2-links -- 7.4. Algebraic Hopf links -- 7.5. Twisted (2 , ) -- 8. Further examples -- 8.1. On cable notations -- 8.2. Colored 2-links -- 8.3. 3-hook for trefoil -- 8.4. Alexander polynomials -- 8.5. Two different paths -- 9. Generalized twisted union -- 9.1. Uncolored trefoil-prime