Results 1  10
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16
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
CFT exercises for the needs of AGT
, 908
"... An explicit check of the AGT relation between the WNsymmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for Walgebra descendants. We collect simplest expressions of this type for N = 3 and for the two lowest de ..."
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An explicit check of the AGT relation between the WNsymmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for Walgebra descendants. We collect simplest expressions of this type for N = 3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 4 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
On the correlation functions of the vector bundle generalization of the bcsystem
 J. Math. Phys
"... ..."
MaxPlanckInstitut für Mathematik in den Naturwissenschaften Leipzig
, 2015
"... Object oriented models vs. data analysis is this the right alternative? by ..."
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Object oriented models vs. data analysis is this the right alternative? by
The Power of Nekrasov Functions
, 908
"... The recent AGT suggestion [1] to use the set of Nekrasov functions [2] as a basis for a linear decomposition of generic conformal blocks works very well not only in the case of Virasoro symmetry, but also for conformal theories with extended chiral algebra. This is rather natural, because Nekrasov f ..."
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The recent AGT suggestion [1] to use the set of Nekrasov functions [2] as a basis for a linear decomposition of generic conformal blocks works very well not only in the case of Virasoro symmetry, but also for conformal theories with extended chiral algebra. This is rather natural, because Nekrasov functions are introduced as expansion basis for generalized hypergeometric integrals, very similar to those which arise in expansion of DotsenkoFateev integrals in powers of alphaparameters. Thus, the AGT conjecture is closely related to the old belief that conformal theory can be effectively described in the free field formalism, and it can actually be a key to clear formulating and proof this longstanding hypothesis. As an application of this kind of reasoning we use knowledge of the exact hypergeometric conformal block for complete proof of the AGT relation for a restricted class of external states. 1
On coalescence of fermions on Riemann surfaces
, 2000
"... We consider coalescing fermions on a Riemann surface and derive generalized determinant formulas, complementing some results of [3]. Possible applications are indicated. 1 Introduction Recall [1] that the 2Npoint function h Q N i=1 b(z i ) Q N i=1 c(y i )i for the fermionic fields b; c on t ..."
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We consider coalescing fermions on a Riemann surface and derive generalized determinant formulas, complementing some results of [3]. Possible applications are indicated. 1 Introduction Recall [1] that the 2Npoint function h Q N i=1 b(z i ) Q N i=1 c(y i )i for the fermionic fields b; c on the Riemann sphere (i.e. Riemann surface of genus g = 0) is determined by the operator product expansions to Q i!j (z i \Gamma z j ) Q i!j (y j \Gamma y i ) Q N i;j=1 (z i \Gamma y j ) \Gamma1 . Since the bcsystem is a free system we may also use Wick's theorem to obtain det( 1 z i \Gammay j ) N i;j=1 , where hb(z)c(y)i = 1 z\Gammay . Comparing the two expressions gives the Cauchy identity Q i!j (z i \Gamma z j ) Q i!j (y j \Gamma y i ) Q N i;j=1 (z i \Gamma y j ) = det / 1 z i \Gamma y j ! N i;j=1 : (1) It was considered in [2, 3] what happens when two (or more) fields coalesce, e.g. z i ! z j . Considering the fields b; c as being charged (with opposite charge),...
unknown title
, 2007
"... Towards physically motivated proofs of the Poincaré and the geometrization conjectures ..."
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Towards physically motivated proofs of the Poincaré and the geometrization conjectures
unknown title
, 2007
"... Towards physically motivated proofs of the Poincaré and the geometrization conjectures ..."
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Towards physically motivated proofs of the Poincaré and the geometrization conjectures