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28
Matrix Model as a Mirror of ChernSimons Theory
, 2002
"... Using mirror symmetry, we show that ChernSimons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of ChernSimons theory. ..."
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Cited by 134 (17 self)
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Using mirror symmetry, we show that ChernSimons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of ChernSimons theory. Moreover, large N dualities in this context lead to computation of all genus Amodel topological amplitudes on toric CalabiYau manifolds in terms of matrix integrals. In the context of type IIA superstring compactifications on these CalabiYau manifolds with wrapped D6 branes (which are dual to Mtheory on G2 manifolds) this leads to engineering and solving Fterms for N = 1 supersymmetric gauge theories with superpotentials involving certain multitrace operators
The WittenReshetikhinTuraev invariants of finite order mapping tori II
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ASYMPTOTICS OF THE QUANTUM INVARIANTS FOR SURGERIES ON THE FIGURE 8 KNOT
, 2006
"... We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a for ..."
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Cited by 15 (3 self)
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We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2, C)representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern–Simons invariants of the corresponding flat SU(2)connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure 8 knot following Kashaev [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].
Analytic asymptotic expansions of the Reshetikhin–Turaev invariants of Seifert 3–manifolds for SU(2)
, 2005
"... We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic fo ..."
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Cited by 7 (3 self)
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We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic formula thereby proving the socalled asymptotic expansion conjecture (AEC) due to J. E. Andersen [An1], [An2] for these manifolds. For the case of Seifert manifolds with base S 2 we actually prove a little weaker result, namely that the asymptotic formula has a form as predicted by the AEC but contains some extra terms which should be zero according to the AEC. We prove that these ‘extra ’ terms are indeed zero if the number of exceptional fibers n is less than 4 and conjecture that this is also the case if n≥4. For the case of Seifert fibered rational homology spheres we identify the Casson–Walker invariant in the asymptotic formula. Our calculations demonstrate a general method for calculating the large r asymptotics of a finite sum Σ r k=1f(k), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to Rozansky [Ro1], [Ro3], is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.
Knot state asymptotics II Witten conjecture and irreducible representations
"... This article pursues the study of the knot state asymptotics in the large level limit initiated in [CM11]. As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of ChernSimons topological qua ..."
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Cited by 4 (2 self)
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This article pursues the study of the knot state asymptotics in the large level limit initiated in [CM11]. As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of ChernSimons topological quantum field theory as a holomorphic section on the SU2character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the ChernSimons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from qdifferential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know. 1
QUANTUM INVARIANTS, MODULAR FORMS, AND LATTICE POINTS II
, 2006
"... ABSTRACT. We study the SU(2) Witten–Reshetikhin–Turaev invariant for the Seifert fibered homology spheres with Mexceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 1/2 and 3/2. By use of nearly modular pr ..."
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Cited by 2 (0 self)
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ABSTRACT. We study the SU(2) Witten–Reshetikhin–Turaev invariant for the Seifert fibered homology spheres with Mexceptional fibers. We show that the WRT invariant can be written in terms of (differential of) the Eichler integrals of modular forms with weight 1/2 and 3/2. By use of nearly modular property of the Eichler integrals we shall obtain asymptotic expansions of the WRT invariant in the largeN limit. We further reveal that the number of the gauge equivalent classes of flat connections, which dominate the asymptotics of the WRT invariant in N→∞, is related to the number of integral lattice points inside the Mdimensional tetrahedron. 1.
CYCLOTOMY AND ENDOMOTIVES
, 901
"... Abstract. We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the BostConnes (BC) quantum statistical mechanical system and the second on the Habiro ..."
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Cited by 2 (0 self)
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Abstract. We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the BostConnes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expansions. Both have compatible endomorphisms actions of the multiplicative semigroup of positive integers. As a higher dimensional generalization, we consider a crossed product ring obtained using Manin’s multivariable generalizations of the Habiro functions and an action by endomorphisms of the semigroup of integer matrices with positive determinant. We then construct a corresponding class of multivariable BC endomotives, which are obtained geometrically from self maps of higher dimensional algebraic tori, and we discuss some of their quantum statistical mechanical properties. These multivariable BC endomotives are universal for (torsion free) Λrings, compatibly with the Frobenius action. Finally, we discuss briefly how Habiro’s universal Witten–Reshetikhin–Turaev invariant of integral homology 3spheres may relate invariants of 3manifolds to gadgets over F1 and semigroup actions on homology 3spheres to endomotives. 1.
CHERNSIMONS THEORY ON L(p, q) LENS SPACES AND GOPAKUMARVAFA DUALITY
, 809
"... Abstract. We consider aspects of ChernSimons theory on L(p, q) lens spaces and its relation with matrix models and topological string theory on CalabiYau threefolds, searching for possible new large N dualities via geometric transition for nonSU(2) cyclic quotients of the conifold. To this aim we ..."
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Abstract. We consider aspects of ChernSimons theory on L(p, q) lens spaces and its relation with matrix models and topological string theory on CalabiYau threefolds, searching for possible new large N dualities via geometric transition for nonSU(2) cyclic quotients of the conifold. To this aim we find, on one hand, some novel matrix integral representations of the SU(N) CS partition function in a generic flat background for the whole L(p, q) family and provide a solution for its large N dynamics; on the other, we perform in full detail the construction of a family of wouldbe dual closed string backgrounds via conifold geometric transition from T ∗ L(p, q). We can then explicitly prove the claim in [2] that GopakumarVafa duality in a fixed vacuum fails in the case q> 1, and briefly discuss how it could be restored in a nonperturbative setting. 1. Overview After the seminal work of Witten [21], ChernSimons (CS) theory has been deeply studied both in Mathematics and Physics. A most attractive property of