3,142 research outputs found
Quantum geometry and quantum algorithms
Motivated by algorithmic problems arising in quantum field theories whose
dynamical variables are geometric in nature, we provide a quantum algorithm
that efficiently approximates the colored Jones polynomial. The construction is
based on the complete solution of Chern-Simons topological quantum field theory
and its connection to Wess-Zumino-Witten conformal field theory. The colored
Jones polynomial is expressed as the expectation value of the evolution of the
q-deformed spin-network quantum automaton. A quantum circuit is constructed
capable of simulating the automaton and hence of computing such expectation
value. The latter is efficiently approximated using a standard sampling
procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The
Quantum Universe'' in honor of G. C. Ghirard
N=2 gauge theories, instanton moduli spaces and geometric representation theory
We survey some of the AGT relations between N=2 gauge theories in four
dimensions and geometric representations of symmetry algebras of
two-dimensional conformal field theory on the equivariant cohomology of their
instanton moduli spaces. We treat the cases of gauge theories on both flat
space and ALE spaces in some detail, and with emphasis on the implications
arising from embedding them into supersymmetric theories in six dimensions.
Along the way we construct new toric noncommutative ALE spaces using the
general theory of complex algebraic deformations of toric varieties, and
indicate how to generalise the construction of instanton moduli spaces. We also
compute the equivariant partition functions of topologically twisted
six-dimensional Yang-Mills theory with maximal supersymmetry in a general
Omega-background, and use the construction to obtain novel reductions to
theories in four dimensions.Comment: 55 pages; v2: typos corrected and reference added; Final version to
appear in the Special Issue "Instanton Counting: Moduli Spaces,
Representation Theory and Integrable Systems" of the Journal of Geometry and
Physics, eds. U. Bruzzo and F. Sal
Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties
We use the gauged linear sigma model introduced by Witten to calculate
instanton expansions for correlation functions in topological sigma models with
target space a toric variety or a Calabi--Yau hypersurface .
In the linear model the instanton moduli spaces are relatively simple objects
and the correlators are explicitly computable; moreover, the instantons can be
summed, leading to explicit solutions for both kinds of models. In the case of
smooth , our results reproduce and clarify an algebraic solution of the
model due to Batyrev. In addition, we find an algebraic relation determining
the solution for in terms of that for . Finally, we propose a
modification of the linear model which computes instanton expansions about any
limiting point in the moduli space. In the smooth case this leads to a (second)
algebraic solution of the model. We use this description to prove some
conjectures about mirror symmetry, including the previously conjectured
``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.Comment: 91 pages and 3 figures, harvmac with epsf (Changes in this version:
one minor correction, one clarification, one new reference
Two-dimensional models as testing ground for principles and concepts of local quantum physics
In the past two-dimensional models of QFT have served as theoretical
laboratories for testing new concepts under mathematically controllable
condition. In more recent times low-dimensional models (e.g. chiral models,
factorizing models) often have been treated by special recipes in a way which
sometimes led to a loss of unity of QFT. In the present work I try to
counteract this apartheid tendency by reviewing past results within the setting
of the general principles of QFT. To this I add two new ideas: (1) a modular
interpretation of the chiral model Diff(S)-covariance with a close connection
to the recently formulated local covariance principle for QFT in curved
spacetime and (2) a derivation of the chiral model temperature duality from a
suitable operator formulation of the angular Wick rotation (in analogy to the
Nelson-Symanzik duality in the Ostertwalder-Schrader setting) for rational
chiral theories. The SL(2,Z) modular Verlinde relation is a special case of
this thermal duality and (within the family of rational models) the matrix S
appearing in the thermal duality relation becomes identified with the
statistics character matrix S. The relevant angular Euclideanization'' is done
in the setting of the Tomita-Takesaki modular formalism of operator algebras.
I find it appropriate to dedicate this work to the memory of J. A. Swieca
with whom I shared the interest in two-dimensional models as a testing ground
for QFT for more than one decade.
This is a significantly extended version of an ``Encyclopedia of Mathematical
Physics'' contribution hep-th/0502125.Comment: 55 pages, removal of some typos in section
Quantum automata, braid group and link polynomials
The spin--network quantum simulator model, which essentially encodes the
(quantum deformed) SU(2) Racah--Wigner tensor algebra, is particularly suitable
to address problems arising in low dimensional topology and group theory. In
this combinatorial framework we implement families of finite--states and
discrete--time quantum automata capable of accepting the language generated by
the braid group, and whose transition amplitudes are colored Jones polynomials.
The automaton calculation of the polynomial of (the plat closure of) a link L
on 2N strands at any fixed root of unity is shown to be bounded from above by a
linear function of the number of crossings of the link, on the one hand, and
polynomially bounded in terms of the braid index 2N, on the other. The growth
rate of the time complexity function in terms of the integer k appearing in the
root of unity q can be estimated to be (polynomially) bounded by resorting to
the field theoretical background given by the Chern-Simons theory.Comment: Latex, 36 pages, 11 figure
Quantum 't Hooft operators and S-duality in N=4 super Yang-Mills
We provide a quantum path integral definition of an 't Hooft loop operator,
which inserts a pointlike monopole in a four dimensional gauge theory. We
explicitly compute the expectation value of the circular 't Hooft operators in
N=4 super Yang-Mills with arbitrary gauge group G up to next to leading order
in perturbation theory. We also compute in the strong coupling expansion the
expectation value of the circular Wilson loop operators. The result of the
computation of an 't Hooft loop operator in the weak coupling expansion exactly
reproduces the strong coupling result of the conjectured dual Wilson loop
operator under the action of S-duality. This paper demonstrates - for the first
time - that correlation functions in N=4 super Yang-Mills admit the action of
S-duality.Comment: 38 pages; v2: references added, typos fixe
Mapping Class Groups and Moduli Spaces of Curves
This is a survey paper that also contains some new results. It will appear in
the proceedings of the AMS summer research institute on Algebraic Geometry at
Santa Cruz.Comment: We expanded section 7 and rewrote parts of section 10. We also did
some editing and made some minor corrections. latex2e, 46 page
Loop models and their critical points
Loop models have been widely studied in physics and mathematics, in problems
ranging from polymers to topological quantum computation to Schramm-Loewner
evolution. I present new loop models which have critical points described by
conformal field theories. Examples include both fully-packed and dilute loop
models with critical points described by the superconformal minimal models and
the SU(2)_2 WZW models. The dilute loop models are generalized to include
SU(2)_k models as well.Comment: 20 pages, 15 figure
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