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50
Elliptic cohomology
 In preparation
"... This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of deri ..."
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Cited by 17 (1 self)
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This paper is an expository account of the relationship between elliptic cohomology and the emerging subject of derived algebraic geometry. We begin in §1 with an overview of the classical theory of elliptic cohomology. In §2 we review the theory of E∞ring spectra and introduce the language of derived algebraic geometry. We apply this theory in §3, where we introduce the notion of an oriented group scheme and describe connection between oriented group schemes and equivariant cohomology theories. In §4 we sketch a proof of our main result, which relates the classical theory of elliptic cohomology to the classification of oriented elliptic curves. In §5 we discuss various applications of these ideas, many of which rely upon a special feature of elliptic cohomology which we call 2equivariance. The theory that we are going to describe lies at the intersection of homotopy theory and algebraic geometry. We have tried to make our exposition accessible to those who are not specialists in algebraic topology; however, we do assume the reader is familiar with the language of algebraic geometry, particularly with the theory of elliptic curves. In order to keep our account readable, we will gloss over many details, particularly where the use of higher category theory is required. A more comprehensive account of the material described here, with complete definitions and proofs, will be given in [21]. In carrying out the work described in this paper, I have benefitted from the ideas of many people. I
Supersymmetric WZW models and twisted Ktheory of SO(3)
, 2004
"... We present an encompassing treatment of D–brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K–t ..."
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Cited by 14 (3 self)
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We present an encompassing treatment of D–brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K–theories and find complete agreement with the CFT analysis of D–brane charges. The K–theoretical computation in particular elucidates some important aspects of N = 1 supersymmetric WZW models on nonsimply connected Lie groups.
Modified Regular Representation OF AFFINE AND VIRASORO ALGEBRAS, VOA STRUCTURE AND SEMIINFINITE COHOMOLOGY.
, 2005
"... We find a counterpart of the classical fact that the regular representation R(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G0 ⊂ G of the Bruhat decomposition is spanned by matrix elements ..."
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Cited by 13 (4 self)
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We find a counterpart of the classical fact that the regular representation R(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G0 ⊂ G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g ⊕ gmodule. The standard regular representation R ( ˆ G) of the affine group ˆ G has two commuting actions of the Lie algebra ˆg with total central charge 0, and carries the structure of a conformal field theory. The modified versions R ′ ( ˆ G) and R ′ ( ˆ G0), originating from the loop version of the Bruhat decomposition, have two commuting ˆgactions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G = SL(2, C). The quantum DrinfeldSokolov reduction transforms the representations of the affine Lie algebras into their Walgebra counterparts, and can be used to produce analogues of the modified regular representations. When g = sl(2, C) the corresponding Walgebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations,
The ring structure for equivariant twisted Ktheory
, 2006
"... We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1: H ∗ (Γ•; A) → H∗−1 ((N ⋊ Γ) •; A) for any crossed module N ..."
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Cited by 13 (0 self)
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We prove, under some mild conditions, that the equivariant twisted Ktheory group of a crossed module admits a ring structure if the twisting 2cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1: H ∗ (Γ•; A) → H∗−1 ((N ⋊ Γ) •; A) for any crossed module N → Γ and prove that any element in the image is ∞multiplicative. As a consequence, we prove that, under some mild conditions, for a crossed module N → Γ and any e ∈ ˇ Z3 (Γ•; S1), that (N) admits a ring structure. As an applithe equivariant twisted Ktheory group K ∗ e,Γ cation, we prove that for a compact, connected and simply connected Lie group G, the equivariant twisted Ktheory group K ∗ [c],G(G) is endowed with a canonical ring structure K i+d [c],G(G)⊗Kj+d [c],G(G) → Ki+j+d [c],G (G), where d = dim G and [c] ∈ H2 ((G⋊G) •; S1).
THOM PROSPECTRA FOR LOOPGROUP REPRESENTATIONS
"... Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of Tspaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new loc ..."
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Cited by 10 (2 self)
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Abstract. This is very much an account of work in progress. We sketch the construction of an Atiyah dual (in the category of Tspaces) for the free loopspace of a manifold; the main technical tool is a kind of Tits building for loop groups, discussed in detail in an appendix. Together with a new localization theorem for Tequivariant Ktheory, this yields a construction of the elliptic genus in the string topology framework of ChasSullivan, CohenJones, Dwyer, Klein, and others. We also show how the Tits building can be used to construct the dualizing spectrum of the loop group. Using a tentative notion of equivariant Ktheory for loop groups, we relate the equivariant Ktheory of the dualizing spectrum to recent work of Freed, Hopkins and Teleman.
Higgs bundles, gauge theories and quantum groups
 Commun. Math. Phys
"... The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding twodimensional topological U(N) gauge theory reproduce quantum wave functions ..."
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The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding twodimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the Nparticle sector. This implies the full equivalence between the above gauge theory and the Nparticle subsector of the quantum theory of Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed
A note on the equality of algebraic and geometric Dbrane charges
 in WZW models.” JHEP 05
, 2004
"... Abstract. The algebraic definition of charges for symmetrypreserving Dbranes in WessZuminoWitten models is shown to coincide with the geometric definition, for all simple Lie groups. The charge group for such branes is computed from the ambiguities inherent in the geometric definition. ..."
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Abstract. The algebraic definition of charges for symmetrypreserving Dbranes in WessZuminoWitten models is shown to coincide with the geometric definition, for all simple Lie groups. The charge group for such branes is computed from the ambiguities inherent in the geometric definition.
STRING TOPOLOGY OF CLASSIFYING SPACES
, 2007
"... Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We ..."
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Cited by 5 (1 self)
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Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG: = map(S 1, BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H∗(LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get on the cohomology H ∗ (LBG) a BValgebra structure.