Quantum Spectral Curve Research Articles

The omega deformation from string and M-theory

Abstract We present a string theory construction of Omega-deformed four-dimensional gauge theories with generic values of ϵ1 and ϵ2. Our solution gives an explicit description of the geometry in the core of Nekrasov and Witten’s realization of the instanton partition function, far from the asymptotic region of their background. This construction lifts naturally to M-theory and corresponds to an M5-brane wrapped on a Riemann surface with a selfdual flux. Via a 9-11 flip, we finally reinterpret the Omega deformation in terms of noncommutative geometry. Our solution generates all modified couplings of the Ω-deformed gauge theory, and also yields a geometric origin for the quantum spectral curve of the associated quantum integrable system.

Loop equations and topological recursion for the arbitrary-β two-matrix model

Abstract We write the loop equations for the β two-matrix model, and we propose a topological recursion algorithm to solve them, order by order in a small parameter. We find that to leading order, the spectral curve is a “quantum” spectral curve, i.e. it is given by a differential operator (instead of an algebraic equation for the hermitian case). Here, we study the case where that quantum spectral curve is completely degenerate, it satisfies a Bethe ansatz, and the spectral curve is the Baxter TQ relation.

The Quiver Matrix Model and 2d-4d Conformal Connection

We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, on the relation between Toda field theories and a class of quiver superconformal gauge theories. On the basis of the CFT representation of the beta deformation of the model, a quantum spectral curve is introduced as << det (x- i g_s \partial \phi(z)) >>=0 at finite N and for beta \neq 1. The planar loop equation in the large N limit follows with the aid of W_n constraints. Residue analysis is provided both for the curve of the matrix model with the "multi-log" potential and for the Seiberg-Witten curve in the case of SU(n) with 2n flavors, leading to the matching of the mass parameters. The isomorphism of the two curves is made manifest.

Manin Matrices, Quantum Elliptic Commutative Families and Characteristic Polynomial of Elliptic Gaudin Model

In this paper we construct the quantum spectral curve for the quantum dynami- cal elliptic gl n Gaudin model. We realize it considering a commutative family corresponding to the Felder's elliptic quantum group E, ~(gln) and taking the appropriate limit. The approach of Manin matrices here suits well to the problem of constructing the generation function of commuting elements which plays an important role in SoV and Langlands con- cept.

Manin matrices and Talalaev's formula

In this paper we study properties of Lax and transfer matrices associated with quantum integrable systems. Our point of view stems from the fact that their elements satisfy special commutation properties, considered by Yu I Manin some 20 years ago at the beginning of quantum group theory. These are the commutation properties of matrix elements of linear homomorphisms between polynomial rings; more explicitly these read: (1) elements of the same column commute; (2) commutators of the cross terms are equal: [Mij, Mkl] = [Mkj, Mil] (e.g. [M11, M22] = [M21, M12]). The main aim of this paper is twofold: on the one hand we observe and prove that such matrices (which we call Manin matrices in short) behave almost as well as matrices with commutative elements. Namely, the theorems of linear algebra (e.g., a natural definition of the determinant, the Cayley–Hamilton theorem, the Newton identities and so on and so forth) have a straightforward counterpart in the case of Manin matrices. On the other hand, we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability. For instance, Manin matrices (and their q-analogs) include matrices satisfying the Yang–Baxter relation ‘RTT=TTR’ and the so-called Cartier–Foata matrices. Also, they enter Talalaev's remarkable formulae: for the ‘quantum spectral curve’, and appear in the separation of variables problem and Capelli identities. We show that theorems of linear algebra, after being established for such matrices, have various applications to quantum integrable systems and Lie algebras, e.g. in the construction of new generators in (and, in general, in the construction of quantum conservation laws), in the Knizhnik–Zamolodchikov equation, and in the problem of Wick ordering. We propose, in the appendix, a construction of quantum separated variables for the XXX-Heisenberg system.

Spectrophotometry of fluorescent pigments

A method is described for measuring the spectral reflexion of fluorescent pigments for incident monochromatic light covering the ultra-violet and visible regions of the spectrum. The results are recorded in the form of a square table from which quantum efficiencies and spectral reflexion curves for sunlight or any illuminant of known energy distribution are calculated. Data are given for a red pigment and an optical bleach. Three curves only, showing the variation of tristimulus values with incident wavelength, are sufficient for the calculation of colour. The curves of tristimulus values may be derived from the reflexion curves or measured directly by a template colorimeter.

Cut-and-join description of generalized Brezin–Gross–Witten model

We investigate the Brezin-Gross-Witten model, a tau-function of the KdV hierarchy, and its natural one-parameter deformation, the generalized Brezin-Gross-Witten tau-function. In particular, we derive the Virasoro constraints, which completely specify the partition function. We solve them in terms of the cut-and-join operator. The Virasoro constraints lead to the loop equations, which we solve in terms of the correlation functions. Explicit expressions for the coefficients of the tau-function and the free energy are derived, and a compact formula for the genus zero contribution is conjectured. A family of polynomial solutions of the KdV hierarchy, given by the Schur functions, is obtained for the half-integer values of the parameter. The quantum spectral curve and its classical limit are discussed.