arXiv subjects

Neetu

Publications and source records attributed to Neetu.

8 recordsLinked to original sources

arXiv2026

On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II

Recently, Mohan et al. [Results Math. 80 (2025), No. 4, 122] answered Freiman's $3k-4$ conjecture in right-ordered groups under certain restrictions. In this paper, we take a step further by investigating the structure of nonempty subsets $S$ of a right-ordered group satisfying the small doubling condition $|S^2| = 3|S|-3$. Moreover, we provide a complete characterization of all nonempty finite subsets $S$ of the Baumslag-Solitar group $\mathrm{BS}(1,q)$ (with $q \in \mathbb{Z}$ and $q \neq -1$) for which $|S^2| = 3|S|-3$ and the identity element is the minimum of $S$.

math.NT↗

arXiv2026

Exact and Asymptotic Counts of MSTD, MDTS, and Balanced Sets in Dicyclic Groups

We investigate the relationship between the sizes of the sum and difference sets of the Dicyclic Group $\mathrm{Dic}_{4n}$. We first determine the exact numbers of MSTD (more sums than differences), MDTS (more differences than sums), and balanced subsets of size two. As a consequence, we show that the numbers of MSTD and balanced subsets of size two are asymptotically equal as $n \to \infty$. For odd $n$, we then obtain exact counts of MSTD, MDTS, and balanced subsets of size three, with the results depending on whether $n$ is divisible by $3$. In this case, we establish that asymptotically the number of MSTD subsets of size three is six times the number of MDTS subsets and also six times the number of balanced subsets. Finally, we establish a lower bound for the number of MSTD, MDTS, and balanced subsets of $\mathrm{Dic}_{4n}$ corresponding to the boundary case of size $2n$.

math.GM↗

arXiv2021

A Matrix Model with BMS$_3$ Constraints

Our goal is to find a matrix model with $BMS_3$ constraints built in. These constraints are imposed through Loop equations. We solve them using a free field realisation of the algebra and write down the partition function in eigenvalue form. We comment on the nature of this partition function and it's relations with integrable systems.

hep-th↗

arXiv2021

A Unitary Matrix Model for $q$-deformed Plancherel Growth

In this paper we construct a unitary matrix model that captures the asymptotic growth of Young diagrams under $q$-deformed Plancherel measure. The matrix model is a $q$ analog of Gross-Witten-Wadia (GWW) matrix model. In the large $N$ limit the model exhibits a third order phase transition between no-gap and gapped phases, which is a $q$-deformed version of the GWW phase transition. We show that the no-gap phase of this matrix model captures the asymptotic growth of Young diagrams equipped with $q$-deformed Plancherel measure. The no-gap solutions also satisfies a differential equation which is the $q$-analogue of the automodel equation. We further provide a droplet description for these growing Young diagrams. Quantising these droplets we identify the Young diagrams with coherent states in the Hilbert space. We also elaborate the connection between moments of Young diagrams and the infinite number of commuting Hamiltonians obtained from the large $N$ droplets and explicitly compute the moments for asymptotic Young diagrams.

hep-th↗

arXiv2020

From 2d Droplets to 2d Yang-Mills

We establish a connection between time evolution of free Fermi droplets and partition function of \emph{generalised} \emph{q}-deformed Yang-Mills theories on Riemann surfaces. Classical phases of $(0+1)$ dimensional unitary matrix models can be characterised by free Fermi droplets in two dimensions. We quantise these droplets and find that the modes satisfy an abelian Kac-Moody algebra. The Hilbert spaces $\mathcal{H}_+$ and $\mathcal{H}_-$ associated with the upper and lower free Fermi surfaces of a droplet admit a Young diagram basis in which the phase space Hamiltonian is diagonal with eigenvalue, in the large $N$ limit, equal to the quadratic Casimir of $u(N)$. We establish an exact mapping between states in $\mathcal{H}_\pm$ and geometries of droplets. In particular, coherent states in $\mathcal{H}_\pm$ correspond to classical deformation of upper and lower Fermi surfaces. We prove that correlation between two coherent states in $\mathcal{H}_\pm$ is equal to the chiral and anti-chiral partition function of $2d$ Yang-Mills theory on a cylinder. Using the fact that the full Hilbert space $\mathcal{H}_+ \otimes \mathcal{H}_-$ admits a \emph{composite} basis, we show that correlation between two classical droplet geometries is equal to the full $U(N)$ Yang-Mills partition function on cylinder. We further establish a connection between higher point correlators in $\mathcal{H}_\pm$ and higher point correlators in $2d$ Yang-Mills on Riemann surface. There are special states in $\mathcal{H}_\pm$ whose transition amplitudes are equal to the partition function of $2d$ \emph{q}-deformed Yang-Mills and in general character expansion of Villain action. We emphasise that the \emph{q}-deformation in the Yang-Mills side is related to special deformation of droplet geometries without deforming the gauge group associated with the matrix model.

hep-th↗

arXiv2019

Quantum Mechanics of Plancherel Growth

Growth of Young diagrams, equipped with Plancherel measure, follows the automodel equation of Kerov. Using the technology of unitary matrix model we show that such growth process is exactly same as the growth of gap-less phase in Gross-Witten and Wadia (GWW) model. The limit shape of asymptotic Young diagrams corresponds to GWW transition point. Our analysis also offers an alternate proof of limit shape theorem of Vershik-Kerov and Logan-Shepp. Using the connection between unitary matrix model and free Fermi droplet description, we map the Young diagrams in automodel class to different shapes of two dimensional phase space droplets. Quantising these droplets we further set up a correspondence between automodel diagrams and coherent states in the Hilbert space. Thus growth of Young diagrams are mapped to evolution of coherent states in the Hilbert space. Gaussian fluctuations of large $N$ Young diagrams are also mapped to quantum (large $N$) fluctuations of the coherent states.

hep-th↗

arXiv2019

New N=2 SuperBMS$_3$ algebra and Invariant Dual Theory for 3D Supergravity

We have constructed a two dimensional theory dual to 3D asymptotically flat Supergravity in presence of two supercharges with(out) internal $R-$symmetry. The duals in both the cases are identified with chiral Wess-Zumino-Witten models. Further gauging the theories, we show that the dual of the one without $R-$symmetry is invariant under the well known ${\cal{N}}=2$ SuperBMS$_3$ introduced in Banerjee et.al. 1609.09210 , while for the one with $R-$symmetry, the dual is invariant under the most generic, so far unknown, quantum ${\cal{N}}=2$ SuperBMS$_3$ symmetry. We have also commented on the phase space description of the duals.

hep-th↗

arXiv2019

Chern-Simons Theory on Seifert Manifold and Matrix Model

Chern-Simons (CS) theories with rank $N$ and level $k$ on Seifert manifold are discussed. The partition functions of such theories can be written as a function of modular transformation matrices summed over different integrable representations of affine Lie algebra $u(N)_k$ associated with boundary Wess-Zumino-Witten (WZW) model. Using properties of modular transform matrices we express the partition functions of these theories as a unitary matrix model. We show that, the eigenvalues of unitary matrices are discrete and proportional to hook lengths of the corresponding integrable Young diagram. As a result, in the large $N$ limit, the eigenvalue density develops an upper cap. We consider CS theory on $S^2\times S^1$ coupled with fundamental matters and express the partition functions in terms of modular transformation matrices. Solving this model at large $N$ we find the dominant integrable representations and show how large $N$ representations are related to each other by transposition of Young diagrams as a result of level rank duality. Next we consider $U(N)$ CS theory on $S^3$ and observed that in Seifert framing the dominant representation is no longer an integrable representation after a critical value of 't Hooft coupling. We also show that CS on $S^3$ admits multiple (two-gap phase) large $N$ phases with the same free energy.

hep-th↗