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Register a new userTopological quantum field theory and polynomial identities for graphs on the torus
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Added by
Vyacheslav Krushkal
Publication date
2019
fields
Physics
and research's language is
English
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We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.
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We prove that the topological locally flat slice genus of large torus knots takes up less than three quarters of the ordinary genus. As an application, we derive the best possible linear estimate of the topological slice genus for torus knots with non-maximal signature invariant.
We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoretic framework allowing its application to other situations.
We derive generalizations of McShanes identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.
We discuss some basic aspects of quantum fields on star graphs, focusing on boundary conditions, symmetries and scale invariance in particular. We investigate the four-fermion bulk interaction in detail. Using bosonization and vertex operators, we solve the model exactly for scale invariant boundary conditions, formulated in terms of the fermion current and without dissipation. The critical points are classified and determined explicitly. These results are applied for deriving the charge and spin transport, which have interesting physical features.
We show that no torus knot of type $(2,n)$, $n>3$ odd, can be obtained from a polynomial embedding $t mapsto (f(t), g(t), h(t))$ where $(deg(f),deg(g))leq (3,n+1) $. Eventually, we give explicit examples with minimal lexicographic degree.
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