The purpose of this note is to provide a short invitation to the universal algebraic approach to topological string theory. In the first section we make an attempt to explain the origin of this approach and how it fits into the bigger picture of full string theory, while in the second half of this note we will introduce the relevant notions in more detail and discuss some of our main results on bulk-deformed open topological string amplitudes.
It is hoped that these lectures will give a point of entry into that vast web of related ideas that go under the name "string theory". I start with a more or less qualitative introduction to gravity as a field theory and sketch how one might try to quantize it. Quantizing gravity using the usual techniques of field theory will turn out to be unsuccessful, and that will be a motivation for trying an indirect approach: string theory. I present bosonic string theory, quantize the closed string in the light cone gauge and show that one of the states in the closed string Hilbert space can be interp…
Poisson-Lie (PL) T-duality has received much attention over the last five years in connection with integrable string worldsheet theories. At the level of the worldsheet, the algebraic structure underpinning these connections is made manifest with the $\mathcal{E}$-model, a first order Hamiltonian description of the string. The $\mathcal{E}$-model shares many similarities with Double Field Theory (DFT). We report on recent progress in establishing a precise linkage with DFT as the target space description of the $\mathcal{E}$-model. There are three important outcomes of this endeavor: 1) PL sym…
In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge '2-group'. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group…
For the homogeneous configuration given by the long string limit of the folded string with a spin in AdS3 and a spin and a winding number in S1, we solve the auxiliary linear problem in the finite-gap method and construct the Lax operator to obtain an algebraic curve. We show that the long spiky string in AdS_3 has the same algebraic curve as the null cusp Wilson loop. The algebraic curve for the circular winding string in AdS3xS1 is evaluated. The Virasoro constraint is discussed to characterize the branch points of each curve.
We apply stochastic quantization method to Kostov's matrix-vector models for the second quantization of orientable strings with Chan-Paton like factors, including both open and closed strings. The Fokker-Planck hamiltonian deduces an orientable open-closed string field theory at the double scaling limit. There appears an algebraic structure in the continuum F-P hamiltonian including a Virasoro algebra and a SU(r) current algebra.
We apply stochastic quantization method to real symmetric matrix-vector models for the second quantization of non-orientable strings, including both open and closed strings. The Fokker-Planck hamiltonian deduces a well-defined non-orientable open-closed string field theory at the double scaling limit of the matrix model. There appears a new algebraic structure in the continuum F-P hamiltonian including a Virasoro algebra and a $SO(r)$ current algebra.
There have been recent advances in the construction of algebraic curves for certain classes of string solutions in the context of the AdS/CFT correspondence. In this paper we obtain the Lax operators and associated spectral curves for circular and folded string solutions in $\,\textrm{AdS}_5\times S^5$. In addition, we provide an original approach for the reconstruction of string solutions in $\,S^3$ from their corresponding curves.
We give a brief introduction to the study of the algebraic structures -- and their geometrical interpretations -- which arise in the BRST construction of a conformal string background. Starting from the chiral algebra $\cA$ of a string background, we consider a number of elementary but universal operations on the chiral algebra. From these operations we deduce a certain fundamental odd Poisson structure, known as a Gerstenhaber algebra, on the BRST cohomology of $\cA$. For the 2D string background, the correponding G-algebra can be partially described in term of a geometrical G-algebra of the …
We show how to construct an algebraic curve for factorized string solution in the context of the AdS/CFT correspondence. We define factorized solutions to be solutions where the flat-connection becomes independent of one of the worldsheet variables by a similarity transformation with a matrix $S$ satisfying $S^{-1}d S=const$. Using the factorization property we construct a well defined Lax operator and an associated algebraic curve. The construction procedure is local and does not require the introduction of a monodromy matrix. The procedure can be applied for string solutions with any boundar…
We investigate algebraic structure of light-cone string field theory which respects the spacetime supersymmetry su(2|2). Extracting building blocks from the explicit oscillator expressions of light-cone superstring field theory on the pp-wave and the flat background, we propose a model for more generic backgrounds by giving an algebraic procedure to construct supercharges and Hamiltonian without referring to explicit oscillator expressions. Based on this strategy, we systematically construct interaction terms of light-cone string field theory on the pp-wave background and some examples of its …
The special geometry ($(t,{\bar t})$-equations) for twisted $N=2$ strings are derived as consistency conditions of a new contact term algebra. The dilaton appears in the contact terms of topological and antitopological operators. The holomorphic anomaly, which can be interpreted as measuring the background dependence, is obtained from the contact algebra relations.
Many Ramond-Ramond backgrounds which arise in the AdS/CFT correspondence are described by integrable sigma-models. The equations of motion for classical spinning strings in these backgrounds are exactly solvable by finite-gap integration techniques. We review the finite-gap integral equations and algebraic curves for coset sigma-models, and then apply the results to the AdS(d+1) backgrounds with d=4,3,2,1.
Loop amplitudes in (p,q) minimal string theory are studied in terms of the continuum string field theory based on the free fermion realization of the KP hierarchy. We derive the Schwinger-Dyson equations for FZZT disk amplitudes directly from the W_{1+\infty} constraints in the string field formulation and give explicitly the algebraic curves of disk amplitudes for general backgrounds. We further give annulus amplitudes of FZZT-FZZT, FZZT-ZZ and ZZ-ZZ branes, generalizing our previous D-instanton calculus from the minimal unitary series (p,p+1) to general (p,q) series. We also give a detailed …
In this short talk we review basic ideas of string field theory with the emphasis on the recent developments. We show how without too much technicality one can look for analytic solutions to Witten's open string field theory. This is an expanded version of a talk given by the author over the last year at a number of occasions and notably at the conference "Selected Topics in Mathematical and Particle Physics" in honor of Prof. Jiri Niederle's 70th birthday.
We formulate a linear difference equation which yields averaged semi-inclusive decay rates for arbitrary, not necessarily large, values of the masses. We show that the rates for decays $M \to m+\M'$ of typical heavy open strings are independent of the masses $M$ and $m$, and compute the ``mass deffect''$M-m-M'$. For closed strings we find decay rates proportional to $M m_{R}^{(1-D)/2}$, where $m_{R}$ is the reduced mass of the decy products. Our method yields exact interaction rates valid for all mass ranges and may provide a fully microscopic basis, not limited to the long string approximatio…
We survey some algebraic geometric aspects of mirror symmetry and duality in string theory. Some applications of computer algebra to algebraic geometry and string theory are shortly reviewed.
A careful treatment of closed string BRST cohomology shows that there are more discrete states and associated symmetries in $D=2$ string theory than has been recognized hitherto. The full structure, at the $SU(2)$ radius, has a natural description in terms of abelian gauge theory on a certain three dimensional cone $Q$. We describe precisely how symmetry currents are constructed from the discrete states, explaining the role of the ``descent equations.'' In the uncompactified theory, we compute the action of the symmetries on the tachyon field, and isolate the features that lead to nonlinear te…
The propagation differential for bosonic strings on a complex torus with three symmetric punctures is investigated. We study deformation aspects between two point and three point differentials as well as the behaviour of the corresponding Krichever-Novikov algebras. The structure constants are calculated and from this we derive a central extension of the Krichever-Novikov algebras by means of b-c systems. The defining cocycle for this central extension deforms to the well known Virasoro cocycle for certain kinds of degenerations of the torus. AMS subject classification (1991): 17B66, 17B90, 14…
We establish explicitely the relation between the algebraic and Nambu-Goto strings when the target space is a four dimensional flat space. We find that the two theories are exactly equivalent only when the algebraic string is restricted to the self-dual or anti self-dual sectors. In its Hamiltonian formulation, the algebraic string defines a constrained system with first and second class constraints. In the self-dual case, we exhibit the appropriate set of second class constraints such that the resulting physical phase space is formulated in the same way as it is in the standard Nambu-Goto str…
We investigate the algebraic curve for string in $Sch_5\times S^5$. We compute the semiclassical spectrum for BMN string in $Sch_5\times S^5$ from the algebraic curve. We compare our results with the anomalous dimensions in $sl(2)$ sector of the null dipole deformation of $\mathcal{N} = 4$ super Yang-Mills theory.
We study generic properties of string theory effective actions obtained by classically integrating out massive excitations from string field theories based on cyclic homotopy algebras of $A_\infty$ or $L_\infty$ type. We construct observables in the UV theory and we discuss their fate after integration-out. Furthermore, we discuss how to compose two subsequent integrations of degrees of freedom (horizontal composition) and how to integrate out degrees of freedom after deforming the UV theory with a new consistent interaction (vertical decomposition). We then apply our general results to the op…
I consider an algebraic construction of creation and annihilation operators for superstring and p-brane parton models. The result can be interpreted as a realisation of multiple quantisation and suggests a relationship between quantisation and dimension. The most general algebraic form of quantisation may eventually be expressed in the language of category theory.
The algebraic curve (finite-gap) classification of rotating string solutions was very important in the development of integrability through comparison with analogous structures at weak coupling. The classification was based on the analysis of monodromy around the closed string cylinder. In this paper we show that certain classical Wilson loop minimal surfaces corresponding to the null cusp and qqbar potential with trivial monodromy can, nevertheless, be described by appropriate algebraic curves. We also show how a correlation function of a circular Wilson loop with a local operator fits into t…
The algebraic curve for the psu (2,2|4) quantum spin chain is determined from the thermodynamic limit of the algebraic Bethe ansatz. The Hamiltonian of this spin chain has been identified with the planar 1-loop dilatation operator of N=4 SYM. In the dual AdS_5 x S^5 string theory, various properties of the data defining the curve for the gauge theory are compared to the ones obtained from semiclassical spinning-string configurations, in particular for the case of strings on AdS_5 x S^1 and the su(2,2) spin chain agreement of the curves is shown.
We propose a method for computing the energy level spacing around classical string solutions in AdS(5)xS(5). This method is based on the integrable structure of the string and can be applied to an arbitrary classical configuration. Our approach treats in equal footing the bosonic and fermionic excitations and provides an unambiguous prescription for the labeling of the fluctuation frequencies. Finally we revisit the computation of these frequencies for the SU(2) and SL(2) circular strings and compare our results to the existing ones.
Lian and Zuckerman proved that the homology of a topological chiral algebra can be equipped with the structure of a BV-algebra; \ie one can introduce a multiplication, an odd bracket, and an odd operator $Δ$ having the same properties as the corresponding operations in Batalin-Vilkovisky quantization procedure. We give a simple proof of their results and discuss a generalization of these results to the non chiral case. To simplify our proofs we use the following theorem giving a characterization of a BV-algebra in terms of multiplication and an operator $Δ$: {\em If $A$ is a supercommutative, …
We construct a Batalin-Vilkovisky (BV) algebra on moduli spaces of Riemann surfaces. This algebra is background independent in that it makes no reference to a state space of a conformal field theory. Conformal theories define a homomorphism of this algebra to the BV algebra of string functionals. The construction begins with a graded-commutative free associative algebra $\C$ built from the vector space whose elements are orientable subspaces of moduli spaces of punctured Riemann surfaces. The typical element here is a surface with several connected components. The operation $Δ$ of sewing two p…
The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and combinatorics of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. The…
We study various methods for computing the one-loop correction to the energy of classical solutions to type IIA string theory in AdS4 x CP3. This involves computing the spectrum of fluctuations and then adding up the fluctuation frequencies. We focus on two classical solutions with support in CP3: a rotating point-particle and a circular spinning string with two angular momenta equal to J. For each of these solutions, we compute the spectrum of fluctuations using two techniques, known as the algebraic curve approach and the world-sheet approach. If we use the same prescription for adding fluct…
We present the OSp(2,2|6) symmetric algebraic curve for the AdS4/CFT3 duality recently proposed in arXiv:0806.1218. It encodes all classical string solutions at strong t'Hooft coupling and the full two loop spectrum of long single trace gauge invariant operators in the weak coupling regime. This construction can also be used to compute the complete superstring semi-classical spectrum around any classical solution. We exemplify our method on the BMN point-like string.
Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we call {\em the Gerstenhaber bracket}. This bracket is compatible with the graded commutative product in cohomology, and hence gives rise to a new class of examples of what mathematicians call a {\em Gerstenhaber algebra}. The latter structure was first discussed in the context of Hochschild cohomology theory \cite{Gers1}. Off-shell in the (chiral) BRST complex, all the identities of a Gerstenhaber algebra hold up to homotopy. Applying…
On the base of the distinction between covariant and contravariant metric tensor components, a new (multivariable) cubic algebraic equation for reparametrization invariance of the gravitational Lagrangian has been derived and parametrized with complicated non - elliptic functions, depending on the (elliptic) Weierstrass function and its derivative. This is different from standard algebraic geometry, where only two-dimensional cubic equations are parametrized with elliptic functions and not multivariable ones. Physical applications of the approach have been considered in reference to theories w…
An algebraic proof of the Gluing Theorem at tree level of perturbation theory in String Field Theory is given. Some applications of the theorem to closed string non-polynomial action are briefly discussed
Deformations of topological open string theories are described, with an emphasis on their algebraic structure. They are encoded in the mixed bulk-boundary correlators. They constitute the Hochschild complex of the open string algebra -- the complex of multilinear maps on the boundary Hilbert space. This complex is known to have the structure of a Gerstenhaber algebra (Deligne theorem), which is also found in closed string theory. Generalising the case of function algebras with a B-field, we identify the algebraic operations of the bulk sector, in terms of the mixed correlators. This gives a ph…
We revisit the duality between type I' and heterotic strings in 9 dimensions. We resolve a puzzle about the validity of type I' perturbation theory and show that there are regions in moduli which are not within the reach of type I' perturbation theory. We find however, that all regions of moduli are described by a special class of real elliptic $K3$'s in the limit where the $K3$ shrinks to a one dimensional interval. We find a precise map between the geometry of dilaton and branes of type I' on the one hand and the geometry of real elliptic $K3$ on the other. We also argue more generally that …
This paper studies giant magnons in AdS_4 x CP^3 using both the string sigma-model and the algebraic curve. We complete the dictionary of solutions by finding the dyonic generalisation of the CP^1 string solution, which matches the `small' giant magnon in the algebraic curve, and by pointing out that the solution recently constructed by the dressing method is the `big' giant magnon. We then use the curve to compute finite-J corrections to all cases, which for the non-dyonic cases always match the AFZ result. For the dyonic RP^3 magnon we recover the S^5 answer, but for the `small' and `big' gi…
The purpose of the present paper is the communication of some results and observations which shed new light on the algebraic structure of the algebra of string observables both in the classical and in the quantum theory.
We investigate the canonical aspects of the algebraic first order formulation of strings introduced two decades ago by Balachandran and collaborators. We slightly enlarge the Lagrangian framework and show the existence of a self-dual formulation and of an Immirzi-type parameter reminiscent of four-dimensional first order gravity. We perform a full Hamiltonian analysis of the self-dual case: we extract the first class constraints and construct the Dirac bracket associated to the second class constraints. The first class constraints contain the diffeomorphisms algebra on the world-sheet, and the…
In this talk we give a brief review of the algebraic structure behind the open and closed topological strings and $D$-branes and emphasize the role of tensor category and the Frobenius algebra. Also, we speculate on the possibility of generalizing the topological strings and the $D$-branes through the subfactor theory.
We construct a universal envelope for any Poisson- and Gerstenhaber algebra. While the deformation theory of Poisson algebras seems to be partially trivial, results from string- and M-theory suggest a rich deformation theory of Gerstenhaber algebras. We apply our construction in this case to well known questions on the topological closed string BRST-complex. Finally, we find a similar algebraic structure, as for the universal envelope, in the SU(2)-WZW model.
In this letter we develope an operator formalism for the $b-c$ systems with conformal weight $λ=1$ defined on a general closed and orientable Riemann surface. The advantage of our approach is that the Riemann surface is represented as an affine algebraic curve. In this way it is possible to perform explicit calculations in string theory at any perturbative order. Besides the obvious applications in string theories and conformal field theories, (the $b-c$ systems at $λ=1$ are intimately related to the free scalar field theory), the operator formalism presented here sheds some light also on the …
We describe the twist quantization of string worldsheet theory, which unifies the description of quantization and the target space symmetry, based on the twisting of Hopf and module algebras. We formulate a method of decomposing a twist into successive twists to analyze the twisted Hopf and module algebra structure, and apply it to several examples, including finite twisted diffeomorphism and extra treatment for zero modes.
We prove some algebraic relations on the translationally invariant solutions and the lump solutions in vacuum string field theory. We show that up to the subtlety at the midpoint the definition of the half-string projectors of the known sliver solution can be generalized to other solutions. We also find that we can embed the translationally invariant solution into the matrix equation of motion with the zero mode.
We examine the algebraic structure of the matrix regularization for the wrapped membrane on $R^{10}\times S^1$ in the light-cone gauge. We give a concrete representation for the algebra and obtain the matrix string theory having the boundary conditions for the matrix variables corresponding to the wrapped membrane, which is referred to neither Seiberg and Sen's arguments nor string dualities. We also embed the configuration of the multi-wrapped membrane in matrix string theory.
The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe …
We discuss a link between the topological recursion relations derived algebraically by Witten and the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa. This is obtained through the definition of an operator ${\cal{W}}_s$ that reproduces the recursion relations for topological string theory coupled to worldsheet gravity a la BCOV. This operator is contained inside an algebra that generalizes the tt* equations and whose direct consequence is the holomorphic anomaly equation itself.
These are notes of a talk to the International Conference on Algebra in honor of A. I. Mal'tsev, Novosibirsk, USSR, 1989 (to appear in Contemporary Mathematics). The concept of a divisor with complex coefficients on an algebraic curve is introduced. We consider families of complex divisors, or, equivalently, families of invertible sheaves and define Arakelov-type metrics on some invertible sheaves produced from them on the base. We apply this technique to obtain a formula for the measure on the moduli space that gives tachyon correlators in string theory.
This paper studies semiclassical strings in AdS3 x S3 x S3 x S1 using the algebraic curve. Calculating one-loop corrections to the energy of the giant magnon fixes the constant term c in the expansion of the coupling h(λ). Comparing these to similar corrections for long spinning strings gives a prediction for the one-loop term f_1 in the expansion of the cusp anomalous dimension f(h), for all α(where α--> 1 is the AdS3 x S3 x T4 limit). For these semiclassical mode sums there is a similar choice of regularisation prescriptions to that encountered in AdS4 x CP3. However at α\neq 1/2 they lead t…
In this sequel to my previous paper, "Is String Theory in Knots?" I explore ways of constructing symmetries through an algebraic stepping process using knotted graphs. The hope is that this may lead to an algebraic formulation of string theory. In the conclusion I speculate that the stepping process is a form of quantisation for which the most general form must be sought. By applying the quantisation step a sufficient number (possibly infinite) of times we may construct an algebra which is equivalent to its own quantisation.
The four dimensional gauged supergravities descending from non-geometric string compactifications involve a wide class of flux objects which are needed to make the theory invariant under duality transformations at the effective level. Additionally, complex algebraic conditions involving these fluxes arise from Bianchi identities and tadpole cancellations in the effective theory. In this work we study a simple T and S-duality invariant gauged supergravity, that of a type IIB string compactified on a $T^6/(Z_2 x Z_2)$ orientifold with O3/O7-planes. We build upon the results of recent works and d…
The gauge invariant observables of the closed bosonic string are quantized without anomalies in four space-time dimensions by constructing their quantum algebra in a manifestly covariant approach. The quantum algebra is the kernel of a derivation on the universal envelopping algebra of an infinite-dimensional Lie algebra. The search for Hilbert space representations of this algebra is separated from its construction, and postponed.
In this writing we shall address certain beautiful inter-relations between the construction of 4-dimensional supersymmetric gauge theories and resolution of algebraic singularities, from the perspective of String Theory. We review in some detail the requisite background in both the mathematics, such as orbifolds, symplectic quotients and quiver representations, as well as the physics, such as gauged linear sigma models, geometrical engineering, Hanany-Witten setups and D-brane probes. We investigate aspects of world-volume gauge dynamics using D-brane resolutions of various Calabi-Yau singular…
We propose a consistently algebraic formulation of the extended KP (supersymmetric) integrable -hierarchy systems. We exploit the results already established in [14] and which consist in a framework suspected to unify in a fascinating way all the possible supersymmetric KP-hierarchies and then their underlying supergravity theories . This construction leads among other to built explicit non standard integrable Lax evolution equations suspected to reduce to the well known KP integrable equation. We present also a contribution of our construction to the subject of string equation and solitons. O…
String backgrounds are described as purely geometric objects related to moduli spaces of Riemann surfaces, in the spirit of Segal's definition of a conformal field theory. Relations with conformal field theory, topological field theory and topological gravity are studied. For each field theory, an algebraic counterpart, the (homotopy) algebra satisfied by the tree level correlators, is constructed.
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.
