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Monodromy
About: Monodromy is a research topic. Over the lifetime, 4751 publications have been published within this topic receiving 93918 citations.
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TL;DR: In this paper, the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories is studied, where WDVV equations and Frobenius manifolds are discussed.
Abstract: These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.
1,379 citations
TL;DR: In this article, a series of τ functions parametrized by integers are introduced and their ratios to the original τ function are then shown to be explicit rational expressions in terms of the coefficients of A(x).
1,083 citations
TL;DR: In this article, a general mechanism for chaotic inflation driven by monodromy-extended closed-string axions is proposed, compatible with moduli stabilization and can be realized in many types of compactifications, including warped Calabi-Yau manifolds and more general Ricci-curved spaces.
Abstract: Wrapped branes in string compactifications introduce a monodromy that extends the field range of individual closed-string axions to beyond the Planck scale. Furthermore, approximate shift symmetries of the system naturally control corrections to the axion potential. This suggests a general mechanism for chaotic inflation driven by monodromy-extended closed-string axions. We systematically analyze this possibility and show that the mechanism is compatible with moduli stabilization and can be realized in many types of compactifications, including warped Calabi-Yau manifolds and more general Ricci-curved spaces. In this broad class of models, the potential is linear in the canonical inflaton field, predicting a tensor to scalar ratio $r\ensuremath{\approx}0.07$ accessible to upcoming cosmic microwave background observations.
1,007 citations
TL;DR: In this paper, a unified treatment of monodromy and spectrum-preserving deformations is presented, in particular a general procedure is described to reduce the latter into the former consistently, and the concept of the τ-function, previously introduced for the former, is extended to the isospectral context.
906 citations
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24 Nov 1998TL;DR: In this paper, the main results of the main theorem were reformulated and reduction steps in proving the main theorems were taken in the following order: Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several variables Equidistribution Monodromy of families of curves Monodromes of some other families GUE discrepancies in various families Distribution of low-lying Frobenius eigenvalues in different families Appendix AD: Densities Appendix AG: Graphs References.
Abstract: Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants Several variables Equidistribution Monodromy of families of curves Monodromy of some other families GUE discrepancies in various families Distribution of low-lying Frobenius eigenvalues in various families Appendix AD: Densities Appendix AG: Graphs References.
838 citations