Discrete Light-cone Quantization Research Articles

Worldsheet formalism for decoupling limits in string theory

We study the bosonic sector of a decoupling limit of type IIA superstring theory, where a background Ramond-Ramond one-form is fined tuned to its critical value, such that it cancels the associated background D0-brane tension. The light excitations in this critical limit are D0-branes, whose dynamics is described by the Banks-Fischler-Shenker-Susskind (BFSS) Matrix theory that corresponds to M-theory in the Discrete Light-Cone Quantization (DLCQ). We develop the worldsheet formalism for the fundamental string in the same critical limit of type IIA superstring theory. We show that the fundamental string develops singularities on its worldsheet, whose topology is described by nodal Riemann spheres as in ambitwistor string theory. We study the T-duality transformations of this string sigma model and provide a worldsheet derivation for the recently revived and expanded duality web that unifies a zoo of decoupling limits in type II superstring theories. By matching the string worldsheet actions, we demonstrate how some of these decoupling limits are related to tensionless (and ambitwistor) string theory, Carrollian string theory, the Spin Matrix limits of the AdS/CFT correspondence, and more.

Lessons from discrete light-cone quantization for physics at null infinity: bosons in two dimensions

Motivated by issues in the context of asymptotically flat spacetimes at null infinity, we discuss in the simplest example of a free massless scalar field in two dimensions several subtleties that arise when setting up the canonical formulation on a single or on two intersecting null hyperplanes with a special emphasis on the infinite-dimensional global and conformal symmetries and their canonical generators, the free data, a consistent treatment of zero modes, matching conditions, and implications for quantization of massless versus massive fields.

Anisotropic compactification of nonrelativistic M-theory

We study a decoupling limit of M-theory where the three-form gauge potential becomes critical. This limit leads to nonrelativistic M-theory coupled to a non-Lorentzian spacetime geometry. Nonrelativistic M-theory is U-dual to M-theory in the discrete light cone quantization, a non-perturbative approach related to the Matrix theory description of M-theory. We focus on the compactification of nonrelativistic M-theory over a two-torus that exhibits anisotropic behaviors due to the foliation structure of the spacetime geometry. We develop a frame covariant formalism of the toroidal geometry, which provides a geometrical interpretation of the recently discovered polynomial realization of SL(2 , ℤ) duality in nonrelativistic type IIB superstring theory. We will show that the nonrelativistic IIB string background fields transform as polynomials of an effective Galilean “boost velocity” on the two-torus. As an application, we construct an action principle describing a single M5-brane in nonrelativistic M-theory and study its compactification over the anisotropic two-torus. This procedure leads to a D3-brane action in nonrelativistic IIB string theory that makes the SL(2 , ℤ) invariance manifest in the polynomial realization.

Adjoint Majorana QCD2 at finite N

The mass spectrum of 1 + 1-dimensional SU(N) gauge theory coupled to a Majorana fermion in the adjoint representation has been studied in the large N limit using Light-Cone Quantization. Here we extend this approach to theories with small values of N, exhibiting explicit results for N = 2, 3, and 4. In the context of Discretized Light-Cone Quantization, we develop a procedure based on the Cayley-Hamilton theorem for determining which states of the large N theory become null at finite N. For the low-lying bound states, we find that the squared masses divided by g2N, where g is the gauge coupling, have very weak dependence on N. The coefficients of the 1/N2 corrections to their large N values are surprisingly small. When the adjoint fermion is massless, we observe exact degeneracies that we explain in terms of a Kac-Moody algebra construction and charge conjugation symmetry. When the squared mass of the adjoint fermion is tuned to g2N/π, we find evidence that the spectrum exhibits boson-fermion degeneracies, in agreement with the supersymmetry of the model at any value of N.

The supersymmetric Neveu-Schwarz branes of non-relativistic string theory

We construct the basic Neveu-Schwarz (NS) brane solutions of non-relativistic string theory using longitudinal T-duality as a solution generating technique. Extending the NS background fields to a supergravity multiplet, we verify that all solutions we find are half-supersymmetric. The two perturbative solutions we find both have an interpretation as the background geometry outside a string-like object. Correspondingly, we refer to these non-Lorentzian backgrounds as winding string and unwound string solution. Whereas the winding string is part of the on-shell spectrum of non-relativistic string theory, the unwound string only makes sense off-shell where it mediates the instantaneous gravitational force. Seen from the nine-dimensional point of view, we find that the winding string solution is sourced by a non-relativistic massive particle and that the unwound string solution is sourced by a massless Galilean particle of zero colour and spin. We explain how these two string solutions fit into a discrete lightcone quantization of string theory. We shortly discuss the basic NS five-brane and Kaluza-Klein monopole solutions and show that they are both half-supersymmetric.

KLT factorization of nonrelativistic string amplitudes

We continue our study of the Kawai-Lewellen-Tye (KLT) factorization of winding string amplitudes in [1]. In a toroidal compactification, amplitudes for winding closed string states factorize into products of amplitudes for open strings ending on an array of D-branes localized in the compactified directions; the specific D-brane configuration is determined by the closed string data. In this paper, we study a zero Regge slope limit of the KLT relations between winding string amplitudes. Such a limit of string theory requires a critically tuned Kalb-Ramond field in a compact direction, and leads to a self-contained corner called nonrelativistic string theory. This theory is unitary, ultraviolet complete, and its string spectrum and spacetime S-matrix satisfy nonrelativistic symmetry. Moreover, the asymptotic closed string states in nonrelativistic string theory necessarily carry nonzero windings. First, starting with relativistic string theory, we construct a KLT factorization of amplitudes for winding closed strings in the presence of a critical Kalb-Ramond field. Then, in the zero Regge limit, we uncover a KLT relation for amplitudes in nonrelativistic string theory. Finally, we show how such a relation can be reproduced from first principles in a purely nonrelativistic string theory setting. We will also discuss connections to the amplitudes of string theory in the discrete light cone quantization (DLCQ), a method that is relevant for Matrix theory.

Exact symmetries and threshold states in two-dimensional models for QCD

Two-dimensional SU(N) gauge theory coupled to a Majorana fermion in the adjoint representation is a nice toy model for higher-dimensional gauge dynamics. It possesses a multitude of “gluinoball” bound states whose spectrum has been studied using numerical diagonalizations of the light-cone Hamiltonian. We extend this model by coupling it to Nf flavors of fundamental Dirac fermions (quarks). The extended model also contains meson-like bound states, both bosonic and fermionic, which in the large-N limit decouple from the gluinoballs. We study the large-N meson spectrum using the Discretized Light-Cone Quantization (DLCQ). When all the fermions are massless, we exhibit an exact mathfrak{osp} (1|4) symmetry algebra that leads to an infinite number of degeneracies in the DLCQ approach. More generally, we show that many single-trace states in the theory are threshold bound states that are degenerate with multi-trace states. These exact degeneracies can be explained using the Kac-Moody algebra of the SU(N) current. We also present strong numerical evidence that additional threshold states appear in the continuum limit. Finally, we make the quarks massive while keeping the adjoint fermion massless. In this case too, we observe some exact degeneracies that show that the spectrum of mesons becomes continuous above a certain threshold. This demonstrates quantitatively that the fundamental string tension vanishes in the massless adjoint QCD2 without explicit four-fermion operators.

T-duality in nonrelativistic open string theory

Nonrelativistic open string theory is defined by a worldsheet theory that produces a Galilean invariant string spectrum and is described at low energies by a nonrelativistic Yang-Mills theory [1]. We study T-duality transformations in the path integral for the sigma model that describes nonrelativistic open string theory coupled to an arbitrary closed string background, described by a string Newton-Cartan geometry, Kalb-Ramond, and dilaton field. We prove that T-duality transformations map nonrelativistic open string theory to relativistic and noncommutative open string theory in the discrete light cone quantization (DLCQ), a quantization scheme relevant for Matrix string theory. We also show how the worldvolume dynamics of nonrelativistic open string theory described by the Dirac-Born-Infeld type action found in [1] maps to the Dirac-Born-Infeld actions describing the worldvolume theories of the DLCQ of open string theory and noncommutative open string theory.

Nonrelativistic string theory and T-duality

Nonrelativistic string theory in flat spacetime is described by a two-dimensional quantum field theory with a nonrelativistic global symmetry acting on the worldsheet fields. Nonrelativistic string theory is unitary, ultraviolet complete and has a string spectrum and spacetime S-matrix enjoying nonrelativistic symmetry. The worldsheet theory of nonrelativistic string theory is coupled to a curved spacetime background and to a Kalb-Ramond two-form and dilaton field. The appropriate spacetime geometry for nonrelativistic string theory is dubbed string Newton-Cartan geometry, which is distinct from Riemannian geometry. This defines the sigma model of nonrelativistic string theory describing strings propagating and interacting in curved background fields. We also implement T-duality transformations in the path integral of this sigma model and uncover the spacetime interpretation of T-duality. We show that T-duality along the longitudinal direction of the string Newton-Cartan geometry describes relativistic string theory on a Lorentzian geometry with a compact lightlike isometry, which is otherwise only defined by a subtle infinite boost limit. This relation provides a first principles definition of string theory in the discrete light cone quantization (DLCQ) in an arbitrary background, a quantization that appears in nonperturbative approaches to quantum field theory and string/M-theory, such as in Matrix theory. T-duality along a transverse direction of the string Newton-Cartan geometry equates nonrelativistic string theory in two distinct, T-dual backgrounds.

Anomalies for Galilean fields

We initiate a systematic study of ‘t Hooft anomalies in Galilean field theories, focusing on two questions therein. In the first, we consider the non-relativistic theories obtained from a discrete light-cone quantization (DLCQ) of a relativistic theory with flavor or gravitational anomalies. We find that these anomalies survive the DLCQ, becoming mixed flavor/boost or gravitational/boost anomalies. We also classify the pure Weyl anomalies of Schrödinger theories, which are Galilean conformal field theories (CFTs) with z=2z=2. There are no pure Weyl anomalies in even spacetime dimension, and the lowest-derivative anomalies in odd dimension are in one-to-one correspondence with those of a relativistic CFT in one dimension higher. These results classify many of the anomalies that arise in the field theories dual to string theory on Schrödinger spacetimes.

Light-front and conformal field theories in two dimensions

Light-front (LF) quantization of massless fields in two spacetime dimensions is a long-standing and much debated problem. Even though the classical wave-equation is well-documented for almost two centuries, either as problems with initial values in spacetime variables or with initial data on characteristics in light-cone variables, the way to a consistent quantization in both types of frames is still a puzzle in many respects. This is in contrast to the most successful Conformal Field Theoretic (CFT) approach in terms of complex variables [Formula: see text], [Formula: see text] pioneered by Belavin, Polyakov and Zamolodchikov in the ’80s. It is shown here that the 2D-massless canonical quantization in both reference frames is completely consistent provided that quantum fields are treated as Operator-Valued Distributions (OPVD) with Partition of Unity (PU) test functions. We recall first that classical fields have to be considered as distributions (e.g. generalized functions in the Russian literature). Then, a necessary condition on the PU test function follows from the required matching of the classical solutions of the massless differential equations in both types of reference frame. Next we use a mathematical formulation for OPVD, developed in the recent past. Specifically, smooth [Formula: see text] fields are introduced through the convolution operation in the distributional context. Due to the specific behavior of the Fourier-transform of the initial test function, this convolution transform has a well-defined integral in the dual space, whatever the initial choice of the reference frame. The relation to the conformal fields method follows immediately from the transition to Euclidean time and leads directly to explicit calculations of a few correlation functions of the scalar field and its energy–momentum tensor. The LF derivation of the Virasoro algebra is then obtained from the [Formula: see text] and [Formula: see text] expansions of the canonical fields as distributional Laplace-transform in these variables. Finally, the popular and problematic Discretized Light Cone Quantization (DLCQ) method is scrutinized with respect to its zero mode and ultraviolet content as encompassed in the continuum OPVD formulation.

Heat kernel and Weyl anomaly of Schrödinger invariant theory

We propose a method inspired by discrete light cone quantization to determine the heat kernel for a Schr\"odinger field theory (Galilean boost invariant with $z=2$ anisotropic scaling symmetry) living in $d+1$ dimensions, coupled to a curved Newton-Cartan background, starting from a heat kernel of a relativistic conformal field theory ($z=1$) living in $d+2$ dimensions. We use this method to show that the Schr\"odinger field theory of a complex scalar field cannot have any Weyl anomalies. To be precise, we show that the Weyl anomaly ${\mathcal{A}}_{d+1}^{G}$ for Schr\"odinger theory is related to the Weyl anomaly of a free relativistic scalar CFT ${\mathcal{A}}_{d+2}^{R}$ via ${\mathcal{A}}_{d+1}^{G}=2\ensuremath{\pi}\ensuremath{\delta}(m){\mathcal{A}}_{d+2}^{R}$, where $m$ is the charge of the scalar field under particle number symmetry. We provide further evidence of the vanishing anomaly by evaluating Feynman diagrams in all orders of perturbation theory. We present an explicit calculation of the anomaly using a regulated Schr\"odinger operator, without using the null cone reduction technique. We generalize our method to show that a similar result holds for theories with a single time-derivative and with even $z>2$.

True muonium on the light front

Applying Discretized Light Cone Quantization, we perform the first calculation of the spectrum of true muonium, the atom, as modified by the inclusion of an Fock component. The shift in the mass eigenvalue is found to be largest for triplet states. If me is taken to be a substantial fraction of , the integrated probability of the electronic component of the state is found to be as large as . Initial studies of the Lamb shift for the atom are performed. Directions for making the simulations fully realistic are discussed.

Symmetric multivariate polynomials as a basis for three-boson light-front wave functions

We develop a polynomial basis to be used in numerical calculations of light-front Fock-space wave functions. Such wave functions typically depend on longitudinal momentum fractions that sum to unity. For three particles, this constraint limits the two remaining independent momentum fractions to a triangle, for which the three momentum fractions act as barycentric coordinates. For three identical bosons, the wave function must be symmetric with respect to all three momentum fractions. Therefore, as a basis, we construct polynomials in two variables on a triangle that are symmetric with respect to the interchange of any two barycentric coordinates. We find that, through the fifth order, the polynomial is unique at each order, and, in general, these polynomials can be constructed from products of powers of the second- and third-order polynomials. The use of such a basis is illustrated in a calculation of a light-front wave function in two-dimensional ϕ(4) theory; the polynomial basis performs much better than the plane-wave basis used in discrete light-cone quantization.

Compression algorithm for discrete light-cone quantization

We adapt the compression algorithm of Weinstein, Auerbach, and Chandra from eigenvectors of spin lattice Hamiltonians to eigenvectors of light-front field-theoretic Hamiltonians. The latter are approximated by the standard discrete light-cone quantization technique, which provides a matrix representation of the Hamiltonian eigenvalue problem. The eigenvectors are represented as singular value decompositions of two-dimensional arrays, indexed by transverse and longitudinal momenta, and compressed by truncation of the decomposition. The Hamiltonian is represented by a rank-four tensor that is decomposed as a sum of contributions factorized into direct products of separate matrices for transverse and longitudinal interactions. The algorithm is applied to a model theory to illustrate its use.

Dual 2d CFT identification of extremal black rings from holes

Five dimensional Einstein gravity vacuum solutions in general fall into two classes of black rings with horizon topology S^2 \times S^1, and black holes with horizon topology S^3. These solutions are specified by their mass and two spins. There are "overlapping" regions of this parameter space where one has extremal rings and holes of the same spins. We show that for such regions the hole has generically a larger entropy than the ring, and likewise, the central charge of the proposed chiral 2d CFT dual to the hole is larger than that of the ring. For special places of this overlapping region where one of the spins tends to zero, the entropies of the extremal ring and hole also tend to zero and essentially become equal. In this case we are dealing with Extremal Vanishing Horizon (EVH) black holes or rings. The near horizon geometry of the near-EVH hole and rings both contain locally AdS_3 throats, providing a basis for the EVH/CFT proposal, a 2d CFT description of the low energy excitations of EVH hole or ring. We argue how the near-EVH hole and near-EVH ring can be distinguished from this dual 2d CFT viewpoint: The hole is a thermal state with zero temperature in the left sector and finite temperature in the right, while the ring is a generic state in the ground state (of the CFT on the plane) in the left sector and a thermal state in the right. The latter is part of the Hilbert space of the 2d CFT obtained in the Discrete Light Cone Quantization (DLCQ).

LIGHT-FRONT HAMILTONIAN AND PATH INTEGRAL QUANTIZATION OF VECTOR SCHWINGER MODEL WITH A PHOTON MASS TERM

Vector Schwinger model with a mass term for the photon, describing 2D electrodynamics with massless fermions, studied by us recently [U. Kulshreshtha, Mod. Phys. Lett. A22, 2993 (2007); U. Kulshreshtha and D. S. Kulshreshtha, Int. J. Mod. Phys. A22, 6183 (2007); U. Kulshreshtha, PoS LC2008, 008 (2008)], represents a new class of models. This theory becomes gauge-invariant when studied on the light-front. This is in contrast to the instant-form theory which is gauge-non-invariant. In this work, we study the light-front Hamiltonian and path integral quantization of this theory under appropriate light-cone gauge-fixing. The discretized light-cone quantization of the theory where we wish to make contact with the experimentally observational aspects of the theory would be presented in a separate paper.

Supersymmetry and the discrete light-cone quantization limit of the Lie 3-algebra model of M theory

In arXiv:1003.4694, we proposed two models of M-theory, Hermitian 3-algebra model and Lie 3-algebra model. In this paper, we study the Lie 3-algebra model with a Lorentzian Lie 3-algebra. This model is ghost-free despite the Lorentzian 3-algebra. We show that our model satisfies two criteria as a model of M-theory. First, we show that the model possesses N=1 supersymmetry in eleven dimensions. Second, we show the model reduces to BFSS matrix theory with finite size matrices in a DLCQ limit.

Near-horizon limits of massless BTZ and their CFT duals

We consider the massless Bañados–Teitelboim–Zanelli (BTZ) black hole and show that it is possible to take its ‘near-horizon’ limit in two distinct ways. The first one leads to a null self-dual orbifold of AdS3 and the second to a singular spacelike AdS orbifold in the large K limit, the ‘pinching orbifold’. We show that from the dual 2D conformal field theory (CFT) viewpoint, the null orbifold corresponds to the p+ = 0 sector of the discrete light-cone quantization of the 2D CFT where a chiral sector of the CFT is decoupled, while the pinching orbifold corresponds to taking an infinite mass gap limit on both the right and left sectors of the 2D CFT, essentially leaving us with the states only. In the latter case, one can combine the near-horizon limit with sending the 3D Planck length ℓP to zero or equivalently the dual 2D CFT central charge c to infinity. We provide preliminary evidence that in the latter case some nontrivial dynamics may survive the limit.

Holography for chiral scale-invariant models

Deformation of any d-dimensional conformal field theory by a constant null source for a vector operator of dimension (d + z -1) is exactly marginal with respect to anisotropic scale invariance, of dynamical exponent z. The holographic duals to such deformations are AdS plane waves, with z=2 being the Schrodinger geometry. In this paper we explore holography for such chiral scale-invariant models. The special case of z=0 can be realized with gravity coupled to a scalar, and is of particular interest since it is related to a Lifshitz theory with dynamical exponent two upon dimensional reduction. We show however that the corresponding reduction of the dual field theory is along a null circle, and thus the Lifshitz theory arises upon discrete light cone quantization of an anisotropic scale invariant field theory.