guard games on graphs a team of mobile agents called guards tries to keep an intruder out of an assigned area by blocking all possible attacks in a graph model for this setting the guards and the intruder are located on the vertices of a graph and they move from node to node via connecting edges the area protected by the guards is an induced subgraph of the given graph we investigate the algorithmic aspects of the guarding problem which is to find the minimum number of guards sufficient to patrol the area we show that the guarding problem is pspace hard and provide a set of approximation algorithms all approximation algorithms are based on the study of a variant of the game where the intruder must reach the guarded area in a single step in order to win this variant of the game appears to be a 2 approximation for the guarding problem and for graphs without cycles of length 5 the minimum number of required guards in both games coincides we give a polynomial time algorithm for solving the one step guarding problem in graphs of bounded treewidth and complement this result by showing that the problem is w1 hard parameterized by the treewidth of the input graph we also show that the problem is fixed parameter tractable fpt parameterized by the treewidth and maximum degree of the input graph finally we turn our attention to a large class of sparse graphs including planar graphs and graphs of bounded genus namely apex minor free graphs we prove that the one step guarding problem is fpt and possess a ptas on apex minor free graphs

invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

the complexity of pursuit on a graph abstract a robber and k cops choose starting vertices in a graph and move in alternation from vertex to vertex along the edges of the graph capture occurs if a cop ever shares a vertex with the robber if k is not fixed and if either the graph is directed or initial positions are given we show that the problem is exptime complete similar techniques lead to the pspace and exptime completeness of some other combinatorial games that were previously only known to be np and pspace hard

generalized kahler potentials from supergravity we consider supersymmetric mathcal n 2 solutions with non vanishing ns three form building on worldsheet results we reduce the problem to a single generalized monge ampere equation on the generalized kahler potential k recently interpreted geometrically by lindstrom rocek von unge and zabzine one input in the procedure is a holomorphic function w that can be thought of as the effective superpotential for a d3 brane probe the procedure is hence likely to be useful for finding gravity duals to field theories with non vanishing abelian superpotential such as leigh strassler theories we indeed show that a purely ns precursor of the lunin maldacena dual to the deformed mathcal n 4 super yang mills falls in our class

the guarding game is e complete the guarding game is a game in which several cops try to guard a region in a directed or undirected graph against robber robber and the cops are placed on the vertices of the graph they take turns in moving to adjacent vertices or staying cops inside the guarded region robber on the remaining vertices the robber region the goal of robber is to enter the guarded region at a vertex with no cop on it the problem is to determine whether for a given graph and given number of cops the cops are able to prevent robber from entering the guarded region fomin et al 2011 7 proved that the problem is np complete when the robber region is restricted to a tree further they prove that is it pspace complete when the robber region is restricted to a directed acyclic graph and they ask about the problem complexity for arbitrary graphs in this paper we prove that the problem is e complete for arbitrary directed graphs

the complexity of searching a graph t parsons originally proposed and studied the following pursuit evasion problem on graphs members of a team of searchers traverse the edges of a graph g in pursuit of a fugitive who moves along the edges of the graph with complete knowledge of the locations of the pursuers what is the smallest number s g of searchers that will suffice for guaranteeing capture of the fugitive it is shown that determining whether s g k for a given integer k is np complete for general graphs but can be solved in linear time for trees we also provide a structural characterization of those graphs g with s g k for k 1 2 3

the pivotal cover and frobenius schur indicators abstract in this paper we introduce the notion of the pivotal cover c piv of a left rigid monoidal category c to develop a theoretical foundation for the theory of frobenius schur fs indicators in non pivotal settings for an object v c piv the n r th fs indicator n r v is defined by generalizing that of an object of a pivotal monoidal category this notion gives a categorical viewpoint to some recent results on generalizations of fs indicators based on our framework we also study the fs indicators of the adjoint object in a finite tensor category which can be considered as a generalization of the adjoint representation of a hopf algebra the indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor

recent results and questions in combinatorial game complexities abstract aim present a systematic development of part of the theory of combinatorial games from the ground up approach computational complexity combinatorial games are completely determined the questions of interest are efficiencies of strategies methodology divide and conquer isolate the various difficulties separating hard from easy games and attack them individually presentation informal examples of games sampled from various levels illustrate the theory with emphasis on formulating and motivating new and old research problems

pursuing a fast robber on a graph the cops and robbers game as originally defined independently by quilliot and by nowakowski and winkler in the 1980s has been much studied but very few results pertain to the algorithmic and complexity aspects of it in this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is np hard and that the parameterized version of the problem is w 2 hard the proof extends to the case where the robber moves s time faster than the cops we show that on split graphs the problem is polynomially solvable if s 1 but is np hard if s 2 we further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for email protected 2 finally we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops

moduli spaces of higgs bundles on degenerating riemann surfaces abstract we prove a gluing theorem for solutions a 0 0 of hitchin s self duality equations with logarithmic singularities on a rank 2 vector bundle over a noded riemann surface 0 representing a boundary point of teichmuller moduli space we show that every nearby smooth riemann surface 1 carries a smooth solution a 1 1 of the self duality equations which may be viewed as a desingularization of a 0 0

geometry of twisted kahler einstein metrics and collapsing we prove that the twisted kahler einstein metrics that arise on the base of certain holomorphic fiber space with calabi yau fibers have conical type singularities along the discriminant locus these fiber spaces arise naturally when studying the collapsing of ricci flat kahler metrics on calabi yau manifolds and of the kahler ricci flow on compact kahler manifolds with semiample canonical bundle and intermediate kodaira dimension our results allow us to understand their collapsed gromov hausdorff limits when the base is smooth and the discriminant has simple normal crossings

on the complexity of connection games in this paper we study three connection games among the most widely played havannah twixt and slither we show that determining the outcome of an arbitrary input position is pspace complete in all three cases our reductions are based on the popular graph problem generalized geography and on hex itself we also consider the complexity of generalizations of hex parameterized by the length of the solution and establish that while short generalized hex is w1 hard short hex is fpt finally we prove that the ultra weak solution to the empty starting position in hex cannot be fully adapted to any of these three games the games of havannah twixt and slither are pspace complete the game of slither does not admit draws on finite planar grid boards short generalized hex is w1 hard and short hex is fixed parameter tractable

picard fuchs uniformization and modularity of the mirror map arithmetic properties of mirror symmetry type iia iib string duality are studied we give criteria for the mirror map q series of certain families of calabi yau manifolds to be automorphic functions for families of elliptic curves and lattice polarized k3 surfaces with surjective period mappings global torelli theorems allow one to present these criteria in terms of the ramification behavior of natural algebraic invariants the functional and generalized functional invariants respectively in particular when applied to one parameter families of rank 19 lattice polarized k3 surfaces our criterion demystifies the mirror moonshine phenomenon of lian and yau and highlights its non monstrous nature the lack of global torelli theorems and presence of instanton corrections makes calabi yau threefold families more complicated via the constraints of special geometry the picard fuchs equations for one parameter families of calabi yau threefolds imply a differential equation criterion for automorphicity of the mirror map in terms of the yukawa coupling in the absence of instanton corrections the projective periods map to a twisted cubic space curve a hierarchy of algebraic instanton corrections correlated with the differential galois group of the picard fuchs equation is proposed

superstrings and topological strings at large n we embed the large n chern simons topological string duality in ordinary superstrings this corresponds to a large n duality between generalized gauge systems with n 1 supersymmetry in four dimensions and superstrings propagating on noncompact calabi yau manifolds with certain fluxes turned on we also show that in a particular limit of the n 1 gauge theory system certain superpotential terms in the n 1 system including deformations if spacetime is noncommutative are captured to all orders in 1 n by the amplitudes of noncritical bosonic strings propagating on a circle with self dual radius we also consider d brane anti d brane system wrapped over vanishing cycles of compact calabi yau manifolds and argue that at large n they induce a shift in the background to a topologically distinct calabi yau which we identify as the ground state system of the brane anti brane system

greedy strategies and larger islands of tractability for conjunctive queries and constraint satisfaction problems most structural decomposition methods can be characterized through hypergraph games that are variations of the robber and cops graph game that characterizes the notion of treewidth decomposition trees correspond to monotone winning strategies of the cops where the escape space of the robber on the hypergraph is shrunk monotonically cops using non monotonic strategies are more powerful but such strategies do not correspond to valid decompositions in general the paper provides a general way to exploit the power of non monotonic strategies by allowing a greedy form of non monotonicity it is shown that deciding the existence of a non monotone greedy winning strategy and compute one if any is tractable moreover from greedy strategies we can compute valid decomposition trees efficiently as a consequence we are able to add power to structural methods and to obtain larger islands of tractability such as the one based on the new notion of greedy hypertree decomposition

cubic hypersurfaces and integrable systems together with the cubic and quartic threefolds the cubic fivefolds are the only hypersurfaces of odd dimension bigger than one for which the intermediate jacobian is a nonzero principally polarized abelian variety p p a v in this paper we show that the family of 21 dimensional intermediate jacobians of cubic fivefolds containing a given cubic fourfold x is generically an algebraic integrable system in the proof we apply an integrability criterion introduced and used by donagi and markman to find a similar integrable system over the family of cubic threefolds in x to enter in the conditions of this criterion we write down explicitly the symplectic structure known by beauville and donagi on the family f x of lines on the general cubic fourfold x and prove that the family of planes on a cubic fivefold containing x is embedded as a lagrangian surface in f x by a symplectic reduction we deduce that our integrable system induces on the nodal boundary another integrable system interpreted generically as the family of 20 dimensional intermediate jacobians of fano threefolds of genus four contained in x along the way we prove an abel jacobi type isomorphism for the fano surface of conics in the general fano threefold of genus four and compute the numerical invariants of this surface

singular improper affine spheres from a given lagrangian submanifold abstract given a lagrangian submanifold l of the affine symplectic 2n space one can canonically and uniquely define a center chord and a special improper affine sphere of dimension 2n both of whose sets of singularities contain l although these improper affine spheres ias always present other singularities away from l the off shell singularities studied in 8 they may also present singularities other than l which are arbitrarily close to l the so called singularities on shell these on shell singularities possess a hidden z 2 symmetry that is absent from the off shell singularities in this paper we study these canonical ias obtained from l and their on shell singularities in arbitrary even dimensions and classify all stable lagrangian legendrian singularities on shell that may occur for these ias when l is a curve or a lagrangian surface

generalized complex manifolds and supersymmetry we find a worldsheet realization of generalized complex geometry a notion introduced recently by hitchin which interpolates between complex and symplectic manifolds the two dimensional model we construct is a supersymmetric relative of the poisson sigma model used in the context of deformation quantization

invariants of 3 manifolds and projective representations of mapping class groups via quantum groups at roots of unity an example of a finite dimensional factorizable ribbon hopf algebra is given by a quotienth u q g of the quantized universal enveloping algebrau q g at a root of unityq of odd degree the mapping class groupm g 1 of a surface of genusg with one hole projectively acts by automorphisms in theh moduleh g ifh is endowed with the coadjointh module structure there exists a projective representation of the mapping class groupm g n of a surface of genusg withn holes labeled by finite dimensionalh modulesx 1 x n in the vector space hom h x 1 x n h g an invariant of closed oriented 3 manifolds is constructed modifications of these constructions for a class of ribbon hopf algebras satisfying weaker conditions than factorizability including most ofu q g at roots of unityq of even degree are described

parameterized pursuit evasion games we study the parameterized complexity of four variants of pursuit evasion on graphs seeded pursuit evasion short seeded pursuit evasion directed pursuit evasion and short directed pursuit evasion both seeded pursuit evasion and short seeded pursuit evasion are played on undirected graphs with given starting positions for both the cops and the robber directed pursuit evasion and its short variant are played on directed graphs with the players free to choose their starting positions we show for seeded pursuit evasion and directed pursuit evasion that finding a winning strategy for the cops is aw hard when we parameterize by the number of cops further we show that the short k move variants of these problems short seeded pursuit evasion and short directed pursuit evasion are aw complete when we parameterize by both the number of cops and turns

the lion and man game on polyhedral surfaces with obstacles abstract we study a geometric version of the cops and robbers game known as the lion and man game in this game a group of lions the pursuers try to capture a man the evader the players have equal speed they can observe each other at all times while the lion and man game is well studied in planar domains such as polygons very little is known about its properties in higher dimensions in this paper we study the game when played on the surface of a polyhedron with or without boundary possibly in the presence of obstacles i e forbidden regions in particular in the absence of obstacles the surface has genus zero and it can be homeomorphic to either a disk or a sphere we assume that the input surface is triangulated we show that three lions with non zero capture distance can capture the man in o a 2 l n 2 2 a 2 l n d steps where a is the area l is the total edge length of the polyhedron n is the number of triangular faces used in the representation of the polyhedron and d is the length of the longest shortest path on the surface

mirror symmetric su 3 structure manifolds with ns fluxes when string theory is compactified on a six dimensional manifold with a nontrivial ns flux turned on mirror symmetry exchanges the flux with a purely geometrical composite ns form associated with lack of integrability of the complex structure on the mirror side considering a general class of t3 fibered geometries admitting su 3 structure we find an exchange of pure spinors e ij and in dual geometries under fiberwise t duality and study the transformations of the ns flux and the components of intrinsic torsion a complementary study of action of twisted covariant derivatives on invariant spinors allows to extend our results to generic geometries and formulate a proposal for mirror symmetry in compactifications with ns flux

hyper hamiltonian dynamics we introduce an extension of hamiltonian dynamics defined on hyper kahler manifolds which we call hyper hamiltonian dynamics we show that this has many of the attractive features of standard hamiltonian dynamics we also discuss the prototypical integrable hyper hamiltonian systems i e quaternionic oscillators

from non semisimple hopf algebras to correlation functions for logarithmic cft we use factorizable finite tensor categories and specifically the representation categories of factorizable ribbon hopf algebras h as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories for any ribbon hopf algebra automorphism of h we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric frobenius algebra we derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters as a crucial ingredient this involves the cartan matrix of the category we also show how for any candidate bulk state space of the type we consider correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group

a two person game on graphs where each player tries to encircle his opponent s men we present results on a combinatorial game which was proposed to one of the authors by ingo althofer personal communication let g be an undirected finite graph without loops and multiple edges and let k be a positive integer with k 12 g 1 there are two players called white and black both having k men of their color in turn beginning with white the players position their men one at a time on unoccupied vertices of g when all men are placed the players take turns moving a man of their color along an edge to an unoccupied adjacent vertex again beginning with white a player wins if his opponent cannot carry out his next move since none of his men has an unoccupied neighbor if the game does not stop then the outcome is a draw we always assume that both players play optimal among other questions we deal with the following ones 1 1 is it true that for all g and k white cannot win the game 2 2 does there exist a tree t and a positive integer k for which the outcome is a draw let g denote the covering number of g i e g is the minimum number of vertices covering all edges of g we prove that black wins the game if g k we use this result to show that white never wins the game if gis bipartite thus providing a partial answer to the first question we answer the second question in the affirmative by constructing an infinite series of trees for which the outcome is a draw for some k moreover we present results on extremal problems arising in the context of the game we also completely solve the cases when g is a path or a cycle further we completely settle the case k 2 in the proofs of our results matchings and cycles in graphs play a predominant role

holomorphic structures in hydrodynamical models of nearly geostrophic flow we study complex structures arising in hamiltonian models of nearly geostrophic flows in hydrodynamics in many of these models an elliptic monge ampere equation defines the relationship between a balanced velocity field defined by a constraint in the hamiltonian formalism and the materially conserved potential vorticity elliptic monge ampere operators define an almost complex structure and in this paper we show that a natural extension of the so called geostrophic momentum transformation of semi geostrophic theory which has a special importance in theoretical meteorology defines kahler and special kahler structures on phase space furthermore analogues of the geostrophic momentum coordinates are shown to be special lagrangian coordinates under conditions which depend upon the physical approximations under consideration certain duality properties of the operators are studied within the framework of the kahler geometry

quantum double of hopf monads and categorical centers the center z c of an autonomous category c is monadic over c if certain coends exist in c the notion of a hopf monad naturally arises if one tries to reconstruct the structure of z c in terms of its monad z we show that z is a quasitriangular hopf monad on c and z c is isomorphic to the braided category z c of z modules more generally let t be a hopf monad on an autonomous category c we construct a hopf monad zt on c the centralizer of t and a canonical distributive law tzt ztt by beck s theory this has two consequences on one hand dt zt t is a quasitriangular hopf monad on c called the double of t and z t c dt c as braided categories as an illustration we define the double d a of a hopf algebra a in a braided autonomous category in such a way that the center of the category of a modules is the braided category of d a modules generalizing the drinfeld double on the other hand the canonical distributive law also lifts zt to a hopf monad z t on t c and z t t0 is the coend of t c for t z this gives an explicit description of the hopf algebra structure of the coend of z c in terms of the structural morphisms of c such a description is useful in quantum topology especially when c is a spherical fusion category as z c is then modular

asymptotic geometry of the hitchin metric we study the asymptotics of the natural l2 metric on the hitchin moduli space with group g mathrm su 2 our main result which addresses a detailed conjectural picture made by gaiotto et al adv math 234 239 403 2013 is that on the regular part of the hitchin system this metric is well approximated by the semiflat metric from gaiotto et al 2013 we prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion and hence that the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay new work by dumas neitzke and later fredrickson shows that the convergence is actually exponential

on fusion categories using a variety of methods developed in the literature in particular the theory of weak hopf algebras we prove a number of general results about fusion categories in characteristic zero we show that the global dimension of a fusion category is always positive and that the s matrix of any not necessarily hermitian modular category is unitary we also show that the category of module functors between two module categories over a fusion category is semisimple and that fusion categories and tensor functors between them are undeformable generalized ocneanu rigidity in particular the number of such categories functors realizing a given fusion datum is finite finally we develop the theory of frobenius perron dimensions in an arbitrary fusion category at the end of the paper we generalize some of these results to positive characteristic

linguistic geometry approach for solving the cops and robber problem in grid environments abstract the cops and robber problem is a game played between two adversary groups in which one group cops attempts to capture the other group robber in this paper the optimal capture time problem is addressed which seeks the best motion strategy from the viewpoint of the cops for capturing the robber within the minimum time or number of steps and a new algorithm based on linguistic geometry lg is proposed lg is widely used for solving two player abstract board games and wargaming problems by breaking the system problem into several subsystems in each of which the playing elements and their paths called trajectories are identified and a network consisting of these trajectories is constructed a solution is identified by searching the subsystems by the alpha beta pruning method also to provide a lower bound for the robber s capture time we have enhanced the alpha beta pruning method by incorporating a new admissible heuristic more informed than existing heuristics as a result the branching factor of the search is decreased and the problem is solved more efficiently for experimentation several cops and robber problems on regular and holed square grids were solved by the lg based method and the results were compared with the reverse minimax a rma algorithm according to the results the presented algorithm requires far less computational time and memory than the rma method in fact approximately 95 5 and 97 5 are saved on average respectively and near optimal solutions were obtained also it was observed that while increasing the number of cops severely affects the computational time and required memory of the rma it has insignificant effects on our algorithm

ends of the moduli space of higgs bundles we associate to each stable higgs pair a 0 phi 0 on a compact riemann surface x a singular limiting configuration a infinity phi infinity assuming that det phi has only simple zeroes we then prove a desingularization theorem by constructing a family of solutions a t t phi t to hitchin s equations which converge to this limiting configuration as t infinity this provides a new proof via gluing methods for elements in the ends of the higgs bundle moduli space and identifies a dense open subset of the boundary of the compactification of this moduli space

the geometry of special symplectic representations abstract we show there is a class of symplectic lie algebra representations over any field of characteristic not 2 or 3 that have many of the exceptional algebraic and geometric properties of both symmetric three forms in two dimensions and alternating three forms in six dimensions the main algebraic result is that suitably generic elements of these representation spaces can be uniquely written as the sum of two elements of a naturally defined lagrangian subvariety we give universal explicit formulae for the summands and show how they lead to the existence of geometric structure on appropriate subsets of the representation space over the real numbers this structure reduces to either a conic special pseudo kahler metric or a conic special para kahler metric

non degeneracy conditions for braided finite tensor categories abstract for a braided finite tensor category c with unit object 1 c lyubashenko considered a certain hopf algebra f c endowed with a hopf pairing f f 1 to define the notion of a non semisimple modular tensor category we say that c is non degenerate if the hopf pairing is non degenerate in this paper we show that c is non degenerate if and only if it is factorizable in the sense of etingof nikshych and ostrik if and only if its muger center is trivial if and only if the linear map hom c 1 f hom c f 1 induced by the pairing is invertible as an application we prove that the category of yetter drinfeld modules over a hopf algebra in c is non degenerate if and only if c is

the relative modular object and frobenius extensions of finite hopf algebras abstract for a certain kind of tensor functor f c d we define the relative modular object f d as the difference between a left adjoint and a right adjoint of f our main result claims that if c and d are finite tensor categories then f can be written in terms of a categorical analogue of the modular function on a hopf algebra applying this result to the restriction functor associated to an extension a b of finite dimensional hopf algebras we recover the result of fischman montgomery and schneider on the frobenius type property of a b we also apply our results to obtain a braided version and a bosonization version of the result of fischman et al

frobenius manifolds projective special geometry and hitchin systems we consider the construction of frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved in particular we prove that the underlying f manifold is canonical we then apply this construction to integrable systems of hitchin type

isolated singularities of affine special kahler metrics in two dimensions we prove that there are just two types of isolated singularities of special kahler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities we also construct examples of such metrics

deciding the winner in k rounds for disjoint arrows a new combinatorial partizan game we consider disjoint arrows a new bounded length two player partizan combinatorial game in this game the two players alice and bob alternate in choosing vertices on a directed graph and no player is allowed to select a vertex previously selected at each round alice selects a vertex u and bob has to reply choosing a new vertex in the out neighborhood of u the first player who cannot move loses we prove that deciding whether bob can endure k rounds when k is part of the input is a pspace complete problem moreover we prove that the parameterized version of the problem is aw complete thus we provide a new member for the small set of problems known complete for the class aw

an annotated bibliography on guaranteed graph searching graph searching encompasses a wide variety of combinatorial problems related to the problem of capturing a fugitive residing in a graph using the minimum number of searchers in this annotated bibliography we give an elementary classification of problems and results related to graph searching and provide a source of bibliographical references on this field

cops and robbers from a distance cops and robbers is a pursuit and evasion game played on graphs that has received much attention we consider an extension of cops and robbers distance k cops and robbers where the cops win if at least one of them is of distance at most k from the robber in g the cop number of a graph g is the minimum number of cops needed to capture the robber in g the distance k analogue of the cop number written c k g equals the minimum number of cops needed to win at a given distance k we study the parameter c k from algorithmic structural and probabilistic perspectives we supply a classification result for graphs with bounded c k g values and develop an o n 2 s 3 algorithm for determining if c k g s for s fixed we prove that if s is not fixed then computing c k g is np hard upper and lower bounds are found for c k g in terms of the order of g we prove that nk 1 2 o 1 c k n o nlog 2nk 1 log k 2 k 1 where c k n is the maximum of c k g over all n vertex connected graphs the parameter c k g is investigated asymptotically in random graphs g n p for a wide range of p p n for each k 0 it is shown that c k g as a function of the average degree d n pn forms an intriguing zigzag shape

special kahler manifolds we give an intrinsic definition of the special geometry which arises in global n 2 supersymmetry in four dimensions the base of an algebraic integrable system exhibits this geometry and with an integrality hypothesis any special kahler manifold is so related to an integrable system the cotangent bundle of a special kahler manifold carries a hyperkahler metric we also define special geometry in supergravity in terms of the special geometry in global supersymmetry

completeness in supergravity constructions we prove that the supergravity r and c maps preserve completeness as a consequence any component mathcal h of a hypersurface h 1 defined by a homogeneous cubic polynomial h such that partial 2h is a complete riemannian metric on mathcal h defines a complete projective special kahler manifold and any complete projective special kahler manifold defines a complete quaternionic kahler manifold of negative scalar curvature we classify all complete quaternionic kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions

two sphere partition functions and gromov witten invariants many n 2 2 two dimensional nonlinear sigma models with calabi yau target spaces admit ultraviolet descriptions as n 2 2 gauge theories gauged linear sigma models we conjecture that the two sphere partition function of such ultraviolet gauge theories recently computed via localization by benini et al and doroud et al yields the exact kahler potential on the quantum kahler moduli space for calabi yau threefold target spaces in particular this allows one to compute the genus zero gromov witten invariants for any such calabi yau threefold without the use of mirror symmetry more generally when the infrared superconformal fixed point is used to compactify string theory this provides a direct method to compute the spacetime kahler potential of certain moduli e g vector multiplet moduli in type iia exactly in we compute these quantities for the quintic and for rodland s pfaffian calabi yau threefold and find agreement with existing results in the literature we then apply our methods to a codimension four determinantal calabi yau threefold in p 7 recently given a nonabelian gauge theory description by the present authors for which no mirror calabi yau is currently known we derive predictions for its gromov witten invariants and verify that our predictions satisfy nontrivial geometric checks

geometric u folds in four dimensions we describe a general construction of geometric u folds compatible with the global formulation of four dimensional extended supergravity on a differentiable spin manifold the topology of geometric u folds depends on certain fiber bundles which encode how supergravity fields are globally glued together smooth non trivial u folds of this type can exist only in theories where both the scalar and space time manifolds have non trivial fundamental group and in addition the configuration of scalar fields of the solution is homotopically non trivial nonetheless certain geometric u folds extend to simply connected backgrounds containing localized sources consistency with string theory requires smooth geometric u folds to be glued using subgroups of the effective discrete u duality group implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter we construct simple examples of geometric u folds in a generalization of the axiondilaton model of n 2 supergravity coupled to a single vector multiplet whose scalar manifold is a generally non compact riemann surface of genus at least two endowed with its uniformizing metric we also discuss the relation between geometric u folds and a moduli space of flat connections defined on the scalar manifold which involves certain character varieties not studied in the literature

twist geometry of the c map we discuss the geometry of the c map from projective special kahler to quaternionic kahler manifolds using the twist construction to provide a global approach to hitchin s description as found by alexandrov et al and alekseevsky et al this is related to the quaternionic flip of haydys we prove uniqueness statements for several steps of the construction in particular we show that given a hyperkahler manifold with a rotating symmetry there is essentially only a one parameter degree of freedom in constructing a quaternionic kahler manifold of the same dimension we demonstrate how examples on group manifolds arise from this picture