We give a brief summary of algebraic aspects of string theory arising in the noncommutative geometry setting of foliations called string diagrammatics which we introduced jointly with Bob Penner. We furthermore discuss how this gives rise to actions on the Hochschild complex of a Frobenius algebra. We then explain how this leads to new quantum chains for loop spaces and a stabilization in the semi--simple case.
It is known that the physics of open strings on a D2-brane on a two-torus is realized from the viewpoint of deformation quantization in the Seiberg-Witten limit. We study its T-dual theory, i.e. D1-brane physics on two-tori. Such theory is described by Kronecker foliation. The algebra of open strings on the D1-brane is then identified with the crossed product representation of a noncommutative two-torus. The Morita equivalence of noncommutative two-tori is also realized geometrically along this line. As an application, Heisenberg modules and the tensor product between them are discussed from t…
Motivated by the form of the noncommutative *-product in a system of open strings and Dp-branes with constant nonzero Neveu-Schwarz 2-form, we define a deformed multiplication operation on a quasitriangular Hopf algebra in terms of its R-matrix, and comment on some of its properties. We show that the noncommutative string theory *-product is a particular example of this multiplication, and comment on other possible Hopf algebraic properties which may underlie the theory.
Foliated manifolds are particular examples of noncommutative spaces. In this article we try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible relation with the black hole entropy adopting the superstring theory origin of the black hole entropy in the extremal case. This situation we believe has some striking si…
We present some exact non-uniform black string solutions of 5-dimensional pure Einstein gravity as well as Einstein-Maxwell-dilaton theory at arbitrary dilaton coupling. The solutions share the common property that their 4-dimensional slices are Schwarzchild-(anti-)de Sitter spacetimes. The pure gravity solution is also generalized to spacetimes of dimensions higher than 5 to get non-uniform black branes.
We show how a noncommutative phase space appears in a natural way in noncritical string theory, the noncommutative deformation parameter being the string coupling.
The model of the bosonic string with the noncommutative world-sheet geometry is proposed in the framework of Fedosov's deformation quantization. The re-interpretation of the model in terms of bosonic string coupled to infinite multiplet of background fields is given and the link to the W-symmetry is discussed. For the case of d=4 Euclidean target space, the stringy counterparts of the Yang-Mills instantons are constructed in both commutative and noncommutative regimes.
Compactification of Matrix Model on a Noncommutative torus is obtained from strings ending on D-branes with background B field. The BPS spectrum of the system and a novel SL(2,Z) symmetry are discussed.
The noncommutative string theory is described by embedding open string theory in a constant second rank antisymmetric $B_{μν}$ field and the noncommutative gauge theory is defined by a deformed $\star$ product. As a check, study of various scattering amplitudes in both noncommutative string and noncommutative gauge theory confirm that in the $α^{'}\to 0$ limit, the noncommutative string theoretic amplitude goes over to the noncommutative gauge theoretic amplitude, and the couplings are related as $g_{YM}=G_0\sqrt{\frac{1}{2α^{'}}}$. Furthermore we show that in this limit there will not be any …
The tachyon field of p-adic string theory is made noncommutative by replacing ordinary products with noncommutative products in its exact effective action. The same is done for the boundary string field theory, treated as the p -> 1 limit of the p-adic string. Solitonic lumps corresponding to D-branes are obtained for all values of the noncommutative parameter theta. This is in contrast to usual scalar field theories in which the noncommutative solitons do not persist below a critical value of theta. As theta varies from zero to infinity, the solution interpolates smoothly between the soliton …
We study some non-perturbative aspects of noncommutative gauge theories. We find analytic solutions of the equations of motion, for noncommutative U(1) gauge theory, that describe magnetic monopoles with a finite tension string attached. These solutions are non-singular, finite and sourceless. We identify the string with the projection of a D-string ending on a D3-brane in the presence of a constant B-field.
Noncommutative tori are among historically the oldest and by now the most developed examples of noncommutative spaces. Noncommutative Yang-Mills theory can be obtained from string theory. This connection led to a cross-fertilization of research in physics and mathematics on Yang-Mills theory on noncommutative tori. One important result stemming from that work is the link between T-duality in string theory and Morita equivalence of associative algebras. In this article we give an overview of the basic results in differential geometry of noncommutative tori. Yang-Mills theory on noncommutative t…
We examine several aspects of S-duality of four-dimensional noncommutative gauge theory. By making duality transformation explicit, we find that S-dual of noncommutative gauge theory is defined in terms of dual noncommutative deformation. In `magnetic' noncommutative U(1) gauge theory, we show that, in addition to gauge bosons, open D-strings constitute important low-energy excitations: noncritical open D-strings. Upon S-duality, they are mapped to noncritical open F-strings. We propose that, in dual `electric' noncommutative U(1) gauge theory, the latters are identified with gauge-invariant, …
Closed string dynamics in the presence of noncommutative Dp-branes is investigated. In particular, we compute bulk closed string two-point scattering amplitudes; the bulk space-time geometries encoded in the amplitudes are shown to be consistent with the recently proposed background space-time geometries dual to noncommutative Yang-Mills theories. Three-point closed string absorption/emission amplitudes are obtained to show some features of noncommutative Dp-branes, such as modified pole structures and exponential phase factors linearly proportional to the external closed string momentum.
Free bosonic strings in noncommutative spacetime are investigated. The string spectrum is obtained in terms of light-cone quantization. We construct two different models. In the first model the critical dimension is still required to be 26 while only extreme high energy spectrum is modified by noncommutative effect. In the second model, however, the critical dimension is reduced to be less than 26 while low-energy (massless) spectrum only contains degrees of freedom of our four dimensional physics.
We present a method to develop a Hodge theory for tangential cohomology of foliations by mimicing Witten's approach to ordinary Morse theory by perturbations of the Laplacian
The perturbative dynamics of noncommutative field theory (NCFT) is discussed from a point view of string field theory. As in the commutative case it is inevitable to introduce a closed string, which may be described as a bound state of two noncommutative open strings. We point out that the closed string, interacting nontrivially with the open string, plays an essential role in the ultraviolet region. The contribution of the closed string is responsible for the discrepancy between the NCFT and the string field theory. It clarifies the controversial issues associated with the ultraviolet/infrare…
We show how string theory can be used to reproduce the one-loop two-point photon amplitude in noncommutative U(1) gauge theory. Using a simple realization of the gauge theory in bosonic string theory, we extract from a string cylinder computation in the decoupling limit the exact one loop field theory result. The result is obtained entirely from the region of moduli space where massless open strings dominate. Our computation indicates that the unusual IR/UV singularities of noncommutative field theory do not come from closed string modes in any simple way.
The M5-brane is investigated near critical field-strength. We show that this limit on the M5-brane reduces to the noncommutative open string limit on the D4-brane. The reduction on a two-torus leads to both the noncommutative open string limit and the noncommutative Yang-Mills limit on the D3-brane. The decoupled noncommutative five-brane is identified with the strong coupling limit of the noncommutative open string theory on the D4-brane and S-duality on the noncommutative D3-brane is identified with a modular transformation on the five-brane. We argue that the open membrane metric defines a …
We discuss origin of equivalence between noncommutative and ordinary Yang-Mills from point of view of string theory. Working in BRST/Hamiltonian framework first we investigate string model in the decoupling limit and show that change of variables and applying the conversion of constraints of decoupled string theory gives commuting coordinates on the D-brane. Also, we discuss algebra of constraints in general case and show the ways of having commutative coordinates without going to decoupling limit. It could be argued that noncommutative string in B-field is equivalent to the commutative model.…
We construct a vertex operator which describes an emission of the ground-state tachyon of the closed string out of the noncommutative open string. Such a vertex operator is shown to exist only when the momentum of the closed-string tachyon is subject to some constraints coming from the background $B$ field. The vertex operator has a multiplicative coupling constant $g(σ)$ which depends on $σ$ as $g(σ)=\sin^2 σ$ in $0 \le σ\le π$. This behavior is the same as in the ordinary B=0 case.
Considering the scattering of massless open strings attached to a D2-brane living in the $B$ field background, we show that corresponding scattering upto the order of $\a'^2$ is exactly given by the gauge theory on noncommutative background, which is characterized by the Moyal bracket.
We examine the noncommutative cylinder solution to a matrix model with a Minkowski background metric. It can be regarded as the noncommutative analogue of a static circular string. Perturbations about the solution yield a tachyonic scalar field (and an additional tachyonic fermion in the full supersymmetric version of the model) in the commutative limit. The tachyonic mode is attributed to the fact that the circular string is unstable under uniform adiabatic deformations. We obtain a stabilizing term which when added to the matrix model removes the tachyonic mass.
We investigate the dynamics of open membrane boundaries in a constant C-field background. We follow the analysis for open strings in a B-field background, and take some approximations. We find that open membrane boundaries do show noncommutativity in this case by explicit calculations. Membrane boundaries are one dimensional strings, so we face a new type of noncommutativity, that is, noncommutative strings.
Refining previous work by Iso, Kawai and Kitazawa, we discuss bi-local string states as a tool for loop computations in noncommutative field theory and matrix models. Defined in terms of coherent states, they exhibit the stringy features of noncommutative field theory. This leads to a closed form for the 1-loop effective action in position space, capturing the long-range non-local UV/IR mixing for scalar fields. The formalism applies to generic fuzzy spaces. The non-locality is tamed in the maximally supersymmetric IKKT or IIB model, where it gives rise to supergravity. The linearized supergra…
We discuss the breakdown of perturbative unitarity of noncommutative quantum field theories in electric-type background in the light of string theory. We consider the analytic structure of string loop two-point functions using a suitable off-shell continuation and then study the zero slope limit of Seiberg and Witten. In this way we pick up how the unphysical tachyonic branch cut appears in the noncommutative field theory. We briefly discuss discontinuities and cutting rules for the full string theory amplitude and relate them to the noncommutative field theoretical results, and also discuss t…
In this talk, based on work done in collaboration with G. Landi and R.J Szabo, I will review how string theory can be considered as a noncommutative geometry based on an algebra of vertex operators. The spectral triple of strings is introduced, and some of the string symmetries, notably target space duality, are discussed in this framework.
Starting from tree and one-loop tachyon amplitudes of open string theory in the presence of a constant B-field, we explore two problems. First we show that in the noncommutative field theory limit the amplitudes reduce to tree and one-loop diagrams of the noncommutative phi-three theory. Next, we check factorization of the one-loop amplitudes in the long cylinder limit.
We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutati…
The interpretation of closed fundamental strings as solitons in open string field theory is reviewed. Noncommutativity is introduced to facilitate an explicit construction. The tension is computed exactly and the correct spectrum is recovered at long wave length.
In this thesis we will discuss various aspects of noncommutative geometry and compactified Little-String theories. First we will give an introduction to the use of noncommutative geometry in string theory. Thereafter we will present a proof of the connection between D-brane dynamics and noncommutative geometry. Then we will explain the concept of instantons in noncommutative gauge theories. The last chapters shift the focus to Little-String- and $(2,0)$-theories. We study compactifications of these theories on tori with twists. First we study the case of two coinciding branes in detail. Afterw…
We consider the ordinary and noncommutative Dirac-Born-Infeld theories within the open string sigma-model. First, we propose a renormalization scheme, hybrid point splitting regularization, that leads directly to the Seiberg-Witten description including their two-form. We also show how such a form appears within the standard renormalization scheme just by some freedom in changing variables. Second, we propose a Wilson factor which has the noncommutative gauge invariance on the classical level and then compute the sigma-model partition function within one of the known renormalization scheme tha…
This is a summary of a series of papers hep-th/9910263,0005283,0010066 written with B. Chen, T. Matsuo and K. Murakami on a p-p', (p<p^{\prime}) open string with B_{ij} field, which has led us to the explicit identification of the Dp-brane with the noncommutative projector soliton via the gaussian damping factor. A lecture given at Summer Institute 2000, FujiYoshida, Yamanashi, Japan, at August 7-14, 2000.
We propose a model action in 1+1 flat space-time (compact in spatial dimension) embedded in D flat space-time with a non dynamical space-time dependent two vector. For the above constrained system Dirac brackets of suitably defined co-ordinates turn out to be non zero and under specific choice of the two vector the model action reduces to a general action of open string in a noncommutative background where we find the natural embedding of the open string action of in presence of magnetic field on D-brane.
We develop a method in which it is possible to calculate one loop corrections to the noncommutativity parameter for open strings in a background F field. We reproduce known disk level results, and find that annulus corrections give the same structure multiplied by a constant due to the tachyon.
Noncommutative torus compactification of Matrix model is shown to be a direct consequence of quantization of the open strings attached to a D-membrane with a non-vanishing background $B$ field. We calculate the BPS spectrum of such a brane system using both string theory results and DBI action. The DBI action leads to a new transformation property of the compactification radii under the $SL(2,Z)_N$ transformations.
It has been shown recently that by turning on a large noncommutativity parameter, the description of tachyon condensation in string theory can be drastically simplified. We reconsider these issues from the standpoint of string field theory, showing that, from this point of view, the key fact is that in the limit of a large B-field, the string field algebra factors as the product of an algebra that acts on the string center of mass only and an algebra that acts on all other degrees of freedom carried by the string.
We study Witten's open string field theory in the presence of a constant B field. We construct the string field theory in the operator formalism and find that, compared to the ordinary theory with no B field, the vertices in the resulting theory has an additional factor. The factor makes the zero modes of strings noncommutative. This is in agreement with the results in the first-quantized formulation. We also discuss background independence of the purely cubic action derived from the above string field theory and then find a redefinition of string fields to remove the additional factor from th…
We study the perturbative unitarity of noncommutative scalar field theories. Field theories with space-time noncommutativity do not have a unitary S-matrix. Field theories with only space noncommutativity are perturbatively unitary. This can be understood from string theory, since space noncommutative field theories describe a low energy limit of string theory in a background magnetic field. On the other hand, there is no regime in which space-time noncommutative field theory is an appropriate description of string theory. Whenever space-time noncommutative field theory becomes relevant massiv…
In the search for the exact minimum of the tachyon potential in the Witten's cubic string field theory we try to learn as much as possible from the string field theory in the large B-field background. We offer a simple alternative proof of the Witten's factorization, carry out the analysis of string field equations also for the noncommutative torus and find some novel relations to the algebraic K-theory. We note an intriguing relation between Chern-Simons and Chern classes of two noncommutative bundles. Finally we observe a certain pattern which enables us to make a plausible conjecture about …
We try to give a qualitative description of the Godbillon-Vey class and its relation on the one hand to the holonomy and on the other hand to the topological entropy of a foliation, using a remarkable theorem proved recently by G. Duminy (which still remains unpublished), relating these three notions in the case of codim-1 foliations. Moreover we shall investigate its possible consequences on string theory. In particular we shall present a conceptual argument according to which the curvature of the B-field (rank two antisymmetric tensor) of open strings might be related to the Godbillon-Vey cl…
A new Lagrangian description that interpolates between the Nambu--Goto and Polyakov version of interacting strings is given. Certain essential modifications in the Poission bracket structure of this interpolating theory generates noncommutativity among the string coordinates for both free and interacting strings. The noncommutativity is shown to be a direct consequence of the nontrivial boundary conditions. A thorough analysis of the gauge symmetry is presented taking into account the new modified constraint algebra, which follows from the noncommutative structures and finally a smooth corresp…
We study the mass and different RR charge distributions of the BPS (p,p-2)-brane bound states in the closed string brane-like $σ$-model. We show that such brane bound states can be realized by introducing a constant B field in the closed string theory. In addition we show that the worldvolume coordinates of these brane bound states turn out to be noncommutative.
We investigate new compactifications of OM theory giving rise to a 3+1 dimensional open string theory with noncommutative $x^0$-$x^1$ and $x^2$-$x^3$ coordinates. The theory can be directly obtained by starting with a D3 brane with parallel (near critical) electric and magnetic field components, in the presence of a RR scalar field. The magnetic parameter permits to interpolate continuously between the $x^0$-$x^1$ noncommutative open string theory and the $x^2$-$x^3$ spatial noncommutative U(N) super Yang-Mills theory. We discuss $SL(2,Z)$ transformations of this theory. Using the supergravity…
We review the quantization of open string in NS-NS background and demonstrate that its endpoint becomes noncommutative. The same approach allows us to determine the noncommutativity that arises for a charged open string in background gauge fields. While NS-NS background is relevant for ``worldvolume'' noncommutativity, a simple argument suggests that RR background is likely to be relevant for ``spacetime'' noncommutativity.
We study the renormalization of non-commutative gauge theories with matter. As in the scalar field theory cases, there are logarithmic infrared divergences resulting from integrating out high momentum modes. In order to reproduce the correct infrared behaviour, the Wilsonian effective action has to include certain ''closed string`` modes with prescribed couplings. In the case of quiver gauge theories, realized in string theory on orbifolds, we identify the required modes with a set of twisted sector fields. These closed string modes have exactly the prescribed couplings to correct the Wilsonia…
The recently found non-critical open string theories is reviewed. These open strings, noncommutative open string theories (NCOS), arise as consistent quantum theories describing the low energy theory of D-branes in a background electric B-field in the critical limit. Focusing on the D3-brane case, we construct the most general (3+1) NCOS, which is described by four parameters. We study S and T -dualities of these theories and argue the existence of a U-duality group.
In a previous paper (hep-th/0509067) using matrix model, we showed that closed string tachyons can resolve spacelike singularity in one particular class of Misner space (with anti-periodic boundary conditions for fermions around the spatial circle). In this note, we show that for Misner space without closed string tachyons, there also exists a mechanism to resolve the singularity in the context of the matrix model, namely cosmological winding string production. We show that here space and time also become noncommutative due to these winding strings. Employing optical theorem, we study the bulk…
In this paper we study the phenomenon of UV/IR mixing in noncommutative field theories from the point of view of world-sheet open-closed duality in string theory. New infrared divergences in noncommutative field theories arise as a result of integrating over high momentum modes in the loops. These are believed to come from integrating out additional bulk closed string modes. We analyse this issue in detail for the bosonic theory and further for the supersymmetric theory on the $C^2/Z_2$ orbifold. We elucidate on the exact role played by the constant background $B$-field in this correspondence.
Motivated by recent discussions of IR/UV mixing in noncommutative field theories, we perform a detailed analysis of the non-planar amplitudes of the bosonic open string in the presence of an external B-field at the one-loop level. We carefully isolate, at the string theory level, the contribution which is responsible for the IR/UV behavior in the field theory limit. We show that it is a pure open string effect by deriving it from the factorization of the one-loop amplitude into the disk amplitudes of intermediate open string insertions. We suggest that it is natural to understand IR/UV mixing …
The `winding state' behavior appears in the two-loop nonplanar contribution to the partition function in thermal noncommutative field theories. We derive this feature directly from the purely open string theory analysis in the presence of the constant background $B$-field; we compute the two-loop partition function for worldsheets with a handle and a boundary when the time direction of the Euclideanized target space is compactified. In contrast to the closed-string-inspired approach, it is not necessary to add infinite number of extra degrees of freedom. Furthermore, we find a piece of support…
We explicitly evaluate one-loop (annulus) planar and nonplanar open string amplitudes in the presence of the background NS-NS two-form field. In the decoupling limit of Seiberg and Witten, we find that the nonplanar string amplitudes reproduce the UV/IR mixing of noncommutative field theories. In particular, the investigation of the UV regime of the open string amplitudes shows that certain IR closed string degrees of freedom survive the decoupling limit as previously predicted from the noncommutative field theory analysis. These degrees of freedom are responsible for the quadratic, linear and…
The background independent formulation of the gauge theories on D-branes in flat space-time is considered, some examples of the solutions of their equations of motion are presented, the solutions of Dirac equation in these backgrounds are analyzed, and the generalizations to the curved spaces, like orbifolds, conifolds, and K3 surfaces, are discussed.
We perform canonical quantization of open strings in the $D$-brane background with a $B$-field. Treating the mixed boundary condition as a primary constraint, we get a set of secondary constraints. Then these constraints are shown to be equivalent to orbifold conditions to be imposed on normal string modes. These orbifold conditions are a generalization of the familiar orbifold conditions which arise when we describe open strings in terms of closed strings. Solving the constraints explicitly, we obtain a simple Hamiltonian for the open string, which reveals the nature of noncommutativity trans…
We analyze the tachyon field in the bosonic open string theory in a constant B-field background using the background independent open string field theory. We show that in the large noncommutativity limit the action of tachyon field is given exactly by the potential term which has the same form as in the case without B-field but the product of tachyon field is taken to be the star product.
We construct six- and four-dimensional toroidal compactifications of the Type I string with magnetic flux on the D-branes. The open strings in this background probe a noncommutative internal geometry. Phenomenologically appealing features such as chiral fermions and supersymmetry breaking in the gauge sector are naturally realized by these vacua. We investigate the spectra of such noncommutative string compactifications and in a bottom-up approach discuss the possibility to obtain the standard or some GUT like model.
Noncommutative ϕ^3 field theory in six dimensions exhibits the logarithmic UV/IR mixing at the two-loop order. We show that open string theory in the presence of constant background NS-NS two-form field yields the same amplitude upon taking a decoupling limit. The stretched string picture proposed on the basis of one-loop analysis naturally generalizes to the two-loop amplitudes in consideration. Our string theory formulation can incorporate the closed string insertions as well as open string insertions. Furthermore, the analysis of the world-sheet partition function and propagators can be str…
We analyze open and mixed sector tree-level amplitudes of N=2 strings in a space-time with (2,2) signature, in the presence of constant B field. The expected topological nature of string amplitudes in the open sector is shown to impose nontrivial constraints on the corresponding noncommutative field theory. In the mixed sector, we first compute a 3-point function and show that the corresponding field theory is written in terms of a generalized *-product. We also analyze a 4-point function (A_{oooc}) of the mixed sector in Theta ---> infinity limit.
We identify the effective string scale of noncommutative Yang-Mills theory (NCYM) with the noncommutativity scale through its dual supergravity description. We argue that Newton's force law may be obtained with 4 dimensional NCYM with maximal SUSY. It provides a nonperturbative compactification mechanism of IIB matrix model. We can associate NCYM with the von Neumann lattice by the bi-local representation. We argue that it is superstring theory on the von Neumann lattice. We show that our identification of its effective string scale is consistent with exact T-duality (Morita equivalence) of NC…
The open string ending on a D-brane with a constant B-field in a pp-wave Ramond-Ramond background is exactly solvable. The theory is controlled by three dimensionful parameters: alpha', the mass parameter (RR background times the lightcone momentum) and the B-field. We quantize the open string theory and determine the full noncommutative structure. In particular, we find a fully noncommutative phase space whose noncommutativity depends on all these parameters. The lightcone Hamiltionian is obtained, and as a consequence of the nontrivial commutation relations of the theory, new features of the…
In this lecture notes we explain and discuss some ideas concerning noncommutative geometry in general, as well as noncommutative field theories and string field theories. We consider noncommutative quantum field theories emphasizing an issue of their renormalizability and the UV/IR mixing. Sen's conjectures on open string tachyon condensation and their application to the D-brane physics have led to wide investigations of the covariant string field theory proposed by Witten about 15 years ago. We review main ingredients of cubic (super)string field theories using various formulations: functiona…
