The Ising model and Special Geometries

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals $\chi^{(n)}$ of the magnetic susceptibility of the Ising model ($n \le 6$) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms, is equivalent to the fact that their symmetric square, or their exterior square, have rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators we already know to be Derived from Geometry, are special globally nilpotent operators: they correspond to "Special Geometries". Beyond the small order factor operators (occurring in the linear differential operators associated with $\chi^{(5)}$ and $\chi^{(6)}$), and, in particular, those associated with modular forms, we focus on the quite large order-twelve and order-23 operators. We show that the order-twelve operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group $Sp(12, \mathbb{C})$. The order-23 operator is shown to factorize in an order-two operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group $SO(21, \mathbb{C})$.Comment: 33 page

Symmetries of L\'evy processes on compact quantum groups, their Markov semigroups and potential theory

Strongly continuous semigroups of unital completely positive maps (i.e. quantum Markov semigroups or quantum dynamical semigroups) on compact quantum groups are studied. We show that quantum Markov semigroups on the universal or reduced C${}^*$-algebra of a compact quantum group which are translation invariant (w.r.t. to the coproduct) are in one-to-one correspondence with L\'evy processes on its $*$-Hopf algebra. We use the theory of L\'evy processes on involutive bialgebras to characterize symmetry properties of the associated quantum Markov semigroup. It turns out that the quantum Markov semigroup is GNS-symmetric (resp. KMS-symmetric) if and only if the generating functional of the L\'evy process is invariant under the antipode (resp. the unitary antipode). Furthermore, we study L\'evy processes whose marginal states are invariant under the adjoint action. In particular, we give a complete description of generating functionals on the free orthogonal quantum group $O_n^+$ that are invariant under the adjoint action. Finally, some aspects of the potential theory are investigated. We describe how the Dirichlet form and a derivation can be recovered from a quantum Markov semigroup and its L\'evy process and we show how, under the assumption of GNS-symmetry and using the associated Sch\"urmann triple, this gives rise to spectral triples. We discuss in details how the above results apply to compact groups, group C$^*$-algebras of countable discrete groups, free orthogonal quantum groups $O_n^+$ and the twisted $SU_q (2)$ quantum group.Comment: 54 pages, thoroughly revised, to appear in the Journal of Functional Analysi

Brane and string field structure of elementary particles

The two quantizations of QFT,as well as the attempt of unifying it with general relativity,lead us to consider that the internal structure of an elementary fermion must be twofold and composed of three embedded internal (bi)structures which are vacuum and mass (physical) bosonic fields decomposing into packets of pairs of strings behaving like harmonic oscillators characterized by integers mu corresponding to normal modes at mu (algebraic) quanta.Comment: 50 page

The Hodge theory of Soergel bimodules

We prove Soergel's conjecture on the characters of indecomposable Soergel bimodules. We deduce that Kazhdan-Lusztig polynomials have positive coefficients for arbitrary Coxeter systems. Using results of Soergel one may deduce an algebraic proof of the Kazhdan-Lusztig conjecture.Comment: 44 pages. v2: many minor changes, final versio

Families of L-functions and their Symmetry

In [90] the first-named author gave a working definition of a family of automorphic L-functions. Since then there have been a number of works [33], [107], [67] [47], [66] and especially [98] by the second and third-named authors which make it possible to give a conjectural answer for the symmetry type of a family and in particular the universality class predicted in [64] for the distribution of the zeros near s=1/2. In this note we carry this out after introducing some basic invariants associated to a family

Whitham Deformations of Seiberg-Witten Curves for Classical Gauge Groups

Gorsky et al. presented an explicit construction of Whitham deformations of the Seiberg-Witten curve for the $SU(N+1)$ \calN = 2 SUSY Yang-Mills theory. We extend their result to all classical gauge groups and some other cases such as the spectral curve of the $A^{(2)}_{2N}$ affine Toda Toda system. Our construction, too, uses fractional powers of the superpotential $W(x)$ that characterizes the curve. We also consider the $u$-plane integral of topologically twisted theories on four-dimensional manifolds $X$ with $b_2^{+}(X) = 1$ in the language of these explicitly constructed Whitham deformations and an integrable hierarchy of the KdV type hidden behind.Comment: latex, 39pp, no figure; some more comments and references on integrable systems are added, and many typos are correcte

Vector-valued covariant differential operators for the M\"obius transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb C), SL(2,\mathbb R))$ that arise from conformal geometry.Comment: To appear in Springer Proceedings in Mathematics and Statistic

Some Cases of Kudla’s Modularity Conjecture for Unitary Shimura Varieties

A common theme of the thesis is the interplay of symmetry and rigidity, which is a general phenomenon in mathematics. Symmetry is a notion related to the degree to which an object remains unchanged under transformations, and rigidity is a notion that in terms of physics can be thought of as a lack of freedom, which leads to stronger properties of an object than we normally expect. An object of higher symmetry often also exhibits a higher extent of rigidity, and vice versa. In the introduction of the thesis, we provide some background on modular forms, number theory, and geometry in a way that does not require familiarity with these subjects. The contributions of this thesis are presented in three articles.In Article I, we establish the existence of rational geometric designs for rational polytopes via the circle method and convex geometry, and discuss the existence of rational spherical designs which relates to Lehmer\u27s conjecture on the Ramanujan tau function. In Article II, we break the barrier of expressing weight-2 modular forms of higher level whose central L-values vanish by products of at most two Eisenstein series. This work shows the power of Rankin--Selberg method and also contributes to the computation of elliptic modular forms.In Preprint III, we prove unconditionally some cases of Kudla\u27s conjecture on the modularity of generating functions of special cycles on unitary Shimura varieties, for norm-Euclidean imaginary quadratic fields. Our method is based on a result of Liu and work of Bruinier--Raum, who confirmed the orthogonal Kudla conjecture over Q