Topological properties of punctual Hilbert schemes of almost-complex fourfolds (I)

In this article, we study topological properties of Voisin's punctual Hilbert schemes of an almost-complex fourfold $X$. In this setting, we compute their Betti numbers and construct Nakajima operators. We also define tautological bundles associated with any complex bundle on $X$, which are shown to be canonical in $K$-theory

Verlinde formulae on complex surfaces: K-theoretic invariants

We conjecture a Verlinde type formula for the moduli space of Higgs sheaves on a surface with a holomorphic 2-form. The conjecture specializes to a Verlinde formula for the moduli space of sheaves. Our formula interpolates between $K$-theoretic Donaldson invariants studied by the first named author and Nakajima-Yoshioka and $K$-theoretic Vafa-Witten invariants introduced by Thomas and also studied by the first and second named authors. We verify our conjectures in many examples (e.g. on K3 surfaces).Comment: Published version. 37 page

Block-Goettsche invariants from wall-crossing

We show how some of the refined tropical counts of Block and Goettsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative q-deformed Gromov-Witten invariants. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined

Vertex Operators, Grassmannians, and Hilbert Schemes

We describe a well-known collection of vertex operators on the infinite wedge representation as a limit of geometric correspondences on the equivariant cohomology groups of a finite-dimensional approximation of the Sato grassmannian, by cutoffs in high and low degrees. We prove that locality, the boson-fermion correspondence, and intertwining relations with the Virasoro algebra are limits of the localization expression for the composition of these operators. We then show that these operators are, almost by definition, the Hilbert scheme vertex operators defined by Okounkov and the author in \cite{CO} when the surface is $\mathbb{C}^2$ with the torus action $z\cdot (x,y) = (zx,z^{-1}y)$.Comment: 20 pages, 0 figure

Refined Topological Vertex and Instanton Counting

It has been proposed recently that topological A-model string amplitudes for toric Calabi-Yau 3-folds in non self-dual graviphoton background can be caluculated by a diagrammatic method that is called the ``refined topological vertex''. We compute the extended A-model amplitudes for SU(N)-geometries using the proposed vertex. If the refined topological vertex is valid, these computations should give rise to the Nekrasov's partition functions of N=2 SU(N) gauge theories via the geometric engineering. In this article, we verify the proposal by confirming the equivalence between the refined A-model amplitude and the K-theoretic version of the Nekrasov's partition function by explicit computation.Comment: 22 pages, 6 figures, minor correction

Post-imperialism, postcolonialism and beyond: towards a periodisation of cultural discourse about colonial legacies

Taking German history and culture as a starting point, this essay suggests a historical approach to reconceptualising different forms of literary engagement with colonial discourse, colonial legacies and (post-) colonial memory in the context of Comparative Postcolonial Studies. The deliberate blending of a historical, a conceptual and a political understanding of the ‘postcolonial’ in postcolonial scholarship raises problems of periodisation and historical terminology when, for example, anti-colonial discourse from the colonial period or colonialist discourse in Weimar Germany are labelled ‘postcolonial’. The colonial revisionism of Germany’s interwar period is more usefully classed as post-imperial, as are particular strands of retrospective engagement with colonial history and legacy in British, French and other European literatures and cultures after 1945. At the same time, some recent developments in Francophone, Anglophone and German literature, e.g. Afropolitan writing, move beyond defining features of postcolonial discourse and raise the question of the post-postcolonial

Crossings, Motzkin paths and Moments

Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain $q$-analogues of Laguerre and Charlier polynomials. The moments of these orthogonal polynomials have combinatorial models in terms of crossings in permutations and set partitions. The aim of this article is to prove simple formulas for the moments of the $q$-Laguerre and the $q$-Charlier polynomials, in the style of the Touchard-Riordan formula (which gives the moments of some $q$-Hermite polynomials, and also the distribution of crossings in matchings). Our method mainly consists in the enumeration of weighted Motzkin paths, which are naturally associated with the moments. Some steps are bijective, in particular we describe a decomposition of paths which generalises a previous construction of Penaud for the case of the Touchard-Riordan formula. There are also some non-bijective steps using basic hypergeometric series, and continued fractions or, alternatively, functional equations.Comment: 21 page

Validation of Sentinel-3 SLSTR Land Surface Temperature Retrieved by the Operational Product and Comparison with Explicitly Emissivity-Dependent Algorithms

Land surface temperature (LST) is an essential climate variable (ECV) for monitoring the Earth climate system. To ensure accurate retrieval from satellite data, it is important to validate satellite derived LSTs and ensure that they are within the required accuracy and precision thresholds. An emissivity-dependent split-window algorithm with viewing angle dependence and two dual-angle algorithms are proposed for the Sentinel-3 SLSTR sensor. Furthermore, these algorithms are validated together with the Sentinel-3 SLSTR operational LST product as well as several emissivity-dependent split-window algorithms with in-situ data from a rice paddy site. The LST retrieval algorithms were validated over three different land covers: flooded soil, bare soil, and full vegetation cover. Ground measurements were performed with a wide band thermal infrared radiometer at a permanent station. The coefficients of the proposed split-window algorithm were estimated using the Cloudless Land Atmosphere Radiosounding (CLAR) database: for the three surface types an overall systematic uncertainty (median) of −0.4 K and a precision (robust standard deviation) 1.1 K were obtained. For the Sentinel-3A SLSTR operational LST product, a systematic uncertainty of 1.3 K and a precision of 1.3 K were obtained. A first evaluation of the Sentinel-3B SLSTR operational LST product was also performed: systematic uncertainty was 1.5 K and precision 1.2 K. The results obtained over the three land covers found at the rice paddy site show that the emissivity-dependent split-window algorithms, i.e., the ones proposed here as well as previously proposed algorithms without angular dependence, provide more accurate and precise LSTs than the current version of the operational SLSTR product

Enumerative geometry of Calabi-Yau 4-folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.Comment: 44 page

D3-instantons, Mock Theta Series and Twistors

The D-instanton corrected hypermultiplet moduli space of type II string theory compactified on a Calabi-Yau threefold is known in the type IIA picture to be determined in terms of the generalized Donaldson-Thomas invariants, through a twistorial construction. At the same time, in the mirror type IIB picture, and in the limit where only D3-D1-D(-1)-instanton corrections are retained, it should carry an isometric action of the S-duality group SL(2,Z). We prove that this is the case in the one-instanton approximation, by constructing a holomorphic action of SL(2,Z) on the linearized twistor space. Using the modular invariance of the D4-D2-D0 black hole partition function, we show that the standard Darboux coordinates in twistor space have modular anomalies controlled by period integrals of a Siegel-Narain theta series, which can be canceled by a contact transformation generated by a holomorphic mock theta series.Comment: 42 pages; discussion of isometries is amended; misprints correcte