Kontsevich Formula Research Articles

Counting rational curves with an m-fold point

We obtain a recursive formula for the number of rational curves of degree d in CP2, that pass through 3d+1−m generic points and that have an m-fold singular point. The special case of counting curves with a triple point was solved earlier by other authors. We obtain the formula by considering a family version of Kontsevich's recursion formula, in contrast to the excess intersection theoretic approach of others. A large number of low degree cases have been worked out explicitly.

Generalizing Tropical Kontsevich's Formula to Multiple Cross-Ratios

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio.
 This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that, we use a correspondence theorem of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet.
 We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

Multiple zeta values in deformation quantization

Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.

Dirac’s magnetic monopole and the Kontsevich star product

We examine relationships between various quantization schemes for an electrically charged particle in the field of a magnetic monopole. Quantization maps are defined in invariant geometrical terms, appropriate to the case of nontrivial topology, and are constructed for two operator representations. In the first setting, the quantum operators act on the Hilbert space of sections of a nontrivial complex line bundle associated with the Hopf bundle, whereas the second approach uses instead a quaternionic Hilbert module of sections of a trivial quaternionic line bundle. We show that these two quantizations are naturally related by a bundle morphism and, as a consequence, induce the same phase-space star product. We obtain explicit expressions for the integral kernels of star-products corresponding to various operator orderings and calculate their asymptotic expansions up to the third order in the Planck constant . We also show that the differential form of the magnetic Weyl product corresponding to the symmetric ordering agrees completely with the Kontsevich formula for deformation quantization of Poisson structures and can be represented by Kontsevich’s graphs.

Flow Polynomials as Feynman Amplitudes and their $\alpha$-Representation

Let $G$ be a connected graph; denote by $\tau(G)$ the set of its spanning trees. Let $\mathbb F_q$ be a finite field, $s(\alpha,G)=\sum_{T\in\tau(G)} \prod_{e \in E(T)} \alpha_e$, where $\alpha_e\in \mathbb F_q$. Kontsevich conjectured in 1997 that the number of nonzero values of $s(\alpha, G)$ is a polynomial in $q$ for all graphs. This conjecture was disproved by Brosnan and Belkale. In this paper, using the standard technique of the Fourier transformation of Feynman amplitudes, we express the flow polynomial $F_G(q)$ in terms of the "correct" Kontsevich formula. Our formula represents $F_G(q)$ as a linear combination of Legendre symbols of $s(\alpha, H)$ with coefficients $\pm 1/q^{(|V(H)|-1)/2}$, where $H$ is a contracted graph of $G$ depending on $\alpha\in \left(\mathbb F^*_q \right)^{E(G)}$, and $|V(H)|$ is odd.

Abelian Quiver Invariants and Marginal Wall-Crossing

We prove the equivalence of (a slightly modified version of) the wall-crossing formula of Manschot, Pioline and Sen and the wall-crossing formula of Kontsevich and Soibelman. The former involves abelian analogues of the motivic Donaldson-Thomas type invariants of quivers with stability introduced by Kontsevich and Soibelman, for which we derive positivity and geometricity properties.

Membrane sigma-models and quantization of non-geometric flux backgrounds

We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M. Starting from a suitable Courant sigma-model on an open membrane with target space M, regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T^*M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's deformation quantization formula for the twisted Poisson manifolds. For constant R-flux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3-product of fields on R-space. We develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group associated to the T-dual doubled geometry, and hence clarify the relation to the twisted convolution products for topological nonassociative torus bundles. We further demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets.

Equivalence of three wall-crossing formulæ

The wall crossing formula of Kontsevich and Soibelman gives an implicit relation between the BPS indices on two sides of the wall of marginal stability by equating two symplectomorphisms constructed from the indices on two sides of the wall. The wall crossing formulae of Manschot, Pioline and the author give two apparently different explicit expressions for the BPS index on one side of the wall in terms of the BPS indices on the other side. We prove the equivalence of all the three formulae.

Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence

When formulated in twistor space, the D-instanton corrected hypermultiplet moduli space in N=2 string vacua and the Coulomb branch of rigid N=2 gauge theories on $R^3 \times S^1$ are strikingly similar and, to a large extent, dictated by consistency with wall-crossing. We elucidate this similarity by showing that these two spaces are related under a general duality between, on one hand, quaternion-Kahler manifolds with a quaternionic isometry and, on the other hand, hyperkahler manifolds with a rotational isometry, further equipped with a hyperholomorphic circle bundle with a connection. We show that the transition functions of the hyperholomorphic circle bundle relevant for the hypermultiplet moduli space are given by the Rogers dilogarithm function, and that consistency across walls of marginal stability is ensured by the motivic wall-crossing formula of Kontsevich and Soibelman. We illustrate the construction on some simple examples of wall-crossing related to cluster algebras for rank 2 Dynkin quivers. In an appendix we also provide a detailed discussion on the general relation between wall-crossing and the theory of cluster algebras.

Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics

We consider a topological quantum mechanics described by a phase space path integral and study the 1-dimensional analog for the path integral representation of the Kontsevich formula. We see that the naive bosonic integral possesses divergences, that it is even naively non-invariant and thus is ill-defined. We then consider a superextension of the theory which eliminates the divergences and makes the theory naively invariant. This super-extension is equivalent to the correct choice of measure and was discussed in the literature. We then investigate the behavior of this extended theory under diffeomorphisms of the extended phase space and despite of its naive invariance find out that the theory possesses anomaly under nonlinear diffeomorphisms. We localize the origin of the anomaly and calculate the lowest nontrivial anomalous contribution.

Rank two ADHM invariants and wallcrossing

Generalized Donaldson-Thomas invariants corresponding to local D6-D2-D0 configurations are defined applying the formalism of Joyce and Song to ADHM sheaves on curves. A wallcrossing formula for invariants of D6-rank two is proven and shown to agree with the wallcrossing formula of Kontsevich and Soibelman. Using this result, the asymptotic D6-rank two invariants of (-1,-1) and (0,-2) local rational curves are computed in terms of the D6-rank one invariants.

D-instantons and twistors

Finding the exact, quantum corrected metric on the hypermultiplet moduli space in Type II string compactifications on Calabi-Yau threefolds is an outstanding open problem. We address this issue by relating the quaternionic-Kähler metric on the hypermultiplet moduli space to the complex contact geometry on its twistor space. In this framework, Euclidean D-brane instantons are captured by contact transformations between different patches. We derive those by recasting the previously known A-type D2-instanton corrections in the language of contact geometry, covariantizing the result under electro-magnetic duality, and using mirror symmetry. As a result, we are able to express the effects of all D-instantons in Type II compactifications concisely as a sum of dilogarithm functions. We conclude with some comments on the relation to microscopic degeneracies of four-dimensional BPS black holes and to the wall-crossing formula of Kontsevich and Soibelman, and on the form of the yet unknown NS5-brane instanton contributions.

Transferring homotopy commutative algebraic structures

We show that the sum over planar tree formula of Kontsevich and Soibelman transfers C ∞ -structures along a contraction. Applying this result to a cosimplicial commutative algebra A • over a field of characteristic zero, we exhibit a canonical C ∞ -structure on Tot ( A • ) , which is unital if A • is; in particular, we obtain a canonical C ∞ -structure on the cochain complex of a simplicial set.

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov–Witten theory the numbers of complex plane rational curves of degree d through 3 d − 1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.

The Hopf algebra of renormalization, normal coordinates and Kontsevich deformation quantization

Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.

Cyclicity of non-associative products on D-branes

The non-commutative geometry of deformation quantization appears in string theory through the effect of a B-field background on the dynamics of D-branes in the topological limit. For arbitrary backgrounds, associativity of the star product is lost, but only cyclicity is necessary for a description of the effective action in terms of a generalized product. In previous work we showed that this property indeed emerges for a non-associative product that we extracted from open string amplitudes in curved background fields. In the present note we extend our investigation through second order in a complete derivative expansion. We establish cyclicity with respect to the Born--Infeld measure and find a logarithmic correction that modifies the Kontsevich formula in an arbitrary background satisfying the generalized Maxwell equation. This equation is the physical equivalent of a divergence-free non-commutative parameter, which is required for cyclicity already in the associative case.

Quantization of Linear Poisson Structures and Degrees of Maps

Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circe). We explain the geometric meaning of this series as degrees of maps of some grand configuration spaces; the associativity proof is also interpreted in purely homological terms. An interpretation in terms of degrees of maps shows that any other 1-form on the circle also leads to a *-product and allows one to compare these products.

On the Globalization of Kontsevich's Star Product and the Perturbative Poisson Sigma Model

The globalization of Kontsevich's local formula (resp., the perturbative expansion of the Poisson sigma model) is described in down-to-earth terms.

D-branes on group manifolds and deformation quantization

Recently M. Kontsevich found a combinatorial formula defining a star-product of deformation quantization for any Poisson manifold. Kontsevich's formula has been reinterpreted physically as quantum correlation functions of a topological sigma model for open strings as well as in the context of D-branes in flat backgrounds with a Neveu–Schwarz B-field. Here the corresponding Kontsevich's formula for the dual of a Lie algebra is derived in terms of the formalism of D-branes on group manifolds. In particular we show that that formula is encoded at the two-point correlation functions of the Wess–Zumino–Witten effective theory with Dirichlet boundary conditions. The B-field entering in the formalism plays an important role in this derivation.

From local to global deformation quantization of Poisson manifolds

We give an explicit construction of a deformation quantization of the algebra of functions on a Poisson manifolds, based on Kontsevich's local formula. The deformed algebra of functions is realized as the algebra of horizontal sections of a vector bundle with flat connection.