Random Unitary Representations of Surface Groups I: Asymptotic Expansions

In this paper, we study random representations of fundamental groups of surfaces into special unitary groups. The random model we use is based on a symplectic form on moduli space due to Atiyah, Bott and Goldman. Let $$\Sigma _{g}$$ Σ g denote a topological surface of genus $$g\ge 2$$ g ≥ 2 . We establish the existence of a large n asymptotic expansion, to any fixed order, for the expected value of the trace of any fixed element of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) under a random representation of $$\pi _{1}(\Sigma _{g})$$ π 1 ( Σ g ) into $$\mathsf {SU}(n)$$ SU ( n ) . Each such expected value involves a contribution from all irreducible representations of $$\mathsf {SU}(n)$$ SU ( n ) . The main technical contribution of the paper is effective analytic control of the entire contribution from irreducible representations outside finite sets of carefully chosen rational families of representations.

Quaternionic Grassmannians and Borel classes in algebraic geometry

The quaternionic Grassmannian H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) is the affine open subscheme of the usual Grassmannian parametrizing those 2 r 2r -dimensional subspaces of a 2 n 2n -dimensional symplectic vector space on which the symplectic form is nondegenerate. In particular, we have HP n = H Gr ⁡ ( 1 , n + 1 ) \operatorname {HP}^n = \operatorname {H Gr}(1,n+1) . For a symplectically oriented cohomology theory A A , including oriented theories but also the Hermitian K \operatorname {K} -theory, Witt groups, and algebraic symplectic cobordism, we have A ( HP n ) = A ( pt ) [ p ] / ( p n + 1 ) A(\operatorname {HP}^n) = A(\operatorname {pt})[p]/(p^{n+1}) . Borel classes for symplectic bundles are introduced in the paper. They satisfy the splitting principle and the Cartan sum formula, and they are used to calculate the cohomology of quaternionic Grassmannians. In a symplectically oriented theory the Thom classes of rank 2 2 symplectic bundles determine Thom and Borel classes for all symplectic bundles, and the symplectic Thom classes can be recovered from the Borel classes. The cell structure of the H Gr ⁡ ( r , n ) \operatorname {H Gr}(r,n) exists in cohomology, but it is difficult to see more than part of it geometrically. An exception is HP n \operatorname {HP}^n where the cell of codimension 2 i 2i is a quasi-affine quotient of A 4 n − 2 i + 1 \mathbb {A}^{4n-2i+1} by a nonlinear action of G a \mathbb {G}_a .

p-form surface charges on AdS: renormalization and conservation

Surface charges of a p-form theory on the boundary of an AdSd+1 spacetime are computed. Counter-terms on the boundary produce divergent corner-terms which holographically renormalize the symplectic form. Different choices of boundary conditions lead to various expressions for the charges and the associated fluxes. With the usual standard AdS boundary conditions, there are conserved zero-mode charges. Moreover, we explore two leaky boundary conditions which admit an infinite number of charges forming an Abelian algebra and non-vanishing flux. Finally, we discuss magnetic p-form charges and electric/magnetic duality.

Tau-Functions and Monodromy Symplectomorphisms

We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical r-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock–Goncharov coordinates are log-canonical for the symplectic form. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the Jimbo–Miwa–Ueno tau-function as the generating function of the monodromy map. This, in particular, solves a recent conjecture by A. Its, O. Lisovyy and A. Prokhorov.

The Tropical Symplectic Grassmannian

We launch the study of the tropicalization of the symplectic Grassmannian, that is, the space of all linear subspaces isotropic with respect to a fixed symplectic form. We formulate tropical analogues of several equivalent characterizations of the symplectic Grassmannian and determine all implications between them. In the process, we show that the Plücker and symplectic relations form a tropical basis if and only if the rank is at most 2. We provide plenty of examples that show that several features of the symplectic Grassmannian do not hold after tropicalizing. We show exactly when do conormal fans of matroids satisfy these characterizations, as well as doing the same for a valuated generalization. Finally, we propose several directions to extend the study of the tropical symplectic Grassmannian.

THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

3D gravity in a box

The quantization of pure 3D gravity with Dirichlet boundary conditions on a finite boundary is of interest both as a model of quantum gravity in which one can compute quantities which are ``more local" than S-matrices or asymptotic boundary correlators, and for its proposed holographic duality to T\overline{T}TT¯-deformed CFTs. In this work we apply covariant phase space methods to deduce the Poisson bracket algebra of boundary observables. The result is a one-parameter nonlinear deformation of the usual Virasoro algebra of asymptotically AdS_33 gravity. This algebra should be obeyed by the stress tensor in any T\overline{T}TT¯-deformed holographic CFT. We next initiate quantization of this system within the general framework of coadjoint orbits, obtaining — in perturbation theory — a deformed version of the Alekseev-Shatashvili symplectic form and its associated geometric action. The resulting energy spectrum is consistent with the expected spectrum of T\overline{T}TT¯-deformed theories, although we only carry out the explicit comparison to \mathcal{O}(1/\sqrt{c})𝒪(1/c) in the 1/c1/c expansion.

Integrable Systems on Singular Symplectic Manifolds: From Local to Global

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a $b$-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [34] and [35] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and $b$-symplectic forms in [34]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set $Z$ of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on $Z$.

Irreducibility of Lagrangian Quot schemes over an algebraic curve

Let C be a complex projective smooth curve and W a symplectic vector bundle of rank 2n over C. The Lagrangian Quot scheme $$LQ_{-e}(W)$$ L Q - e ( W ) parameterizes subsheaves of rank n and degree $$-e$$ - e which are isotropic with respect to the symplectic form. We prove that $$LQ_{-e}(W)$$ L Q - e ( W ) is irreducible and generically smooth of the expected dimension for all large e, and that a generic element is saturated and stable.

On forms, cohomology and BV Laplacians in odd symplectic geometry

We study the cohomology of the complexes of differential, integral and a particular class of pseudo-forms on odd symplectic manifolds taking the wedge product with the symplectic form as a differential. We thus extend the result of Ševera and the related results of Khudaverdian–Voronov on interpreting the BV odd Laplacian acting on half-densities on an odd symplectic supermanifold. We show that the cohomology classes are in correspondence with inequivalent Lagrangian submanifolds and that they all define semidensities on them. Further, we introduce new operators that move from one Lagragian submanifold to another and we investigate their relation with the so-called picture changing operators for the de Rham differential. Finally, we prove the isomorphism between the cohomology of the de Rham differential and the cohomology of BV Laplacian in the extended framework of differential, integral and a particular class of pseudo-forms.