the cauchy problem for the homogeneous monge ampere equation ii legendre transform abstract we continue our study of the cauchy problem for the homogeneous real and complex monge ampere equation hrma hcma in the prequel y a rubinstein and s zelditch 27 a quantum mechanical approach for solving the hcma was developed and was shown to coincide with the well known legendre transform approach in the case of the hrma in this article that uses tools of convex analysis and can be read independently we prove that the candidate solution produced by these methods ceases to solve the hrma even in a weak sense as soon as it ceases to be differentiable at the same time we show that it does solve the equation on its dense regular locus and we derive an explicit a priori upper bound on its monge ampere mass the technique involves studying regularity of legendre transforms of families of non convex functions

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

complex symplectomorphisms and pseudo kahler islands in the quantization of toric manifolds let p be a delzant polytope we show that the quantization of the corresponding toric manifold x p in toric kahler polarizations and in the toric real polarization are related by analytic continuation of hamiltonian flows evaluated at time t sqrt 1 s we relate the quantization of x p in two different toric kahler polarizations by taking the time sqrt 1 s hamiltonian flow of strongly convex functions on the moment polytope p by taking s to infinity we obtain the quantization of x p in the singular real toric polarization recall that x p has an open dense subset which is biholomorphic to mathbb c n the quantization of x p in a toric kahler polarization can also be described by applying the complexified hamiltonian flow of the abreu guillemin symplectic potential g at time t sqrt 1 to an appropriate finite dimensional subspace of quantum states in the quantization of t mathbb t n in the vertical polarization by taking other imaginary times t k sqrt 1 k in mathbb r we describe toric kahler metrics with cone singularities along the toric divisors in x p for convex hamiltonian functions and sufficiently negative imaginary part of the complex time we obtain degenerate kahler structures which are negative definite in some regions of x p we show that the pointwise and l 2 norms of quantum states are asymptotically vanishing on negative definite regions

coherent state transforms and the mackey stone von neumann theorem mackey showed that for a compact lie group k the pair k c 0 k has a unique non trivial irreducible covariant pair of representations we study the relevance of this result to the unitary equivalence of quantizations for an infinite dimensional family of k k invariant polarizations on t k the kahler polarizations in the family are generated by complex time hamiltonian flows applied to the schrodinger vertical real polarization the unitary equivalence of the corresponding quantizations of t k is then studied by considering covariant pairs of representations of k defined by geometric prequantization and of representations of c 0 k defined via heisenberg time evolution followed by time geometric quantization induced evolution we show that in the semiclassical and large imaginary time limits the unitary transform whose existence is guaranteed by mackey s theorem can be approximated by composition of the time geometric quantization induced evolution with the time evolution

coherent state transforms and abelian varieties we extend the coherent state transform cst of hall to the context of abelian varieties by considering them as quotients of the complexification of the abelian group k u 1 g we show that this transform applied to appropriate distributions on k gives all classical theta functions and that by defining on this space of theta functions an inner product related to the k averaged heat kernel the unitarity of the cst transform is still preserved

kahler metrics on toric orbifolds a theorem of e lerman and s tolman generalizing a result of t delzant states that compact symplectic toric orbifolds are classified by their moment polytopes together with a positive integer label attached to each of their facets in this paper we use this result and the existence of global actionangle coordinates to give an effective parametrization of all compatible toric complex structures on a compact symplectic toric orbifold by means of smooth functions on the corresponding moment polytope this is equivalent to parametrizing all toric kahler metrics and generalizes an analogous result for toric manifolds a simple explicit description of interesting families of extremal kahler metrics arising from recent work of r bryant is given as an application of the approach in this paper the fact that in dimension four these metrics are self dual and conformally einstein is also discussed this gives rise in particular to a one parameter family of self dual einstein metrics connecting the well known eguchi hanson and taub nut metrics

adapted complex structures and the geodesic flow in this paper we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact real analytic riemannian manifold motivated by the complexifier approach of t thiemann as well as certain formulas of v guillemin and m stenzel we obtain the polarization associated to the adapted complex structure by applying the imaginary time geodesic flow to the vertical polarization meanwhile at the level of functions we show that every holomorphic function is obtained from a function that is constant along the fibers by composition with the imaginary time geodesic flow we give several equivalent interpretations of this composition including a convergent power series in the vector field generating the geodesic flow

on the bks pairing for kahler quantizations of the cotangent bundle of a lie group a natural one parameter family of kahler quantizations of the cotangent bundle t k of a compact lie group k taking into account the half form correction was studied in fmmn in the present pa per it is shown that the associated blattner kostant sternberg bks pairing map is unitary and coincides with the parallel transport of the quantum connection introduced in our previous work from the point of view of adpw the bks pairing map is a composition of uni tary coherent state transforms of k introduced in ha1 continuity of the hermitian structure on the quantum bundle in the limit when one of the kahler polarizations degenerates to the vertical real polar ization leads to the unitarity of the corresponding bks pairing map this is in agreement with the unitarity up to scaling with respect to a rescaled inner product of this pairing map established by hall

direct images fields of hilbert spaces and geometric quantization geometric quantization often produces not one hilbert space to represent the quantum states of a classical system but a whole family h s of hilbert spaces and the question arises if the spaces h s are canonically isomorphic axelrod et al j diff geo 33 787 902 1991 and hitchin commun math phys 131 347 380 1990 suggest viewing h s as fibers of a hilbert bundle h introduce a connection on h and use parallel transport to identify different fibers here we explore to what extent this can be done first we introduce the notion of smooth and analytic fields of hilbert spaces and prove that if an analytic field over a simply connected base is flat then it corresponds to a hermitian hilbert bundle with a flat connection and path independent parallel transport second we address a general direct image problem in complex geometry pushing forward a hermitian holomorphic vector bundle e to y along a non proper map y to s we give criteria for the direct image to be a smooth field of hilbert spaces third we consider quantizing an analytic riemannian manifold m by endowing tm with the family of adapted kahler structures from lempert and sz ke bull lond math soc 44 367 374 2012 this leads to a direct image problem when m is homogeneous we prove the direct image is an analytic field of hilbert spaces for certain such m but not all the direct image is even flat which means that in those cases quantization is unique

geometric quantization parallel transport and the fourier transform in quantum mechanics the momentum space and position space wave functions are related by the fourier transform we investigate how the fourier transform arises in the context of geometric quantization we consider a hilbert space bundle mathcal h over the space mathcal j of compatible complex structures on a symplectic vector space this bundle is equipped with a projectively flat connection we show that parallel transport along a geodesic in the bundle mathcal h to mathcal j is a rescaled orthogonal projection or bogoliubov transformation we then construct the kernel for the integral parallel transport operator finally by extending geodesics to the boundary for which the metaplectic correction is essential we obtain the segal bargmann and fourier transforms as parallel transport in suitable limits

coherent state transforms and vector bundles on elliptic curves abstract we extend the coherent state transform cst of hall to the context of the moduli spaces of semistable holomorphic vector bundles with fixed determinant over elliptic curves we show that by applying the cst to appropriate distributions we obtain the space of level k rank n and genus one non abelian theta functions with the unitarity of the cst transform being preserved furthermore the shift in the level k k n appears in a natural way in this finite dimensional framework

geometric quantization complex structures and the coherent state transform abstract it is shown that the heat operator in the hall coherent state transform for a compact lie group k j funct anal 122 1994 103 151 is related with a hermitian connection associated to a natural one parameter family of complex structures on t k the unitary parallel transport of this connection establishes the equivalence of geometric quantizations of t k for different choices of complex structures within the given family in particular these results establish a link between coherent state transforms for lie groups and results of hitchin comm math phys 131 1990 347 380 and axelrod et al j differential geom 33 1991 787 902

system size resonance in a binary attractor neural network system size resonance ssr is a phenomenon in which the response of a system is optimal for a certain finite size but poorer as the size goes to zero or infinity in order to show ssr effects in binary attractor neural networks we study the response of a network in the ferromagnetic phase to an external time dependent stimulus under the presence of such a stimulus the network shows ssr as is demonstrated by the measure of the signal amplification both analytically and by simulation

generalized singular values interlacing inequalities and monotonic norms abstract we complete the results of an earlier paper by sodupe where interlacing properties for generalized singular values of matrices and submatrices are studied we show that there is a close relationship between these interlacing properties and the orthant monotonicity of the norms involved in the definition of the generalized singular values we briefly discuss interlacing properties for other kinds of a s numbers

the cauchy problem for the homogeneous monge ampere equation i toeplitz quantization the cauchy problem for the homogeneous real complex monge ampere equation hrma hcma arises from the initial value problem for geodesics in the space of kahler metrics equipped with the mabuchi metric this cauchy problem is believed to be ill posed and a basic problem is to characterize initial data of weak solutions which exist up to time t in this article we use a quantization method to construct a subsolution of the hcma on a general projective variety and we conjecture that it solves the equation for as long as the unique solution exists the subsolution called the quantum analytic continuation potential is obtained by i toeplitz quantizing the hamiltonian flow determined by the cauchy data ii analytically continuing the quantization and iii taking a certain logarithmic classical limit we then prove that in the case of torus invariant metrics where the hcma reduces to the hrma the quantum analytic continuation potential coincides with the well known legendre transform potential and hence solves the equation as long as it is smooth in the sequel the cauchy problem for the homogeneous monge ampere equation ii legendre transform adv math 228 2011 2989 3025 it is proved that the legendre transform potential ceases to solve the hcma once it ceases to be smooth the results here and in the sequels in particular characterize the initial data of smooth geodesic rays