Mapping between Witten and Lightcone String Field Theories
Theodore Erler, Hiroaki Matsunaga
备注：
74 pages, 23 figures. Added appendix on the lightcone SFT action. Simplified discussion of the field transformation and the singular nature of lightcone gauge in generic SFTs, which has been called the "soft string problem."
We propose a transformation between the off-shell field variables of Witten's open bosonic string field theory and the traditional lightcone string field theory of Kaku and Kikkawa, based on Mandelstam's interacting string picture. This is accomplished by deforming the Witten vertex into lightcone cubic and quartic vertices, followed by integrating out the ghost and lightcone oscillator excitations from the string field. Surprisingly, the last step does not alter the cubic and quartic interactions and does not generate effective vertices, and leads precisely to Kaku and Kikkawa's lightcone string field theory.
The equivalence of quantum field theory and string theory as exemplified by the AdS/CFT correspondence is explored from the point of view of lightcone quantization. On the string side we discuss the lightcone version of the static string connecting a heavy external quark source to a heavy external antiquark source, together with small oscillations about the static string configuration. On the field theory side we analyze the weak/strong coupling transition in a ladder diagram model of the quark antiquark system, also from the point of view of the lightcone. Our results are completely consisten…
We quantize in light cone gauge the bosonic sector of string theory on Anti-de Sitter space in the zero curvature radius limit. We find that the worldsheet falls apart into a theory of free partons and map the Hilbert space of the string theory to the Hilbert space of a free scalar in light-front description. We outline how the string worldsheet reproduces the field theory at weak coupling.
We explore all the possible ways of fixing the kappa symmetry for both branes and strings by means of a constant projector. We find that they can be classified according to their behaviour under Dirac conjugation and conjugation. This latter controls the maximum power of the fermionic variables in which the (super)vielbein can be expanded while the former determines which states cannot be described in the gauge. In particular there exists an interesting class of projectors for which vielbein are at most quadratic in the fermionic variables. As an example we compute the action for the type IIb …
We study the one loop correction to the closed bosonic string propagator, including the possibile presence of D-branes, by discretizing the light cone worldsheet on an M times N rectangular lattice, with M proportional to P^+ and N+1 proportional to ix^+. The integrals over the moduli then become sums which we evaluate numerically. The main purpose of this study is to assess the reliability of the worldsheet lattice as a regulator of the divergences in string perturbation theory. There are two natural geometrical counterterms for the lightcone worldsheet, one proportional to the area of the wo…
We continue our study of open string perturbation theory on the lightcone worldsheet lattice, which is an $M\times N$ rectangular grid. Here $M$ is the number of $P^+$ units and $N$ is the number of $ix^+$ units. We extend our previous analysis to the bosonic open string one planar loop self-energy. We find that, when all open string coordinates satisfy Neumann conditions, the ultraviolet worldsheet divergences associated with the closed string tachyon and boundary effects can be cancelled by renormalization of bulk ($AM^1$) and boundary ($BM^0$) worldsheet "cosmological constants". The bulk d…
A recent paper by Gross and Erler (hep-th/0406199) showed that by making a certain well-defined, unitary transformation on the mode basis for the open bosonic string--one that identifies the lightcone component of position with the string midpoint--it is possible to render the action for cubic string field theory local in lightcone time. In this basis, then, cubic string field theory possesses a well-defined initial value formulation and a conserved Hamiltonian. With this new understanding it seems natural to study time dependent solutions representing the the decay of an unstable D-branes. In…
We investigate the issues of holography and string interactions in the duality between SYM and the pp wave background. We argue that the Penrose diagram of the maximally supersymmetric pp-wave has a one dimensional boundary. This fact suggests that the holographic dual of the pp-wave can be described by a quantum mechanical system. We believe this quantum mechanical system should be formulated as a matrix model. From the SYM point of view this matrix model is built out of the lowest lying KK modes of the SYM theory on an $S^3$ compactification, and it relates to a wave which has been compactif…
Although the energy spectrum of the Heisenberg spin chain on a circle defined by $H=\frac{1}{4}\sum_{k=1}^M(σ_k^xσ_{k+1}^x+σ_k^yσ_{k+1}^y +Δσ_k^zσ_{k+1}^z)$ is well known for any fixed $M$, the boundary conditions vary according to whether $M\in 4\mathbb{N}+r$, where $r=-1,0,1,2$, and also according to the parity of the number of overturned spins in the state, In string theory all these cases must be allowed because interactions involve a string with $M$ spins breaking into strings with $M_1<M$ and $M-M_1$ spins (or vice versa). We organize the energy spectrum and degeneracies of $H$ in the ca…
We explore the large spin spectrum in two-dimensional conformal field theories with a finite twist gap, using the modular bootstrap in the lightcone limit. By recursively solving the modular crossing equations associated to different $PSL(2,\mathbb{Z})$ elements, we identify the universal contribution to the density of large spin states from the vacuum in the dual channel. Our result takes the form of a sum over $PSL(2,\mathbb{Z})$ elements, whose leading term generalizes the usual Cardy formula to a wider regime. Rather curiously, the contribution to the density of states from the vacuum beco…
In this paper, we explore the questions of time, locality and causality in the framework of covariant open bosonic string field theory. We show that if an open string field is expressed as a certain local function on spacetime--in particular, a function of the lightcone component of the midpoint and the transverse center of mass degrees of freedom--that cubic string field theory is nonsingular and local in lightcone time. In particular, the theory has a well defined initial value formulation resembling that of an ordinary second order relativistic field theory in lightcone frame. This descript…
We study the flight time fluctuations in an anisotropic medium inspired by a cosmic string with an effective fluctuating refractive index caused by fluctuating vacuum electric fields, which are analogous to the lightcone fluctuations due to fluctuating spacetime metric when gravity is quantized. The medium can be realized as a metamaterial that mimics a cosmic string in the sense of transformation optics. For a probe light close to the analogue string, the flight time variance is $ν$ times that in a normal homogeneous and isotropic medium, where $ν$ is a parameter characterizing the deficit an…
The purpose of this article is to extend the lightcone worldsheet lattice description of string theory to include the Neveu-Schwarz model. We model each component of the fermionic worldsheet field by a critical Ising model. We show that a simple choice of boundary conditions for the Ising variables leads to the half integer modes required by the model. We identify the G-parity operation within the Ising model and formulate the procedure for projecting onto the even G-parity sector. We construct the lattice version of the three open string vertex, with the necessary operator insertion at the in…
This paper describes my talk given to the 27th Johns Hopkins Workshop: Symmetries and Mysteries of M Theory, Göteborg, Sweden, 24-26 August, 2003. After a brief introduction to the lightcone worldsheet formalism for summing the planar diagrams of field theory, I explain how the {\it uv} divergences of quantum field theory translate to the new language of string. It is shown through one loop that, at least for scalar cubic vertices, the counter-terms necessary for Poincaré invariance in space-time dimensions $D\leq 6$ are indeed local on the worldsheet. The extension to cover the case of gauge …
In this paper, we extend our previous study of causality and local commutativity of string fields in the pp-wave lightcone string field theory to include interaction. Contrary to the flat space case result of Lowe, Polchinski, Susskind, Thorlacius and Uglum, we found that the pp-wave interaction does not affect the local commutativity condition. Our results show that the pp-wave lightcone string field theory is not continuously connected with the flat space one. We also discuss the relation between the condition of local commutativity and causality. While the two notions are closely related in…
The Yukawa interaction sector of superstring inspired models that give superconducting strings, can be described in terms of a supersymmetric quantum mechanics algebra. We relate the Witten index of susy quantum mechanics with an index characteristic to superconducting string models.
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…
We investigate instanton expansions of partition functions of several toric E-string models using local mirror symmetry and elliptic modular forms. We also develop a method to obtain the Seiberg--Witten curve of E-string with arbitrary Wilson lines with the help of elliptic functions.
In this note we describe the physics of equivalence of the Seiberg-Witten invariants of 4-manifolds and certain Gromov-Witten invariants defined by pseudo-holomorphic curves. We show that physics of the pseudo-holomorphic curves should be governed by the N=2 Green-Schwarz string.
We initiate the study of M-strings in the thermodynamic limit. In this limit the BPS partition function of M5 branes localizes on configurations with a large number of strings which leads to a reformulation of the partition function in terms of a matrix model. We solve this matrix model and obtain its spectral curve which can be interpreted as the Seiberg-Witten curve associated to the compactified M5 brane theory.
We study Witten's cubic open string field theory on multiple $Dp$-branes. On multiple $Dp$-branes the string fields carry $U(N)$ group indices. Mapping the string world-sheet onto the upper half complex plane and evaluating the Polyakov string path integral of Witten's cubic open string, we obtain a Fock space representation of one, two and three-string vertex operators. In the low energy region, the system is described by massless gauge fields and massless scalar fields, carrying $U(N)$ group indices. The two-string vertex induces mass terms for the gauge fields and the scalar fields. The cub…
I review several old/new approaches to the string/gauge correspondence for the cusped/lightcone Wilson loops. The main attention is payed to SYM perturbation theory calculations at two loops and beyond and to the cusped loop equation. These three introductory lectures were given at the 48 Cracow School of Theoretical Physics: "Aspects of Duality", June 13-22, 2008, Zakopane, Poland.
We study oscillating string solutions in the Klebanov-Witten and its non-Abelian T-dual background dualised along an SU(2) isometry. We find the string energy as the function of oscillation number and angular momentum. We show that for a particular set of T-dual co-ordinates both the background have equal string states. We also study the string states where the strings are expanding and contracting in the T-dual co-ordinate direction. We expect the presence of the superconformal field theory dual operators whose anomalous dimensions depend on T-dual co-ordinate.
We construct the Seiberg-Witten curve for the E-string theory in six-dimensions. The curve is expressed in terms of affine E_8 characters up to level 6 and is determined by using the mirror-type transformation so that it reproduces the number of holomorphic curves in the Calabi-Yau manifold and the amplitudes of N=4 U(n) Yang-Mills theory on 1/2 K3. We also show that our curve flows to known five- and four-dimensional Seiberg-Witten curves in suitable limits.
In view of two-dimensional topological gravity coupled to matter, we study the Seiberg-Witten theory for the low-energy behavior of N=2 supersymmetric Yang-Mills theory with ADE gauge groups. We construct a new solution of the Picard-Fuchs equations obeyed by the Seiberg-Witten periods. Our solution is expressed as the linear sum over the infinite set of one-point functions of gravitational descendants in $d<1$ topological strings. It turns out that our solution provides the power series expansion around the origin of the quantum moduli space of the Coulomb branch. For SU(N) gauge group we sho…
We analyze BPS equations for string-like configurations derived from the M5-brane worldvolume action with a Nambu-Poisson structure constructed in arXiv:0804.3629, arXiv:0805.2898. We solve the BPS equations up to the first order in the parameter $g$ which characterizes the strength of the Nambu-Poisson bracket. We compare our solutions to previously constructed BPS string solitons in the conventional description of M5-brane in a constant three-form background via Seiberg-Witten map, and find agreement.
We present arguments pointing to a behavior of the string free energy in the presence of a black hole horizon similar to the Atick-Witten dependence on temperature beyond the Hagedorn transition. We give some evidence based on orbifold techniques applied to Rindler space and further support is found within a Hamiltonian approach. However, we argue that the interpretation in terms of a reduction of degrees of freedom is confronted by serious problems. Finally, we point out the problems concerning heuristic red-shift arguments and the local interpretation of thermodynamical quantities.
The Becchi-Rouet-Stora-Tyutin (BRST) ghost field is a key element in constructing Witten's cubic open string field theory. However, to date, the ghost sector of the string field theory has not received a great deal of attention. In this study, we address the BRST ghost on multiple $Dp$-branes, which carries non-Abelian indices and couples to a non-Ablelian gauge field. We found that the massless components of the BRST ghost field can play the role of the Faddeev-Popov ghost in the non-Alelian gauge field, such that the string field theory maintains the local non-Abelian gauge invariance.
We show how to evaluate the periods in Seiberg-Witten theories and in K3-fibered Calabi-Yau manifolds by using fibrations of the theories. In the Seiberg-Witten theories, it is shown that the dual pair of fields can be constructed from the classical fields in a simple form. As for Calabi-Yau manifolds which are fibrations of K3 surface, we obtain the solutions of the Picard-Fuchs equations from the periods of K3 surface. By utilizing the expression of periods for two-parameter models of type-II string, we derive the solutions of the Picard-Fuchs equations around the points of enhanced gauge sy…
We study limits of four-dimensional type II Calabi-Yau compactifications with vanishing four-cycle singularities, which are dual to $\IT^2$ compactifications of the six-dimensional non-critical string with $E_8$ symmetry. We define proper subsectors of the full string theory, which can be consistently decoupled. In this way we obtain rigid effective theories that have an intrinsically stringy BPS spectrum. Geometrically the moduli spaces correspond to special geometry of certain non-compact Calabi-Yau spaces of an intriguing form. An equivalent description can be given in terms of Seiberg-Witt…
We construct the Zinn-Justin-Batalin-Vilkovisky action for tachyons and gauge bosons from Witten's 3-string vertex of the bosonic open string without gauge fixing. Through canonical transformations, we find the off-shell, local, gauge-covariant action up to 3-point terms, satisfying the usual field theory gauge transformations. Perturbatively, it can be extended to higher-point terms. It also gives a new gauge condition in field theory which corresponds to the Feynman-Siegel gauge on the world-sheet.
We consider a lightcone wrapped supermembrane compactified on 2-torus in the matrix regularization. We examine the double dimensional reduction technique and deduce the free matrix string of (p,q)-string in type IIB superstring theory explicitly from the matrix-regularized wrapped supermembrane. In addition we obtain the (2+1)-dimensional super Yang-Mills action in curved background. We also examine the duality.
We prove a no-ghost theorem for a bosonic string propagating in Nappi-Witten spacetime. This is achieved in two steps. We first demonstrate unitarity for a class of NW/U(1) modules: the norm of any state which is primary with respect to a chosen timelike U(1) is non-negative. We then show that physical states - states satisfying the Virasoro constraints - in a class of modules of an affinisation of the Nappi-Witten algebra are contained in the NW/U(1) modules. Similar to the case of strings on $AdS_3$, in order to saturate the spectrum obtained in light-cone quantization we are led to include …
The background for string propagation is obtained by a chiral gauging of the $SL(2,R)$ Wess-Zumino-Witten model. It is shown explicitly that the resulting background fields satisfy the field equations of the three dimensional string effective action and the target space has curvature singularity. Close connection of our solution with the three dimensional black string is demonstrated.
In this paper we consider Witten's bosonic open string field theory in the presence of a constant background of the second-rank antisymmetric tensor field $B_{ij}$. We extend the operator formulation of Gross and Jevicki in this situation and construct the overlap vertices explicitly. As a result we find a noncommutative structure of the Moyal type only in the zero-mode sector, which is consistent with the result of the correlation functions among vertex operators in the world sheet formulation. Furthermore we find out a certain unitary transformation of the string field which absorbs the Moya…
We present an algebraic approach to string theory. An embedding of $sl(2|1)$ in a super Lie algebra together with a grading on the Lie algebra determines a nilpotent subalgebra of the super Lie algebra. Chirally gauging this subalgebra in the corresponding Wess-Zumino-Witten model, breaks the affine symmetry of the Wess-Zumino-Witten model to some extension of the $N=2$ superconformal algebra. The extension is completely determined by the $sl(2|1)$ embedding. The realization of the superconformal algebra is determined by the grading. For a particular choice of grading, one obtains in this way,…
We argue that certain BPS states in the D3-brane probe realization of N=2 SU(2) Super-Yang-Mills theory correspond to multi-pronged strings connecting the D3-brane to the background 7-branes. This provides a physical realization of the decay of these states on the curve of marginal stability, and explains their absence in the strong coupling regime.
We study refined B-model via the beta ensemble of matrix models. Especially, for four dimensional N=2 SU(2) supersymmetric gauge theories with N_f=0,1 and 2 fundamental flavors, we discuss the correspondence between deformed disk amplitudes on each Seiberg-Witten curve and the Nekrasov-Shatashvili limit of the corresponding irregular one point block of a degenerate operator via the AGT correspondence. We also discuss the relation between deformed annulus amplitudes and the irregular two point block of the degenerate operator, and check a desired agreement for N_f=0 and 1 cases.
Perturbative spectra and related factorization properties of one-loop open string amplitudes in the presence of a constant external background B are analysed in detail. While the pattern of the closed string spectrum, obtained after a careful study of the properly symmetrized amplitudes, turns out to be unaffected by the presence of B, a series of double open-string poles, which would be absent when B is turned off, can couple owing to a partial symmetry loss. These features are studied first in a bosonic setting and then generalized in the more satisfactory superstring context. When the backg…
We construct complete sets of (open and closed string) covariant coherent state and mass eigenstate vertex operators in bosonic string theory. By minimally extending the standard definition of coherent states so as to include the string theory requirements, we show that the naive construction of the the closed string coherent states requires the existence of a lightlike compactification of spacetime. When the null winding states in the underlying Hilbert space are projected out the resulting vertex operators satisfy the definition of a coherent state and have a classical interpretation. We pre…
Schnabl recently constructed an analytic solution for tachyon condensation in Witten's open string field theory. The solution consists of two pieces. Only the first piece is involved in proving that the solution satisfies the equation of motion when contracted with any state in the Fock space. On the other hand, both pieces contribute in evaluating the kinetic term to reproduce the value predicted by Sen's conjecture. We therefore need to understand why the second piece is necessary. We evaluate the cubic term of the string field theory action for Schnabl's solution and use it to show that the…
We show that a large number of $CY_3$ manifolds are involved in an intricate way in Mathieu moonshine viz. their Gromov--Witten invariants are related to the expansion coefficients of the twined/ twisted--twined elliptic genera of $K3$. We use the string duality between CHL orbifolds of heterotic string theory on $K3 \times T^2$ and type IIA string theory on $CY_3$ manifolds to explicitly show this connection. We then work out two concrete examples where we exactly match the expansion coefficients on both sides of the duality.
It is shown that Witten's star product in string field theory, defined as the overlap of half strings, is equivalent to the Moyal star product involving the relativistic phase space of even string modes. The string field A(x[σ]) can be rewritten as a phase space field of the even modes $x_{2n},x_{0}, p_{2n}$ where $x_{2n}$ are the positions of the even string modes, and $p_{2n}$ are related to the Fourier space of the odd modes $x_{2n+1}$ up to a linear transformation. The $p_{2n}$ play the role of conjugate momenta for the even modes $x_{2n}$ under the string star product. The split string fo…
We show that the degenerate positive-norm physical propagating fields of the open bosonic string can be gauged to the higher rank fields at the same mass level. As a result, their scattering amplitudes can be determined from those of the higher spin fields. This phenomenon arises from the existence of two types of zero-norm states with the same Young representations as those of the degenerate positive-norm states in the old covariant first quantized (OCFQ) spectrum. This is demonstrated by using the lowest order gauge transformation of Witten's string field theory (WSFT) up to the fourth massi…
We explicitly evaluate the disk S-matrix elements of one closed string and an arbitrary number of open string states in the presence of a large background B-flux. From this calculation, we show that in the world-volume action of D-branes in terms of non-commutative fields, the closed string fields must be treated as functionals of the non-commutative gauge fields. We also find the generalized multiplication rule *_N between N open string fields on the world-volume of the D-brane. In particular, this result indicates that the difference between the familiar * and the *_N product is just some to…
We describe a non-perturbative quantization of classical Wilson loops in the WZW model. The quantized Wilson loop is an operator acting on the Hilbert space of closed strings and commuting either with the full Kac-Moody chiral algebra or with one of its subalgebras. We prove that under open/closed string duality, it is dual to a boundary perturbation of the open string theory. As an application, we show that such operators are useful tools for identifying fixed points of the boundary renormalization group flow.
We study Witten open string field theory in the pp-wave background in the tensionless limit, and construct the N-string vertex in the basis which diagonalizes the string perturbative spectrum. We found that the Witten *-product can be viewed as infinite copies of the Moyal product with the same noncommutativity parameter $θ=2$. Moreover, we show that this Moyal structure is universal in the sense that, written in the string bit basis, Witten's *-product for any background can always be given in terms of the above-mentioned Moyal structure. We identify some projective operators in this algebra …
We generalise our previous result\cite{13} to cases with three and four Wilson lines. We show that the Nekrasov-type formula for E-string theory is valid for those cases and the Seiberg-Witten curves are reproduced from the formula. In addition, we show the dependence of the Seiberg-Witten curve on the Wilson lines.
We study the structure of anomalies in general heterotic string theories by considering general 2-dimensional $\mathcal{N}=(0,1)$ supersymmetric quantum field theories (SQFTs), without assuming conformal invariance nor the correct central charges. First we generalize the precise notion of the $B$-field introduced by Witten. Then we express the target space anomalies by invariants of SQFTs. Perturbative anomalies correspond to the Witten index of some class of SQFTs, while global anomalies correspond to a torsion version of the Witten index. The torsion index gives some of the invariants of SQF…
In this paper, we recast the fermionic ghost sector of Witten's open bosonic string field theory in the language of noncommutative field theory. In particular, following the methods of hep-th/0202087, we find that in Siegel gauge Witten's star product roughly corresponds to a continuous tensor product of Clifford Algebras, and we formulate important operators of the theory in this language, notably the kinetic operator of vacuum string field theory and the BRST operator describing the vacuum of the unstable D-25 brane. We find that the BRST operator is singular in this formulation. We explore …
It is pointed out that any conformally transformed of a flat space-time metric $\ball g_{ij} = f(x)\;η_{ij}$ is a solution to Witten's equation of Chern-Simons gravity, which holds outside matter in (2+1) dimensions. It is also shown that a simultaneous exterior solution to both Witten's and Einstein's equations, yields the lensing effect of an isolated cosmic string, if $f(x)$ is reduced to an arbitrary dimensionless constant. However, the solution to Witten's equation with $f(x)$ being an arbitrary and continuous function of space-time coordinates, also leads to an open circular path for a l…
We showed that there is no SU(2) Witten anomaly in a large class of 4d N = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.
We reinterpret U(N) Chern-Simons-Witten theory quantized on a torus as a free fermion system. Its Hilbert space and some observables are simply related to those of group quantum mechanics, even at finite N and k. Its large N limit can be described using techniques developed for matrix quantum mechanics and two-dimensional Yang-Mills theory. We discuss the bosonization of this theory, which for YM_2 gave a precise interpretation of Wilson loop operators in terms of string creation and annihilation operators, and examine its consequences for a string interpretation here. The formalism seems enti…
We find a Nekrasov-type expression for the Seiberg-Witten prepotential for the six-dimensional non-critical E_8 string theory toroidally compactified down to four dimensions. The prepotential represents the BPS partition function of the E_8 strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta. We show that our expression exhibits expected modular properties. In particular, we prove that it obeys the modular anomaly equation known to be satisfied by the prepotential.
We solve the equation of motion of Witten's cubic open string field theory in a series expansion using the regulated butterfly state. The expansion parameter is given by the regularization parameter of the butterfly state, which can be taken to be arbitrarily small. Unlike the case of level truncation, the equation of motion can be solved for an arbitrary component of the Fock space up to a positive power of the expansion parameter. The energy density of the solution is well-defined and remains finite even in the singular butterfly limit, and it gives approximately 68% of the D25-brane tension…
We examine string field algebra which is generated by star product in Witten's string field theory including ghost part. We perform calculations using oscillator representation consistently. We construct wedge like states in ghost part and investigate algebras among them. As a by-product we have obtained some solutions of vacuum string field theory. We also discuss some problems about identity state. We hope these calculations will be useful for further investigation of Witten type string field theory.
