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تسجيل مستخدم جديدSingular vector structure of quantum curves
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We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.
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In modern mathematical and theoretical physics various generalizations, in particular supersymmetric or quantum, of Riemann surfaces and complex algebraic curves play a prominent role. We show that such supersymmetric and quantum generalizations can
be combined together, and construct supersymmetric quantum curves, or super-quantum curves for short. Our analysis is conducted in the formalism of super-eigenvalue models: we introduce $beta$-deformed version of those models, and derive differential equations for associated $alpha/beta$-deformed super-matrix integrals. We show that for a given model there exists an infinite number of such differential equations, which we identify as super-quantum curves, and which are in one-to-one correspondence with, and have the structure of, super-Virasoro singular vectors. We discuss potential applications of super-quantum curves and prospects of other generalizations.
As we have shown in the previous work, using the formalism of matrix and eigenvalue models, to a given classical algebraic curve one can associate an infinite family of quantum curves, which are in one-to-one correspondence with singular vectors of a
certain (e.g. Virasoro or super-Virasoro) underlying algebra. In this paper we reformulate this problem in the language of conformal field theory. Such a reformulation has several advantages: it leads to the identification of quantum curves more efficiently, it proves in full generality that they indeed have the structure of singular vectors, it enables identification of corresponding eigenvalue models. Moreover, this approach can be easily generalized to other underlying algebras. To illustrate these statements we apply the conformal field theory formalism to the case of the Ramond version of the super-Virasoro algebra. We derive two classes of corresponding Ramond super-eigenvalue models, construct Ramond super-quantum curves that have the structure of relevant singular vectors, and identify underlying Ramond super-spectral curves. We also analyze Ramond multi-Penner models and show that they lead to supersymmetric generalizations of BPZ equations.
Relativistic arbitrary spin Hamiltonians are shown to obey the algebraic structure of supersymmetric quantum system if their odd and even parts commute. This condition is identical to that required for the exactness of the Foldy-Wouthuysen transforma
tion. Applied to a massive charged spin-$1$ particle in a constant magnetic field, supersymmetric quantum mechanics necessarily requires a gyromagnetic factor $g=2$.
The relativistic quantum dynamics of scalar bosons in the background of a full vector coupling (minimal plus nonminimal vector couplings) is explored in the context of the Duffin-Kemmer-Petiau formalism. The Coulomb phase shift is determined for a ge
neral mixing of couplings and it is shown that the space component of the nonminimal coupling is a {it sine qua non} condition for the exact closed-form scattering amplitude. It follows that the Rutherford cross section vanishes in the absence of the time component of the minimal coupling. Bound-state solutions obtained from the poles of the partial scattering amplitude show that the time component of the minimal coupling plays an essential role. The bound-state solutions depend on the nonminimal coupling and the spectrum consists of particles or antiparticles depending on the sign of the time component of the minimal coupling without chance for pair production even in the presence of strong couplings. It is also shown that an accidental degeneracy appears for a particular mixing of couplings.
We show that, when a non-integrable rational map changes to an integrable one continuously, a large part of the Julia set of the map approach indeterminate points (IDP) of the map along algebraic curves. We will see that the IDPs are singular loci of the curves.
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