40,948 research outputs found
Topological Representation of Geometric Theories
Using Butz and Moerdijk's topological groupoid representation of a topos with
enough points, a `syntax-semantics' duality for geometric theories is
constructed. The emphasis is on a logical presentation, starting with a
description of the semantical topological groupoid of models and isomorphisms
of a theory and a direct proof that this groupoid represents its classifying
topos. Using this representation, a contravariant adjunction is constructed
between theories and topological groupoids. The restriction of this adjunction
yields a contravariant equivalence between theories with enough models and
semantical groupoids. Technically a variant of the syntax-semantics duality
constructed in [Awodey and Forssell, arXiv:1008.3145v1] for first-order logic,
the construction here works for arbitrary geometric theories and uses a slice
construction on the side of groupoids---reflecting the use of `indexed' models
in the representation theorem---which in several respects simplifies the
construction and allows for an intrinsic characterization of the semantic side.Comment: 32 pages. This is the first pre-print version, the final revised
version can be found at
http://onlinelibrary.wiley.com/doi/10.1002/malq.201100080/abstract (posting
of which is not allowed by Wiley). Changes in v2: updated comment
Mathai-Quillen Formulation of Twisted N=4 Supersymmetric Gauge Theories in Four Dimensions
We present a detailed description of the three inequivalent twists of N=4
supersymmetric gauge theories. The resulting topological quantum field theories
are reobtained in the framework of the Mathai-Quillen formalism and the
corresponding moduli spaces are analyzed. We study their geometric features in
each case. In one of the twists we make contact with the theory of non-abelian
monopoles in the adjoint representation of the gauge group. In another twist we
obtain a topological quantum field theory which is orientation reversal
invariant. For this theory we show how the functional integral contributions to
the vacuum expectation values leading to topological invariants notably
simplify.Comment: 70 pages, macropackage phyzz
Electric-Magnetic duality and the "Loop Representation" in Abelian Gauge Theories
Abelian Gauge Theories are quantized in a geometric representation that
generalizes the Loop Representation and treates electric and magnetic operators
on the same footing. The usual canonical algebra is turned into a topological
algebra of non local operators that resembles the order-disorder dual algebra
of 't Hooft. These dual operators provide a complete description of the
physical phase space of the theories.Comment: 13 pages, LaTex (run twice
Fukaya Categories as Categorical Morse Homology
The Fukaya category of a Weinstein manifold is an intricate symplectic
invariant of high interest in mirror symmetry and geometric representation
theory. This paper informally sketches how, in analogy with Morse homology, the
Fukaya category might result from gluing together Fukaya categories of
Weinstein cells. This can be formalized by a recollement pattern for Lagrangian
branes parallel to that for constructible sheaves. Assuming this structure, we
exhibit the Fukaya category as the global sections of a sheaf on the conic
topology of the Weinstein manifold. This can be viewed as a symplectic analogue
of the well-known algebraic and topological theories of (micro)localization
q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
We characterise the quantum group gauge symmetries underlying q-deformations
of two-dimensional Yang-Mills theory by studying their relationships with the
matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge
theories, together with their refinements and supersymmetric extensions. We
develop uniqueness results for quantum deformations and refinements of gauge
theories in two dimensions, and describe several potential analytic and
geometric realisations of them. We reconstruct standard q-deformed Yang-Mills
amplitudes via gluing rules in the representation category of the quantum group
associated to the gauge group, whose numerical invariants are the usual
characters in the Grothendieck group of the category. We apply this formalism
to compute refinements of q-deformed amplitudes in terms of generalised
characters, and relate them to refined Chern-Simons matrix models and
generalized unitary matrix integrals in the quantum beta-ensemble which compute
refined topological string amplitudes. We also describe applications of our
results to gauge theories in five and seven dimensions, and to the dual
superconformal field theories in four dimensions which descend from the N=(2,0)
six-dimensional superconformal theory.Comment: 71 pages; v2: references added; final version to be published in
Nuclear Physics
Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry
We develop techniques for describing the derived moduli spaces of solutions
to the equations of motion in twists of supersymmetric gauge theories as
derived algebraic stacks. We introduce a holomorphic twist of N=4
supersymmetric gauge theory and compute the derived moduli space. We then
compute the moduli spaces for the Kapustin-Witten topological twists as its
further twists. The resulting spaces for the A- and B-twist are closely related
to the de Rham stack of the moduli space of algebraic bundles and the de Rham
moduli space of flat bundles, respectively. In particular, we find the
unexpected result that the moduli spaces following a topological twist need not
be entirely topological, but can continue to capture subtle algebraic
structures of interest for the geometric Langlands program.Comment: 55 pages; minor correction
Categorification and correlation functions in conformal field theory
A modular tensor category provides the appropriate data for the construction
of a three-dimensional topological field theory. We describe the following
analogue for two-dimensional conformal field theories: a 2-category whose
objects are symmetric special Frobenius algebras in a modular tensor category
and whose morphisms are categories of bimodules. This 2-category provides
sufficient ingredients for constructing all correlation functions of a
two-dimensional rational conformal field theory. The bimodules have the
physical interpretation of chiral data, boundary conditions, and topological
defect lines of this theory.Comment: 16 pages, Invited contribution to the ICM 200
Extended quantum field theory, index theory and the parity anomaly
We use techniques from functorial quantum field theory to provide a geometric
description of the parity anomaly in fermionic systems coupled to background
gauge and gravitational fields on odd-dimensional spacetimes. We give an
explicit construction of a geometric cobordism bicategory which incorporates
general background fields in a stack, and together with the theory of symmetric
monoidal bicategories we use it to provide the concrete forms of invertible
extended quantum field theories which capture anomalies in both the path
integral and Hamiltonian frameworks. Specialising this situation by using the
extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners
due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity
anomaly. We compute explicitly the 2-cocycle of the projective representation
of the gauge symmetry on the quantum state space, which is defined in a
parity-symmetric way by suitably augmenting the standard chiral fermionic Fock
spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that
naturally appear in the index theorem. We describe the significance of our
constructions for the bulk-boundary correspondence in a large class of
time-reversal invariant gauge-gravity symmetry-protected topological phases of
quantum matter with gapless charged boundary fermions, including the standard
topological insulator in 3+1 dimensions.Comment: 63 pages, 3 figures; v2: clarifying comments and references added;
Final version to be published in Communications in Mathematical Physic
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