A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
Develops spectral triples for quantum projective space.
We compute Stokes matrices and monodromy for the quantum cohomology of projective spaces. We prove that the Stokes' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves.
Quantum Kirwan maps between K-theories of G-varieties and GIT quotients.
The abstract discusses financial irreversibility using quantum mechanics and projective geometry.
Explains quantum cohomology of Grassmannians using tt* equations.
Paper solves quantum differential equations for projective bundles using Borel multitransforms.
Researchers create projective representations of Hecke groups using TQFT.
Quantum methods model uncertain volatility in financial markets.
A quantum system can be entirely described by the Kähler structure of the projective space P(H) associated to the Hilbert space H of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional qua…
We give a direct calculation of the curvature of the Hitchin connection, in geometric quantization on a symplectic manifold, using only differential geometric techniques. In particular, we establish that the curvature acts as a first-order operator on the quantum spaces. Projective flatness follows if the Kähler struct…
This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.
We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum…
We show that using the family of adapted Kähler polarizations of the phase space of a compact, simply connected, Riemannian symmetric space of rank-1, the obtained field of quantum Hilbert spaces produced by geometric quantization including the half-form correction is flat if is the 3-dimensional sphere …
We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invarian…
Generalizes quantum integrability to all signatures for projectively equivalent metrics.
Quantum models improve reinforcement learning in complex spaces.
SQS uses quantum kernels to improve credit scoring with fewer data points.
Quantum cohomology gives a finite dimensional integrable system via the Dubrovin connection. Motivated by Givental's work on mirror symmetry, we use gauge theory techniques and the Frobenius Integrability Theorem to find flat sections for the Dubrovin connection. An explicit calculation is given for projective space.
Restricts quantum representations of mapping class groups to integral coefficients.
New braid group action defined on projective quantum sl(2) modules.
We define convex projective structures on 2D surfaces with holes and investigate their moduli space. We prove that this moduli space is canonically identified with the higher Teichmuller space for the group PSL_3 defined in our paper math/0311149. We define the quantum version of the moduli space of convex projective s…
Paper studies Frobenius manifolds and quantum differential equations, proving Dubrovin Conjecture for Hirzebruch surfaces.
Geodesic flows on Kähler manifolds are quantum integrable when metrics are c-projectively equivalent.
We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing…
MPE framework proves universal approximation for quantum data distribution.
Quantum machine learning for 2D classification tasks using optimized feature maps.
New framework for quantum invariants of 3-manifolds using homology.
Unified framework combines trace-induced quantum kernels for improved machine learning models.
Study shows quantum behavior near infinity in metric asymptotics.
Develops a new geometric framework for quantum metrics.
Wigner's theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner's theorem: a holomorph…
We establish various results on the large level limit of projective quantum representations of surface mapping class groups obtained by quantizing moduli spaces of flat SU(n)-bundle. Working with the metaplectic correction, we proved that these projective representations lift to asymptotic representations. We show that…
Holomorphic structures on quantum flag manifolds uniquely defined.
In this paper, we will provide a review of the geometric construction, proposed by Witten, of the SU(n) quantum representations of the mapping class groups which are part of the Reshetikhin-Turaev TQFT for the quantum group U_q(sl(n, C)). In particular, we recall the differential geometric construction of Hitchin's pro…
Projective simulation converges to optimal behavior in Markov decision processes.
We introduce the notion of a "state function" for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states fo…
By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This all…
A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized …
In this paper we propose a geometrization of the non-relativistic quantum mechanics for mixed states. Our geometric approach makes use of the Uhlmann's principal fibre bundle to describe the space of mixed states and as a novelty tool, to define a dynamic-dependent metric tensor on the principal manifold, such that the…
We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-defi…
Quantum ML predicts data with improved speed and accuracy.
We sketch our recent application of a non-commutative version of the Cartan `moving-frame' formalism to the quantum Euclidean space , the space which is covariant under the action of the quantum group . For each of the two covariant differential calculi over based on the -matrix formalism, we…
In the quantum Teichmuller theory, based on Penner coordinates, the mapping class groups of punctured surfaces are represented projectively. The case of a genus three surface with one puncture is worked out explicitly. The projective factor is calculated. It is given by the exponential of the Liouville central charge.
Quantum theory uses modular group representations to assign invariants to 3-manifolds.
FSPA bypasses eigenvalue estimation for quantum PCA, achieving optimal complexity and robustness.
Quantum kernels can be efficiently embedded into classical feature spaces.
We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional and including a Lagrangian formalism in which self-adjoint (Schroedinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we …
In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpret…