Differential Forms in Algebraic Topology

Bott, Raoul; Tu, Loring W.

doi:10.1007/978-1-4757-3951-0

Cited by 2,244 publications

(2,602 citation statements)

References 0 publications

Supporting

Mentioning

2,506

Contrasting

Unclassified

Order By: Relevance

“…The localization formula. Recall (see, e.g., [7]) that if f : X → Y is a smooth proper map between connected oriented manifolds such that f restricted to some open subset of X is a diffeomorphism, then for a compactly supported form µ on Y , we have X f * µ = Y µ.…”

Section: 1mentioning

confidence: 99%

Thom polynomials of Morin singularities

Bérczi¹,

Szenes²

2012

Ann. Math.

172

View full text Add to dashboard Cite

We prove a formula for Thom polynomials of A d singularities in any codimension. We use a combination of the test-curve model of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and are computationally effective up to d = 6. IntroductionWe begin with a quick summary of the notions of global singularity theory and the theory of Thom polynomials. For a more detailed review we refer the reader to [1], [14].Consider a holomorphic map f : N → K between two complex manifolds, of dimensions n ≤ k. We say that p ∈ N is a singular point of f if the rank of the differential df p : T p N → T f (p) K is less than n.Topology often forces f to be singular at some points of N , and we will be interested in studying such situations. Before we proceed, we introduce a finer classification of singular points. Choose local coordinates near p ∈ N and f (p) ∈ K, and consider the resulting map-germf p : (C n , 0) → (C k , 0), which may be thought of as a sequence of k power series in n variables without constant terms. The group of infinitesimal local coordinate changes Diff(C k ) × Diff(C n ) acts on the space J (n, k) of all such map-germs. We will call Diff(C k ) × Diff(C n )-orbits or, more generally, Diff(C k ) × Diff(C n )-invariant subsets O ⊂ J (n, k) singularities. For a singularity O and holomorphic f : N → K, we can define the setwhich is independent of any coordinate choices. Then, under some additional technical assumptions (compact N , appropriately chosen closed O, and sufficiently generic f ),is an analytic subvariety of N . The computation of the Poincaré dual class α O [f ] ∈ H * (N, Z) of this set is one of the fundamental problems of global singularity theory. This is indeed useful: for example, A similar result, which we will call Thom's principle, has been used in the holomorphic category (cf. [14], [9] and §2 of the present paper). To formulate it in more concrete terms, denote by C[λ, θ] Sn×S k the space of those polynomials in the variables (λ 1 , . . . , λ n , θ 1 , . . . , θ k ) which are invariant under the permutations of the λs and the permutations of the θs. According to the structure theorem of symmetric polynomials, C[λ, θ] Sn×S k itself is a polynomial ring in the elementary symmetric polynomialsUsing the Chern-Weil map, a polynomial Q ∈ C[λ, θ] Sn×S k and a pair of bundles (E, F ) over N of ranks n and k, respectively, produces a characteristic class Q(E, F ) ∈ H * (N, C). Then the complex variant of Thom's principle reads:A precise version of this statement is described in Section 2. The polynomial Tp O is called the Thom polynomial of O, and the computation of these polynomials is a central problem of singularity theory.The structure of the Diff(C k ) × Diff(C n )-action on J (n, k) is rather complicated; even the parametrization of the orbits is difficult. There is, however, a simple invariant on the space of orbits: to each map-germf : (C n , 0) → (C k , 0), we can associate the finite-dimensional nilpotent algebra Af defin...

show abstract

Section: 1mentioning

confidence: 99%

Thom polynomials of Morin singularities

Bérczi¹,

Szenes²

2012

Ann. Math.

172

View full text Add to dashboard Cite

show abstract

“…Since H is just an S 1 -bundle over X, its cohomology is determined via the Leray spectral sequence [23] in terms of the cohomology of X. As discussed in [5], the homology and cohomology of H are…”

Section: The Geometry and Topology Of Hmentioning

confidence: 99%

Toric duality is Seiberg duality

Beasley¹,

Plesser²

2001

J. High Energy Phys.

148

258

View full text Add to dashboard Cite

We study four N = 1 SU (N ) 6 gauge theories, with bi-fundamental chiral matter and a superpotential. In the infrared, these gauge theories all realize the low-energy worldvolume description of N coincident D3-branes transverse to the complex cone over a del Pezzo surface dP 3 which is the blowup of P 2 at three generic points. Therefore, the four gauge theories are expected to fall into the same universality class-an example of a phenomenon that has been termed "toric duality." However, little independent evidence has been given that such theories are infrared-equivalent.In fact, we show that the four gauge theories are related by the N = 1 duality of Seiberg, vindicating this expectation. We also study holographic aspects of these gauge theories. In particular we relate the spectrum of chiral operators in the gauge theories to wrapped D3-brane states in the AdS dual description. We finally demonstrate that the other known examples of toric duality are related by N = 1 duality, a fact which we conjecture holds generally.

show abstract

“…As shown in Table 10 of [8], for the lattices in which T 2 3 is of B-type the constraints (20) and (21) restrict the wrapping numbers (n a,b k , m a,b k ) mod 2 for a and b of the basis 1-cycles (π 2k−1 , π 2k ) on T 2 k for a and b to be in one of two classes, whereas for lattices in which T 2 3 is of A-type they must be in one of three classes. This makes the search for solutions satisfying (14) much easier in the former case than than in the latter.…”

Section: Introductionmentioning

confidence: 99%

Constructing the supersymmetric Standard Model from intersecting D6-branes on theZ6′orientifold

Bailin

Love

2009

Nuclear Physics B

View full text Add to dashboard Cite

Intersecting stacks of supersymmetric fractional branes on the Z ′ 6 orientifold may be used to construct the supersymmetric Standard Model. If a, b are the stacks that generate the SU (3) colour and SU (2) L gauge particles, then, in order to obtain just the chiral spectrum of the (supersymmetric) Standard Model (with non-zero Yukawa couplings to the Higgs mutiplets), it is necessary that the number of intersections a ∩ b of the stacks a and b, and the number of intersections a ∩ b ′ of a with the orientifold image b ′ of b satisfy (a∩b, a∩b ′ ) = (2, 1) or (1, 2). It is also necessary that there is no matter in symmetric representations of the gauge group, and not too much matter in antisymmetric representations, on either stack. Fractional branes having all of these properties may be constructed on the Z ′ 6 orientifold. We provide a number of new examples having these properties, some of which may be extended to give the Standard Model spectrum. Specifically, we construct four-stack models with two further stacks, each with just a single brane, which have the matter spectrum of the supersymmetric Standard Model, including a single pair of Higgs doublets, plus three rightchiral neutrino singlets. Ramond-Ramond tadpole cancellation is achieved by the introduction of backgroundH 3 flux, the 3-form field strength associated with the Kalb-Ramond 2-form field B 2 . There remains a single unwanted gauged U (1) B−L .

show abstract

Differential Forms in Algebraic Topology

Cited by 2,244 publications

References 0 publications

Thom polynomials of Morin singularities

Thom polynomials of Morin singularities

Toric duality is Seiberg duality

Constructing the supersymmetric Standard Model from intersecting D6-branes on theZ6′orientifold

Contact Info

Product

Resources

About