172 research outputs found
Extended Seiberg-Witten Theory and Integrable Hierarchy
The prepotential of the effective N=2 super-Yang-Mills theory perturbed in
the ultraviolet by the descendents of the single-trace chiral operators is
shown to be a particular tau-function of the quasiclassical Toda hierarchy. In
the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental
hypermultiplets at the appropriate locus of the moduli space of vacua) or a
theory on a single fractional D3 brane at the ADE singularity the hierarchy is
the dispersionless Toda chain. We present its explicit solutions. Our results
generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support
the prior work hep-th/0302191 which established the equivalence of these N=2
theories with the topological A string on CP^1 and clarify the origin of the
Eguchi-Yang matrix integral. In the higher rank case we find an appropriate
variant of the quasiclassical tau-function, show how the Seiberg-Witten curve
is deformed by Toda flows, and fix the contact term ambiguity.Comment: 49 page
Clustering and the Three-Point Function
We develop analytical methods for computing the structure constant for three
heavy operators, starting from the recently proposed hexagon approach. Such a
structure constant is a semiclassical object, with the scale set by the inverse
length of the operators playing the role of the Planck constant. We reformulate
the hexagon expansion in terms of multiple contour integrals and recast it as a
sum over clusters generated by the residues of the measure of integration. We
test the method on two examples. First, we compute the asymptotic three-point
function of heavy fields at any coupling and show the result in the
semiclassical limit matches both the string theory computation at strong
coupling and the tree-level results obtained before. Second, in the case of one
non-BPS and two BPS operators at strong coupling we sum up all wrapping
corrections associated with the opposite bridge to the non-trivial operator, or
the "bottom" mirror channel. We also give an alternative interpretation of the
results in terms of a gas of fermions and show that they can be expressed
compactly as an operator-valued super-determinant.Comment: 52 pages + a few appendices; v2 typos correcte
On the Direct Construction of MDS and Near-MDS Matrices
The optimal branch number of MDS matrices makes them a preferred choice for
designing diffusion layers in many block ciphers and hash functions.
Consequently, various methods have been proposed for designing MDS matrices,
including search and direct methods. While exhaustive search is suitable for
small order MDS matrices, direct constructions are preferred for larger orders
due to the vast search space involved. In the literature, there has been
extensive research on the direct construction of MDS matrices using both
recursive and nonrecursive methods. On the other hand, in lightweight
cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a
better balance between security and efficiency as a diffusion layer compared to
MDS matrices. However, no direct construction method is available in the
literature for constructing recursive NMDS matrices. This paper introduces some
direct constructions of NMDS matrices in both nonrecursive and recursive
settings. Additionally, it presents some direct constructions of nonrecursive
MDS matrices from the generalized Vandermonde matrices. We propose a method for
constructing involutory MDS and NMDS matrices using generalized Vandermonde
matrices. Furthermore, we prove some folklore results that are used in the
literature related to the NMDS code
Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
We compare finite rank perturbations of the following three ensembles of
complex rectangular random matrices: First, a generalised Wishart ensemble with
one random and two fixed correlation matrices introduced by Borodin and
P\'ech\'e, second, the product of two independent random matrices where one has
correlated entries, and third, the case when the two random matrices become
also coupled through a fixed matrix. The singular value statistics of all three
ensembles is shown to be determinantal and we derive double contour integral
representations for their respective kernels. Three different kernels are found
in the limit of infinite matrix dimension at the origin of the spectrum. They
depend on finite rank perturbations of the correlation and coupling matrices
and are shown to be integrable. The first kernel (I) is found for two
independent matrices from the second, and two weakly coupled matrices from the
third ensemble. It generalises the Meijer -kernel for two independent and
uncorrelated matrices. The third kernel (III) is obtained for the generalised
Wishart ensemble and for two strongly coupled matrices. It further generalises
the perturbed Bessel kernel of Desrosiers and Forrester. Finally, kernel (II),
found for the ensemble of two coupled matrices, provides an interpolation
between the kernels (I) and (III), generalising previous findings of part of
the authors.Comment: 39 pages, 4 figures; v2: 43 pages, presentation of Thm 1.4 improved,
alternative proof of Prop 3.1 and reference added; v3: final typo
corrections, to appear in AIHP Probabilite et Statistiqu
Commuting Quantum Matrix Models
We study a quantum system of commuting matrices and find that such a
quantum system requires an explicit curvature dependent potential in its
Lagrangian for the system to have a finite energy ground state. In contrast it
is possible to avoid such curvature dependence in the Hamiltonian. We study the
eigenvalue distribution for such systems in the large matrix size limit. A
critical r\^ole is played by . For the competition between
eigenvalue repulsion and the attractive potential forces the eigenvalues to
form a sharp spherical shell.Comment: 17 page
't Hooft Operators in Gauge Theory from Toda CFT
We construct loop operators in two dimensional Toda CFT and calculate with
them the exact expectation value of certain supersymmetric 't Hooft and dyonic
loop operators in four dimensional \Ncal=2 gauge theories with SU(N) gauge
group. Explicit formulae for 't Hooft and dyonic operators in \Ncal=2^* and
\Ncal=2 conformal SQCD with SU(N) gauge group are presented. We also briefly
speculate on the Toda CFT realization of arbitrary loop operators in these
gauge theories in terms of topological web operators in Toda CFT.Comment: 49 pages, LaTeX. Typos fixed, references adde
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