arXiv subjects

Nikita Nekrasov

Publications and source records attributed to Nikita Nekrasov.

At least 19 recordsLinked to original sources

arXiv2026

Fluid dynamics as intersection problem

We formulate the equations of fluid dynamics as an intersection-theoretic problem on an infinite-dimensional symplectic manifold naturally associated with spacetime. This perspective separates the structures determined by the equation of state and the spacetime geometry from the differential-topological data of spacetime. It leads to a geometric derivation of the covariant formulation of hydrodynamics due to Lichnerowicz and Carter, clarifies the role of the canonical velocity and hydrodynamic invariants, including the asymptotic Hopf invariant and the Ertel invariant, and yields a generalized Kelvin circulation theorem. We also explain the relation between the canonical velocity, the four-velocity, and other choices of hydrodynamic frame. In addition, we identify a five-dimensional geometric origin of the formalism underlying covariant hydrodynamics. The formalism extends naturally to fluids with additional degrees of freedom, including multicomponent fluids, charged fluids, and superfluids, and incorporates the chiral anomaly and Onsager quantization. It also suggests a possible bridge between hydrodynamics, Poisson sigma models, and topological field theories. We further argue that the same intersection-theoretic viewpoint applies to self-dual fields, including chiral bosons in 1+1 dimensions, tensor fields of the (2,0) theory in 1+5 dimensions, and the self-dual four-form field of type-IIB supergravity in 1+9 dimensions.

hep-th↗

arXiv2026

Vershik-Kerov in higher times

Several generalizations of Vershik-Kerov limit shape problem are motivated by topological string theory and supersymmetric gauge theory instanton count. In this paper specifically we study the circular and linear quiver theories. We also briefly discuss the double-elliptic generalization of the Vershik-Kerov problem, related to six dimensional gauge theory compactified on a torus, and to elliptic cohomology of the Hilbert scheme of points on a plane. We prove that the limit shape in that setting is governed by a genus two algebraic curve, suggesting unexpected dualities between the enumerative and equivariant parameters.

hep-th↗

arXiv2026

Lectures on Gauge theories and Many-Body systems

These lectures study two correspondences between gauge theories and integrable many-body systems. The first arises from infinite-dimensional Hamiltonian reduction and relates gauge-theoretic dynamics directly to Calogero--Moser-type systems and their quantum counterparts. The second emerges in supersymmetric gauge theory through instanton counting and non-perturbative dualities, linking classical problems on one side to quantum problems on the other. A central motivation comes from the observation that conjugacy classes of holonomies in gauge theory can be interpreted as configurations of indistinguishable particles on a circle. In quantum theory these particle positions become random variables, and the correspondence may be either exact or approximate depending on spacetime dimension and supersymmetry. We focus on the Calogero--Moser--Sutherland family associated with root systems of type A and with SU(N) gauge theories in dimensions from one to six. In low dimensions the correspondence is direct and involves matrix quantum mechanics, two-dimensional Yang--Mills theory, and higher-dimensional Chern--Simons-type theories. In four, five, and six dimensions with eight supercharges, the correspondence takes a more indirect form through supersymmetric gauge theory and the Omega-deformation. We introduce non-local observables whose correlation functions satisfy non-perturbative Dyson--Schwinger equations and, in four dimensions, lead to Schrodinger equations for many-body systems. The notes are divided accordingly: the first part develops symplectic-reduction constructions of Calogero--Moser systems, while the second studies localization, partition measures, and order/disorder observables in supersymmetric gauge theory.

math-ph↗

arXiv2026

Parallel surface defects, Hecke operators, and quantum Hitchin system

We examine two types of half-BPS surface defects $-$ regular monodromy surface defect and canonical surface defect $-$ in four-dimensional gauge theory with $\mathcal{N}=2$ supersymmetry and $Ω_{\varepsilon_1,\varepsilon_2}$-background. Mathematically, we investigate integrals over the moduli spaces of parabolic framed sheaves over $\mathbb{P}^2$. Using analytic methods of $\mathcal{N}=2$ theories, we demonstrate that the former gives a twisted $\mathcal{D}$-module on $\text{Bun}_{G_{\mathbb{C}}}$ while the latter acts as a Hecke operator. In the limit $\varepsilon_2 \to 0$, the cluster decomposition implies the Hecke eigensheaf property for the regular monodromy surface defect. The eigenvalues are given by the opers associated to the canonical surface defect. We derive, in our $\mathcal{N}=2$ gauge theoretical framework, that the twisted $\mathcal{D}$-modules assigned to the opers in the geometric Langlands correspondence represent the spectral equations for quantum Hitchin integrable system. A duality to topologically twisted four-dimensional $\mathcal{N}=4$ theory is discussed, in which the two surface defects are mapped to Dirichlet boundary and 't Hooft line defect. This is consistent with earlier works on the $\mathcal{N}=4$ theory approach to the geometric Langlands correspondence.

hep-th↗

arXiv2025

Global magni$4$icence, or: 4G Networks

The global magnificent four theory is the homological version of a maximally supersymmetric $(8+1)$-dimensional gauge theory on a Calabi-Yau fourfold fibered over a circle. In the case of a toric fourfold we conjecture the formula for its twisted Witten index. String-theoretically we count the BPS states of a system of $D0$-$D2$-$D4$-$D6$-$D8$-branes on the Calabi-Yau fourfold in the presence of a large Neveu-Schwarz $B$-field. Mathematically, we develop the equivariant $K$-theoretic DT4 theory, by constructing the four-valent vertex with generic plane partition asymptotics. Physically, the vertex is a supersymmetric localization of a non-commutative gauge theory in $8+1$ dimensions.

hep-th↗

arXiv2025

Yang-Mills flows for multilayered graphene

We clarify the origin of magic angles in twisted multilayered graphene using Yang-Mills flows in two dimensions. We relate the effective Hamiltonian describing the electrons in the multilayered graphene to the ${\bar\partial}_{A}$ operator on a two dimensional torus coupled to an $SU(N)$ gauge field. Despite the absence of a characteristic class such as $c_{1}$ relevant for the quantum Hall effect, we show that there are topological invariants associated with the zero modes occuring in a family of Hamiltonians. The flatbands in the spectrum of the effective Hamiltonian are associated with Yang-Mills connections, studied by M.Atiyah and R.Bott long time ago. The emergent $U(1)$ magnetic field with nonzero flux is presumably responsible for the observed Hall effect in the absence of (external) magnetic field. We provide a numeric algorithm transforming the original single-particle Hamiltonian to the direct sum of ${\bar\partial}_{A}$ operators coupled to abelian gauge fields with non-zero $c_1$'s. Our perspective gives a simple bound for magic angles: if the gauge field $A(α)$ is such that the YM energy $\Vert F_{A(α)} \Vert^2$ is smaller than that of $U(1)$ magnetic flux embedded into $SU(2)$, then $α$ is not magic.

hep-th↗

arXiv2025

Noncommutative Jacobi identity, and gauge theory

We prove the noncommutative analogue of Jacobi triple product identity. As an application we organizing the q-characters of circular quiver gauge theories into an infinite product. We conjecture the gauge origami theory interpretation of the Jacobi identity.

math-ph↗

arXiv2025

di-Langlands correspondence and extended observables

We explore the $\textit{difference Langlands correspondence}$ using the four dimensional ${\mathcal{N}}=2$ super-QCD. Surface defects and surface observables play the crucial role. As an application, we give the first construction of the full set of quantum integrals, i.e. commuting differential operators, such that the partition function of the so-called regular monodromy surface defect is their joint eigenvectors in an evaluation module over the Yangian $Y(\mathfrak{gl}(2))$, making it the wavefunction of a $N$-site $\mathfrak{gl}(2)$ spin chain with bi-infinite spin modules. We construct the $\mathbf{Q}$- and $\tilde{\mathbf{Q}}$-surface observables which are believed to be the $Q$-operators on the bi-infinite module over the Yangian $Y(\mathfrak{gl}(2))$, and compute their eigenvalues, the $Q$-functions, as vevs of the surface observables.

hep-th↗

arXiv2024

A Panorama Of Physical Mathematics c. 2022

What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forward-looking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

hep-th↗

arXiv2024

Elliptic analogue of Vershik-Kerov limit shape

We review the limit shape problem for the Plancherel measure and its generalizations found in supersymmetric gauge theory instanton count. We focus on the measure, interpolating between the Plancherel measure and uniform measure, a U(1) case of N=2* gauge theory. We give the formula for its limit shape in terms of elliptic functions, generalizing the trigonometric ``arcsin'' law of Vershik-Kerov and Logan-Schepp.

math-ph↗

arXiv2023

Femtomolar detection of the heart failure biomarker NT-proBNP in artificial saliva using an immersible liquid-gated aptasensor with reduced graphene oxide

Measuring NT-proBNP biomarker is recommended for preliminary diagnostics of the heart failure. Recent studies suggest a possibility of early screening of biomarkers in saliva for non-invasive identification of cardiac diseases at the point-of-care. However, NT-proBNP concentrations in saliva can be thousand time lower than in blood plasma, going down to pg/mL level. To reach this level, we developed a label-free aptasensor based on a liquid-gated field effect transistor using a film of reduced graphene oxide monolayer (rGO-FET) with immobilized NT-proBNP specific aptamer. We found that, depending on ionic strength of tested solutions, there were different levels of correlation in responses of electrical parameters of the rGO-FET aptasensor, namely, the Dirac point shift and transconductance change. The correlation in response to NT-proBNP was high for 1.6 mM phosphate-buffered saline (PBS) and zero for 16 mM PBS in a wide range of analyte concentrations, varied from 1 fg/mL to 10 ng/mL. The effects of transconductance and Dirac point shift in PBS solutions of different concentrations are discussed. The biosensor exhibited a high sensitivity for both transconductance (2 uS/decade) and Dirac point shift (2.3 mV/decade) in diluted PBS with the linear range from 10 fg/mL to 1 pg/mL. The aptasensor performance has been also demonstrated in undiluted artificial saliva with the achieved limit of detection down to 41 fg/mL (~4.6 fM).

cond-mat.mtrl-sci↗

arXiv2023

Quantum Spin Systems and Supersymmetric Gauge Theories, I

The relation between supersymmetric gauge theories in four dimensions and quantum spin systems is exploited to find an explicit formula for the Jost function of the $N$ site $\mathfrak{sl}_{2}$ $XXX$ spin chain (for infinite dimensional complex spin representations), as well as the $SL_N$ Gaudin system, which reduces, in a limiting case, to that of the $N$-particle periodic Toda chain. Using the non-perturbative Dyson-Schwinger equations of the supersymmetric gauge theory we establish relations between the spin chain commuting Hamiltonians with the twisted chiral ring of gauge theory. Along the way we explore the chamber dependence of the supersymmetric partition function, also the expectation value of the surface defects, giving new evidence for the AGT conjecture.

hep-th↗

arXiv2023

Elliptic Calogero-Moser system, crossed and folded instantons, and bilinear identities

Affine analogues of the Q-functions are constructed using folded instantons partition functions. They are shown to be the solutions of the quantum spectral curve of the N-body elliptic Calogero-Moser (eCM) system, the quantum Krichever curve. They also solve the elliptic analogue of the quantum Wronskian equation. In the companion paper we present the quantum analogue of Krichever's Lax operator for eCM. A connection to crossed instantons on Taub-Nut spaces, and opers on a punctured torus is pointed out.

math-ph↗

arXiv2023

Analytic continuation and supersymmetry

Real-valued parameters of quantum field theory, such as Planck constant, coupling constants, temperature and spacetime metric, chemical potentials or background gauge fields can be made complex. In perturbative string theory, the worldsheet metric is sometime Lorentzian, somewhere Euclidean, and likely complex in between. Less often, integral, or quantized, parameters, such as the number of flavors or species (as in replica method), the number of colors (as in matrix models, master field), spacetime dimension (dimensional regularization), or angular momentum (Regge poles) are continued to complex values. We observe that many, if not all of these analytic continuations can be realized with conventional supersymmetric quantum field (or M-) theory. The supersymmetry is softly broken near a defect supporting the theory being analytically continued in some of its parameters.

hep-th↗

arXiv2023

Seiberg-Witten Geometry of Four-Dimensional $\mathcal N=2$ Quiver Gauge Theories

Seiberg-Witten geometry of mass deformed $\mathcal N=2$ superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space $\mathfrak M$ of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space ${\rm Bun}_{\mathbf G} (\mathcal E)$ of holomorphic $G^{\mathbb C}$-bundles on a (possibly degenerate) elliptic curve $\mathcal E$ defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group $G$. The integrable systems $\mathfrak P$ underlying the special geometry of $\mathfrak M$ are identified. The moduli spaces of framed $G$-instantons on ${\mathbb R}^{2} \times {\mathbb T}^{2}$, of $G$-monopoles with singularities on ${\mathbb R}^{2} \times {\mathbb S}^{1}$, the Hitchin systems on curves with punctures, as well as various spin chains play an important rôle in our story. We also comment on the higher-dimensional theories.

hep-th↗

arXiv2023

Interfaces and Quantum Algebras, II: Cigar Partition Function

The supersymmetric cigar (half-)index or cigar partition function of 3d $\mathcal{N}=2$ gauge theories contains a wealth of information. Physically, it captures the spectrum of BPS states, the non-perturbative corrections to various partition functions, the effective twisted superpotential and the data of supersymmetric vacua. Mathematically, it defines the K-theoretic Vertex counting vortices/quasimaps, and connects to quantum K-theory, as well as elliptic cohomology and stable envelopes. We explore these topics from the physics standpoint, systematically developing the foundations and explaining various mathematical properties using the quantum field theory machinery.

hep-th↗

arXiv2023

Intersecting defects in gauge theory, quantum spin chains, and Knizhnik-Zamolodchikov equations

We propose an interesting BPS/CFT correspondence playground: the correlation function of two intersecting half-BPS surface defects in four-dimensional $\mathcal{N}=2$ supersymmetric $SU(N)$ gauge theory with $2N$ fundamental hypermultiplets. We show it satisfies a difference equation, the fractional quantum T-Q relation. Its Fourier transform is the $5$-point conformal block of the $\widehat{\mathfrak{sl}}_N$ current algebra with one of the vertex operators corresponding to the $N$-dimensional $\mathfrak{sl}_N$ representation, which we demonstrate with the help of the Knizhnik-Zamolodchikov equation. We also identify the correlator with a state of the $XXX_{\mathfrak{sl}_2}$ spin chain of $N$ Heisenberg-Weyl modules over $Y(\mathfrak{sl}_2)$. We discuss the associated quantum Lax operators, and connections to isomonodromic deformations.

hep-th↗

arXiv2023

Interfaces and Quantum Algebras, I: Stable Envelopes

The stable envelopes of Okounkov et al. realize some representations of quantum algebras associated to quivers, using geometry. We relate these geometric considerations to quantum field theory. The main ingredients are the supersymmetric interfaces in gauge theories with four supercharges, relation of supersymmetric vacua to generalized cohomology theories, and Berry connections. We mainly consider softly broken compactified three dimensional $\mathcal{N} =4$ theories. The companion papers will discuss applications of this construction to symplectic duality, Bethe/gauge correspondence, generalizations to higher dimensional theories, and other topics.

hep-th↗