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Bong H. Lian

Publications and source records attributed to Bong H. Lian.

At least 37 records · Page 2Linked to original sources
arXiv2008

Chiral Equivariant Cohomology II

This is the second in a series of papers on a new equivariant cohomology that takes values in a vertex algebra. In an earlier paper, the first two authors gave a construction of the cohomology functor on the category of O(sg) algebras. The new cohomology theory can be viewed as a kind of "chiralization'' of the classical equivariant cohomology, the latter being defined on the category of G^* algebras a la H. Cartan. In this paper, we further develop the chiral theory by first extending it to allow a much larger class of algebras which we call sg[t] algebras. In the geometrical setting, our principal example of an O(sg) algebra is the chiral de Rham complex Q(M) of a G manifold M. There is an interesting subalgebra of Q(M) which does not admit a full O(sg) algebra structure but retains the structure of an sg[t] algebra, enough for us to define its chiral equivariant cohomology. The latter then turns out to have many surprising features that allows us to delineate a number of interesting geometric aspects of the G manifold M, sometimes in ways that are quite different from the classical theory.

math.DG↗
arXiv2007

Howe Pairs in the Theory of Vertex Algebras

For any vertex algebra V and any subalgebra A of V, there is a new subalgebra of V known as the commutant of A in V. This construction was introduced by Frenkel-Zhu, and is a generalization of an earlier construction due to Kac-Peterson and Goddard-Kent-Olive known as the coset construction. In this paper, we interpret the commutant as a vertex algebra notion of invariant theory. We present an approach to describing commutant algebras in an appropriate category of vertex algebras by reducing the problem to a question in commutative algebra. We give an interesting example of a Howe pair (ie, a pair of mutual commutants) in the vertex algebra setting.

math.RT↗
arXiv2007

A Convexity Theorem and Reduced Delzant Spaces

The convexity theorem of Atiyah and Guillemin-Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar-Lerman proved that the Marsden-Weinstein reduction of a connected Hamitonian $G$-manifold is a stratified symplectic space. Suppose $1\ra A\ra G\ra T\ra 1$ is an exact sequence of compact Lie groups and $T$ is a torus. Then the reduction of a Hamiltonian $G$-manifold with respect to $A$ yields a Hamiltonian $T$-space. We show that if the $A$-moment map is proper, then the convexity theorem holds for such a Hamiltonian $T$-space, even when it is singular. We also prove that if, furthermore, the $T$-space has dimension $2dim T$ and $T$ acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case.

math.DG↗
arXiv2006

Genus Zero Modular Functions

This project was sponsored through the Schiff Fellowship program of Brandeis University. This project involved using the power series method to construct a third order nonlinear ordinary differential equation, a Schwarzian equation, for each of the "genus zero" modular functions, described in the Conway-Norton paper. We first use the Borcherd recursion formuli to generate, in each case, a modular function up to whatever degree we desire, and then use the fact that there is a Schwarzian equation, determined by a single rational function we call a Q-value. By similar power series methods, we compute the coefficients of our rational function, and hence have all the necessary data to create a Schwarzian differential equation for each modular function. This equation can, in turn, be used to recover the modular function itself.

math.NT↗
arXiv2006

Chiral Equivariant Cohomology I

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues.

math.DG↗
arXiv2003

c=2 Rational Toroidal Conformal Field Theories via the Gauss Product

We find a concise relation between the moduli $τ, ρ$ of a rational Narain lattice $Γ(τ,ρ)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

hep-th↗
arXiv2003

Counting Unimodular Lattices in $\R^{r,s}$

Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2,2}$ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2,2}$, a new interpretation of the Gauss product of binary forms brings new light to a number of related objects -- rank 4 rational Narain lattices, over-lattices, rank 2 primitive sublattices of an abstract rank 4 even unimodular lattice $U^2$, and isomorphisms of discriminant groups of rank 2 lattices.

math.QA↗
arXiv2002

Autoequivalences of Derived Category of A K3 Surface and Monodromy Transformations

We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry we also consider the mirror counterpart, i.e. symplectic version of it. In the case of $ρ(X)=1$, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence of the mirror side.

math.AG↗
arXiv2001

The $S^1$ fixed points in Quot-schemes and mirror principle computations

We describe the $S^1$-action on the Quot-scheme $\Quot({\cal E}^n)$ associated to the trivial bundle ${\cal E}^n=CP^1\times{\smallBbb C}^n$. In particlular, the topology of the $S^1$-fixed-point components in $\Quot({\cal E}^n)$ and the $S^1$-weights of the normal bundle of these components are worked out. Mirror Principle, as developed by three of the current authors in the series of work [L-L-Y1, I, II, III, IV], is a method for studying certain intersection numbers on a stable map moduli space. As an application, in Mirror Principle III, Sec 5.4, an outline was given in the case of genus zero with target a flag manifold. The results on $S^1$ fixed points in this paper are used here to do explicit Mirror Principle computations in the case of Grassmannian manifolds. In fact, Mirror Principle computations involve only a certain distinguished subcollection of the $S^1$-fixed-point components. These components are identified and are labelled by Young tableaus. The $S^1$-equivariant Euler class $e_{S^1}$ of the normal bundle to these components is computed. A diagrammatic rule that allows one to write down $e_{S^1}$ directly from the Young tableau is given. From this, the aforementioned intersection numbers on the moduli space of stable maps can be worked out. Two examples are given to illustrate our method. Using our method, the A-model for Calabi-Yau complete intersections in a Grassmannian manifold can now also be computed explicitly. This work is motivated by the intention to provide further details of mirror principle and to understand the relation to physical theory. Some related questions are listed for further study.

math.AG↗
arXiv2000

A Survey of Mirror Principle

This note briefly reviews the {\it Mirror Principle} as developed in the series of papers \LLYI\LLYII\LLYIII\LLYIV\LCHY. We illustrate this theory with a few new examples. One of them gives an intriguing connection to a problem of counting holomorphic disks and annuli. This note has been submitted for the proceedings of the Workshop on Strings, Duality and Geometry the C.R.M. in Montreal of March 2000.

math.AG↗
arXiv2000

Maximal Unipotent Monodromy for Complete Intersection CY Manifolds

The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the moduli space is called a large radius limit (or large complex structure limit). In this paper we are going to construct one parameter families of $n$ dimensional Calabi-Yau manifolds, which are complete intersections in toric varieties and which have a monodromy operator $T$ such that (T$^{N}-id)^{n+1}=0$ but (T$^{N}-id)^{n}\neq0,$ i.e the monodromy operator is maximal unipotent.

math.AG↗
arXiv2000

A reconstruction of Euler data

We apply the mirror principle of [L-L-Y] to reconstruct the Euler data $Q=\{Q_d\}_{d\in{\tinyBbb N}\cup\{0\}}$ associated to a vector bundle $V$ on ${\smallBbb C}{\rm P}^n$ and a multiplicative class $b$. This gives a direct way to compute the intersection number $K_d$ without referring to any other Euler data linked to $Q$. Here $K_d$ is the integral of the cohomology class $b(V_d)$ of the induced bundle $V_d$ on a stable map moduli space. A package '{\tt \verb+EulerData_MP.m+}' in Maple V that carries out the actual computation is provided. For $b$ the Chern polynomial, the computation of $K_1$ for the bundle $V=T_{\ast}{\smallBbb C}{\rm P}^2$, and $K_d$, $d=1,2,3$, for the bundles ${\cal O}_{{\tinyBbb C}{\rm P}^4}(l)$ with $6\le l\le 10$ done using the code are also included.

math.AG↗
arXiv1999

Mirror Principle II

We generalize our theorems in "Mirror Principle I" to a class of balloon manifolds. Many of the results are proved for convex projective manifolds. In a subsequent paper, Mirror Principle III, we will extend the results to projective manifolds without the convexity assumption.

math.AG↗
arXiv1995

From String Backgrounds to Topological Field Theories

The BRST formalism has played a fundamental role in the construction of bosonic closed string backgrounds, ie. the stringy analogs of classical solutions to the field equations of general relativity. The concept of a string background has been extended to the notion of $W$-strings, where the BRST symmetry is still largely conjectural. More recently, the BRST formalism has entered the construction of two dimensional topological conformal quantum field theories, such as those that arise from Calabi-Yau varieties. In this lecture, we focus on common features of the BRST cohomology algebras of string backgrounds and topological field theories. In this context, we present some new evidence for a remarkable relationship that transports us from bosonic and $W$-string backgrounds to the B-model topological conformal field theories associated to certain noncompact Calabi-Yau varieties. This paper will appear in the proceedings of the {\it Symposium on BRS Symmetry} held at RIMS, September 18-22, 1995.

hep-th↗