arXiv subjects
Publications and source records attributed to A. Losev.
The modern usage of the words astronomy and astrology is traced back to distinctions, largely ignored in recent scholarship. Three interpretations of celestial phenomena (in a geometric, a substantialist and a prognostic versions) coexisted during the Hellenistic period. From Plato to Isidore of Seville, semantic changes are evidenced and their later development is sketched.
The present paper is the second part of our project in which we describe quantum field theories with instantons in a novel way by using the "infinite radius limit" (rather than the limit of free field theory) as the starting point. The theory dramatically simplifies in this limit, because the correlation functions of all, not only topological (or BPS), observables may be computed explicitly in terms of integrals over finite-dimensional moduli spaces of instanton configurations. In Part I (arXiv:hep-th/0610149) we discussed in detail the one-dimensional (that is, quantum mechanical) models of this type. Here we analyze the supersymmetric two-dimensional sigma models and four-dimensional Yang--Mills theory, using the one-dimensional models as a prototype. We go beyond the topological (or BPS) sectors of these models and consider them as full-fledged quantum field theories. We study in detail the space of states and find that the Hamiltonian is not diagonalizable, but has Jordan blocks. This leads to the appearance of logarithms in the correlation functions. We find that our theories are in fact logarithmic conformal field theories (theories of this type are of interest in condensed matter physics). We define jet-evaluation observables and consider in detail their correlation functions. They are given by integrals over the moduli spaces of holomorphic maps, which generalize the Gromov--Witten invariants. These integrals generally diverge and require regularization, leading to an intricate logarithmic mixing of the operators of the sigma model. A similar structure arises in the four-dimensional Yang--Mills theory as well.
A new kind of aperiodic tiling is introduced. It is shown to underlie a structure obtained as a superposition of waves with incommensurate periods. Its connections to other other tilings and quasicrystals are discussed.
A recursive scheme relying on decagons is used to generate Penrose-like sublattices or tilings. Its relevance for understanding structures with non-crystallographic symmetry is discussed.
We propose a Hodge field theory construction that captures algebraic properties of the reduction of Zwiebach invariants to Gromov-Witten invariants. It generalizes the Barannikov-Kontsevich construction to the case of higher genera correlators with gravitational descendants. We prove the main theorem stating that algebraically defined Hodge field theory correlators satisfy all tautological relations. From this perspective the statement that Barannikov-Kontsevich construction provides a solution of the WDVV equation looks as the simplest particular case of our theorem. Also it generalizes the particular cases of other low-genera tautological relations proven in our earlier works; we replace the old technical proofs by a novel conceptual proof.
This is a brief summary of our studies of quantum field theories in a special limit in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full-fledged quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. In particular, in dimensions two and four our theories are logarithmic conformal field theories.
Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.
We introduce the notion of Zwiebach invariants that generalize Gromov-Witten invariants and homotopical algebra structures. We outline the induction procedure that induces the structure of Zwiebach on the subbicomplex, that gives the structure of Gromov-Witten invariants on subbicomplex with zero diffferentials. We propose to treat Hodge dGBV with 1/12 axiom as the simplest set of Zwiebach invariants, and explicitely prove that it induces WDVV and Getzler equations in genera 0 and 1 respectively.
We consider supersymmetric sigma models on the Kahler target spaces, with twisted mass. The Kahler spaces are assumed to have holomorphic Killing vectors. Introduction of a superpotential of a special type is known to be consistent with N=2 superalgebra (Alvarez-Gaume and Freedman). We show that the algebra acquires central charges in the anticommutators {Q_L, Q_L} and {Q_R, Q_R}. These central charges have no parallels, and they can exist only in two dimensions. The central extension of the N=2 superalgebra we found paves the way to a novel phenomenon -- spontaneous breaking of a part of supersymmetry. In the general case 1/2 of supersymmetry is spontaneously broken (the vacuum energy density is positive), while the remaining 1/2 is realized linearly. In the model at hand the standard fermion number is not defined, so that the Witten index as well as the Cecotti-Fendley-Intriligator-Vafa index are useless. We show how to construct an index for counting short multiplets in internal algebraic terms which is well-defined in spite of the absence of the standard fermion number. Finally, we outline derivation of the quantum anomaly in {\bar Q_L, Q_R}.
This paper is a sequel to [LoMa] where moduli spaces of painted stable curves were introduced and studied. We define the extended modular operad of genus zero, algebras over this operad, and study the formal differential geometric structures related to these algebras: pencils of flat connections and Frobenius manifolds without metric. We focus here on the combinatorial aspects of the picture. Algebraic geometric aspects are treated in [Ma2].
We consider multiplet shortening for BPS solitons in N=1 two-dimensional models. Examples of the single-state multiplets were established previously in N=1 Landau-Ginzburg models. The shortening comes at a price of loosing the fermion parity $(-1)^F$ due to boundary effects. This implies the disappearance of the boson-fermion classification resulting in abnormal statistics. We discuss an appropriate index that counts such short multiplets. A broad class of hybrid models which extend the Landau-Ginzburg models to include a nonflat metric on the target space is considered. Our index turns out to be related to the index of the Dirac operator on the soliton reduced moduli space (the moduli space is reduced by factoring out the translational modulus). The index vanishes in most cases implying the absence of shortening. In particular, it vanishes when there are only two critical points on the compact target space and the reduced moduli space has nonvanishing dimension. We also generalize the anomaly in the central charge to take into account the target space metric.
A recently proposed convolution technique for the calculation of local density of states is described more thouroughly and new results of its application are presented. For separable systems the exposed method allows to construct the ldos for a higher dimensionality out of lower dimensional parts. Some practical and theoretical aspects of this approach are also discussed.
We study realization of the target space diffeomorphisms in the type $C$ topological string. We found that the charges, which generate transformations of the boundary observables, form an algebra, which differs from that of bulk charges by the contribution of the bubbled disks. We discuss applications to noncommutative field theories.
We discuss an N=2 quantum mechanics with or without a central charge. A representation is constructed with the number of bosonic degrees of freedom less that one half of the fermionic degrees of freedom. We suggest a systematic method of reducing the bosonic degrees of freedom called ``dynamical reduction." Our consideration opens a problem of a general classification of nonstandard representations of N=2$ superalgebra.
We consider multiplet shortening for BPS solitons in N=1 two-dimensional models. Examples of the single-state multiplets were established previously in N=1 Landau-Ginzburg models. The shortening comes at a price of loosing the fermion parity $(-1)^F$ due to boundary effects. This implies the disappearance of the boson-fermion classification resulting in abnormal statistics. To count such short multiplets we introduce a new index. We consider the phenomenon of shortening in a broad class of hybrid models which extend the Landau-Ginzburg models to include a nonflat metric on the target space. Our index turns out to be related to the index of the Dirac operator on the soliton moduli space. The latter vanishes in most cases implying the absence of shortening. We also generalize the anomaly in the central charge to take into account the target space metric.
It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad $(H_*(\bar{M}_{0,n+1}))$ of the moduli spaces of $n$--pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series $\bar{L}_n$ of pointed stable curves of genus zero. Whereas $\bar{M}_{0,n+1}$ parametrizes trees of $\bold{P}^1$'s with pairwise distinct nonsingular marked points, $\bar{L}_n$ parametrizes strings of $\bold{P}^1$'s stabilized by marked points of two types. The union of all $\bar{L}_n$'s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union.
Field theory with instantons can be partially regularized by adding degrees of freedom at some scale. These extra degrees of freedom lead to the appearence of the new topological defects. These defects which we call freckles have some characteristic size depending on the scale at which the extra degrees of freedom revive. The examples of two dimensional sigma model, four dimensional gauge theory are studied. The compactification of the four dimensional supersymmetric gauge theory down to two dimensions is also considered and the new phenomena are found.
We discuss topologically stable solitons in two-dimensional theories with the extended supersymmetry assuming that the spatial coordinate is compact. This problem arises in the consideration of the domain walls in the popular theories with compactified extra dimensions. Contrary to naive expectations, it is shown that the solitons on the cylinder can be BPS saturated. In the case of one chiral superfield, a complete theory of the BPS saturated solitons is worked out. We describe the classical solutions of the BPS equations. Depending on the choice of the Kahler metric, the number of such solutions can be arbitrarily large. Although the property of the BPS saturation is preserved order by order in perturbation theory, nonperturbative effects eliminate the majority of the classical BPS states upon passing to the quantum level. The number of the quantum BPS states is found. It is shown that the N=2 field theory includes an auxiliary N=1 quantum mechanics, Witten's index of which counts the number of the BPS particles.