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We review some aspects of Nambu mechanics on the basis of the works previously published separately by the present author. Main focuses are on three themes, its various symmetry structures, their possible relevance to string/M theory, and a Hamilton-Jacobi like reformulation. We try to elucidate the basic ideas, most of which were rooted in more or less the same ground, and to explain motivations behind these works from a unified and vantage viewpoint. Various unsolved questions are mentioned. We also include some historical account on the genesis of the Nambu mechanics, and discuss (in the Appendix) some parallelism of various ideas behind the Nambu's paper with Dirac's old works which are related to the description of vortical flows in terms of gauge potentials., Comment: Written version of an invited talk in the workshop "Space-time topology behind formation of micro-macro magneto-vortical structure by Nambu mechanics", Osaka City University, Sept. 28--Oct.1, 2020, corrected typos, added sentences, and added references, version to be published in a special section of the journal PTEP

This lecture consists of three parts. In part I, an overview is given on the so-called Matrix theory in the light-front gauge as a proposal for a concrete and non-perturbative formulation of M-theory. I emphasize motivations towards its covariant formulation. Then, in part II, I turn the subject to the so-called Nambu bracket and Nambu mechanics, which were proposed by Nambu in 1973 as a possible extension of the ordinary Hamiltonian mechanics. After reviewing briefly Nambu's original work, it will be explained why his idea may be useful in exploring higher symmetries which would be required for covariant formulations of Matrix theory. Then, using this opportunity, some comments on the nature of Nambu mechanics and its quantization are given incidentally: though they are not particularly relevant for our specialized purpose of constructing covariant Matrix theory, they may be of some interests for further developments in view of possible other applications of Nambu mechanics. The details will be relegated to forthcoming publications. In part III, I give an expository account of the basic ideas and main results from my recent attempt to construct a covariantized Matrix theory on the basis of a simple matrix version of Nambu bracket equipped with some auxiliary variables, which characterize the scale of M-theory and simultaneously play a crucial role in realizing (dynamical) supersymmetry in a covariant fashion., Written version of the lectures delivered in the workshop "Higher Structure in String Theory and M-theory", TFC Thematic Program, Fundamental Problems on Quantum Physics, Tohoku University (March 7-11, 2016), to be published in the proceedings (v2:corrected typos and reference)

We reformulate the Matrix theory of D-particles in a manifestly Lorentz-covariant fashion in the sense of 11 dimesnional flat Minkowski space-time, from the viewpoint of the so-called DLCQ interpretation of the light-front Matrix theory. The theory is characterized by various symmetry properties including higher gauge symmetries, which contain the usual SU($N$) symmetry as a special case and are extended from the structure naturally appearing in association with a discretized version of Nambu's 3-bracket. The theory is scale invariant, and the emergence of the 11 dimensional gravitational length, or M-theory scale, is interpreted as a consequence of a breaking of the scaling symmetry through a super-selection rule. In the light-front gauge with the DLCQ compactification of 11 dimensions, the theory reduces to the usual light-front formulation. In the time-like gauge with the ordinary M-theory spatial compactification, it reduces to a non-Abelian Born-Infeld-like theory, which in the limit of large $N$ becomes equivalent with the original BFSS theory., Comment: 47 pages, 1 figure, the version to be published in JHEP, corrected typos, added sentences and references

§1. Remarks on string field theory from a historical perspective In the first part of the present written report, I would like to review the early history of string theory from the viewpoint of its connection to field theory as well as string field theory, including my personal recollections. I feel somewhat reluctantly that this may be justified, in view of the fact that among the participants of this conference I belong to the eldest generation, starting my research career four decades ago almost simultaneously with the birth of string theory. A hope is that these remarks from a personal point of view tell us something about the nature of string field theory, and also that reflections on the historical side are of some help for conveying to the reader the background and motivations for my recent singular attempt toward a ‘field theory’ of D0-branes. String theory started with the Veneziano amplitude which was originally proposed to be a mathematical model for a duality between Regge-pole and resonance behaviors, being established in the late 1960s experimentally as the characteristics of the scattering of hadrons. The Regge-pole behavior corresponds to the exchange of composite states of hadrons exchanged in t-channel, while the resonance behavior corresponds to the formation of resonating states composed of scattering hadrons in s-channel. The duality meant that a single scattering amplitude can be described either by summation over the Regge pole contributions or over the resonances, but not both simultaneously. The Veneziano formula was regarded as an ideal realization of this duality in the limit of narrow resonance approximation, ignoring finite life times of hadronic states.

I discuss my recent attempt in search of a new framework for quantum field theory of D branes. After explaining some motivations in the background of this project, I present, as a first step towards our goal, a second-quantized reformulation of the U(N) Yang-Mills quantum mechanics in which the D0-brane creation-and-annihilation fields connecting theories with different N are introduced. Physical observables are expressed in terms of bilinear forms of the D0 fields. The large N limit is briefly treated using this new formalism., Comment: 8 pages, 2 figures, based on an invited talk presented at the international workshop "Progress of String Theory and Quantum Field Theory" ( Osaka City University, December 7-10, 2007), to be published in the proceedings

We give a brief expository discussion on the holographic correspondence of correlation functions in the large J limit of AdS/CFT conjecture. We first review our proposals on the interpretation of the so-called GKPW relation in the large J limit or BMN limit, which are based upon a tunneling picture in relating the AdS bulk to its boundary. Some concrete results, explicitly confirming our picture, are summarized. We then proceed to comment on various issues related to this subject, such as extension of the present picture to nonconformal Dp-brane backgrounds, the correlators of deformed Wilson loops, spinning-string/spin-chain correspondence, and the inclusion of higher string-loop effects. In particular, as for the deformation of Wilson loops, we present a typical tunneling world-sheet solution which can be used for direct derivation of the expectation values of deformed Wilson loops following our picture., 21 pages, written version of a talk given at the symposium "Frontiers of Quantum Science", YITP, Kyoto, Feb. 2005, corrected typos and some minor revisions of text

The issue of holographic principle in the PP-wave limit of the AdS/CFT correspondence is discussed, in the hope of clarifying some confusions in the literature. We show that, in the plane-wave limit, the relation between the partition function in the bulk and the gauge-invariant correlation functions on the boundary should be interpreted on the basis of a tunneling picture in the semi-classical approximation which is appropriate for the plane-wave limit. This leads to a natural relation between Euclidean S-matrix in the bulk and the short-distance operator-product expansion of the so-called BMN operators on the boundary.

This is an abbreviated written version of pedagogical lectures delivered in the workshop "Noncommutative Differential Geometry and its Application to Physics" (Shonan, May 31-June 4, 1999). The purpose was to give an overview on the nature of string theory as quantum gravity. After giving a brief historical introduction, we discuss how general relativity can be derived as consistent local field theory of massless spin 2 particle in the low-energy limit. Then, several important issues, such as the manner how string theory contains gravity, the stringy resolution of ultraviolet difficulty, duality between open and closed strings, are critically reviewed.

We analyze the nature of space-time nonlocality in string theory. After giving a brief overview on the conjecture of the space-time uncertainty principle, a (semi-classical) reformulation of string quantum mechanics, in which the dynamics is represented by the noncommutativity between temporal and spatial coordinates, is outlined. The formalism is then compared to the space-time noncommutative field theories associated with nonzero electric B-fields., Comment: 10 pages with one figure. Written version of talks presented at the Strings 2000 (Michigan, July, 2000) and the Summer Workshop on Branes (KIAS, Seoul, June, 2000)

We briefly review the present status of string theory from the viewpoint of its implications on the short-distance space-time structure and black hole physics. Special emphases are given on two closely related issues in recent developments towards nonperturbative string theory, namely, the role of the space-time uncertainty relation as a qualitative but universal characterization of the short-distance structure of string theory and the microscopic formulation of black-hole entropies. We will also suggest that the space-time uncertainty relation can be an underlying principle for the holographic property of M theory, by showing that the space-time uncertainty relation naturally explains the UV/IR relation used in a recent derivation of the holographic bound for D3 brane by Susskind and Witten., Comment: 34 pages, no figures, to appear in the special issue of the Journal of Chaos, Solitons and Fractals on "Superstrings, M, F, S...Theory"

We show that the path-integral quantization of relativistic strings with the Schild action is essentially equivalent to the usual Polyakov quantization at critical space-time dimensions. We then present an interpretation of the Schild action which points towards a derivation of superstring theory as a theory of quantized space-time where the squared string scale plays the role of the minimum quantum for space-time areas. A tentative approach towards such a goal is proposed, based on a microcanonical formulation of large N supersymmetric matrix model., 15 pages, Latex, no figures, a note and references are added

We argue that the space-time uncertainty relation of the form $\ensuremath{\Delta}X\ensuremath{\Delta}T\ensuremath{\gtrsim}{\ensuremath{\alpha}}^{\ensuremath{'}}$ for the observability of the distances with respect to time, $\ensuremath{\Delta}T$, and space, $\ensuremath{\Delta}X$, is universally valid in string theory including D-branes. This relation has been previously proposed by one (T. Y.) of the present authors as a simple qualitative representation of the perturbative short-distance structure of fundamental string theory. We show that the relation, combined with the usual quantum mechanical uncertainty principle, explains the key qualitative features of D-particle dynamics.

We present detailed discussions on the stochastic Hamiltonians for non-critical string field theories on the basis of matrix models. Beginning from the simplest $c=0$ case, we derive the explicit forms of the Hamiltonians for the higher critical case $k=3$ (which corresponds to $c=-22/5$) and for the case $c=1/2$, directly from the double-scaled matrix models. In particular, for the two-matrix case, we do not put any restrictions on the spin configurations of the string fields. The properties of the resulting infinite algebras of Schwinger-Dyson operators associated with the Hamiltonians and the derivation of the Virasoro and $W_3$ algebras therefrom are also investigated. Our results suggest certain universal structure of the stochastic Hamiltonians, which might be useful for an attempt towards a background independent string field theory., Comment: 70 pages, LaTeX, typographical errors are corrected, to be published in Phys. Rev. D

We study correlation functions in (0+1)-dimensional maximally supersymmetric U(N) Yang-Mills theory, which was proposed by Banks et al. as a non-perturbative definition of 11-dimensional M-theory in the infinite-momentum frame. We perform first-principle calculations using Monte Carlo simulations, and compare the results against the predictions obtained previously based on the gauge-gravity correspondence from 10 dimensions. After providing a self-contained review on these predictions, we present clear evidence that the predictions in the large-N limit actually hold even at small N such as N=2 and 3. The predicted behavior seems to continue to the far infrared regime, which goes beyond the naive range of validity of the 10D supergravity analysis. This suggests that the correlation functions also contain important information on the M-theory limit., Comment: v1: 43 pages, 16 figures. v2, v3: minor corrections

We study correlation functions in (0+1)-dimensional maximally supersymmetric U(N) gauge theory, which represents the low-energy effective theory of D0-branes. In the large-N limit, the gauge-gravity duality predicts power-law behaviors in the infrared region for the two-point correlation functions of operators corresponding to supergravity modes. We evaluate such correlation functions on the gauge theory side by the Monte Carlo method. Clear power-law behaviors are observed at N=3, and the predicted exponents are confirmed consistently. Our results suggest that the agreement extends to the M-theory regime, where the supergravity analysis in 10 dimensions may not be justified a priori., 4 pages, 3 figures

The consequences of symmetry properties of a previously proposed action principle which describes the theory space of two-dimensional gravity are investigated. The string equation corresponding to the double scaling limit of the one-matrix model with general polynomial potential is expressed as a bilinear equation for the τ function of the KP hierarchy. The Virasoro condition for the τ function is then shown to be a Ward-like identity derived from the string equation corresponding to a conformal symmetry of the action principle. The possibility of a Ginzburg-Landau type integral representation of the τ function as a version of noncritical closed string field theory in less-than-one-dimensional space-time is discussed. Finally, the role of analyticity with respect to the eigenvalue of the puncture operator in interpreting the action principle is emphasized.

We show that the known solutions to O( N ) symmetric vector models in the double-scaling limit are embedded in the KP hierarchy by a consistent truncation. We also discuss a second-quantized topological formulation of the theory.

We present a nonperturbative theory of randomly branching chains (or polymers). The method is based upon the existence of critical points in the large- N limit of O( N ) symmetric vector models. We derive a class of ordinary differential equations governing the behaviour of the partition function with respect to the coupling constants. Solving these equations explicitly, we derive several exact results. Our model is useful to make comparison with the recent nonperturbative studies of two-dimensional quantum gravity since the structure of the differential equations has many parallel features, such as the flow property, the Virasoro structure in the Schwinger-Dyson equation, and so on, with those of random surfaces.

We present a simple action principle for the non-perturbative string theory in less than one dimension. It generates the continuum String equations and the associated KdV flows. Our approach is based on the operator method developed for studying the KP hierarchy.

In connection with the space-time uncertainty principle which gives a simple qualitative characterization of non-local or non-commutative nature of short-distance space-time structure in string theory, author's recent approaches toward field theories for D-branes are briefly outlined, putting emphasis on some key ideas lying in the background. The final section of the present report is devoted partially to a tribute to Yukawa on the occasion of the centennial of his birth., Comment: 12 pages, 3 figures, written version of an invited talk at the 21st Nishinomiya-Yukawa Mermorial Simposium, Nov. 2006, to be published in the proceedings

We propose a new field-theoretic framework for formulating the non-relativistic quantum mechanics of D particles in a Fock space of U(N) Yang-Mills theories with all different N in a unified way. D-particle field operators, creating and annihilating a D particle and hence changing N one by one, are defined. The base space of these D-particle fields is a (complex) vector space of infinite dimensions. The gauge invariance of Yang-Mills quantum mechanics is reinterpreted as a quantum-statistical symmetry, which is taken into account by setting up a novel algebraic and projective structure in the formalism. Ordinary physical observables of Yang-Mills theory, obeying the standard algebra, are expressed as bilinear forms of the D-particle fields. Together with the open-closed string duality, our new formulation suggests a trinity of three different but mutually dual viewpoints in string theory., Comment: corrected typos, corrected references, 39 pages, 2 figures, version to be published in Prog. Theor. Phys

We extend the fermion representation of single-charge 1/2-BPS operators in the four-dimensional N=4 super Yang-Mills theory to general (multi-charge) 1/2-BPS operators such that all six directions of scalar fields play roles on an equal footing. This enables us to construct a field-theorectic representation for a second-quantized system of spherical D3-branes in the 1/2-BPS sector. The Fock space of D3-branes is characterized by a novel exclusion principle (called `Dexclusion' principle), and also by a nonlocality which is consistent with the spacetime uncertainty relation. The Dexclusion principle is realized by composites of two operators, obeying the usual canonical anticommutation relation and the Cuntz algebra, respectively. The nonlocality appears as a consequence of a superselction rule associated with a symmetry which is related to the scale invariance of the super Yang-Mills theory. The entropy of the so-called superstars, with multiple charges, which have been proposed to be geometries corresponding to the condensation of giant gravitons is discussed from our viewpoint and is argued to be consistent with the Dexclusion principle. Our construction may be regarded as a first step towards a possible new framework of general D-brane field theory., Comment: 43 pages, 4 figures; version 2, corrected typos and added references

The issue of holographic mapping between bulk and boundary in the plane-wave limit of AdS/SYM correspondence is reexamined from the viewpoint of correlation functions. We first study the limit of large angular momentum for the so-called GKP-W relation in supergeravity approximation, connecting directly the effective action in the bulk and the generating functional of correlation functions on the boundary. The spacetime tunneling picture which has been proposed in our previous works naturally emerges. This gives not only a justification of our previous proposal, with some important refinements, on the mapping between bulk effective interaction and the CFT coefficients on the boundary in the plane-wave limit, but also implies various insights on the interpretation of holography in the plane-wave limit. Based on this result, we construct a new `holographic' string field theory. We confirm for several nontrivial examples that this gives the CFT coefficients derived by perturbation theory on the gauge-theory side. Our results are useful for understanding how apparently different duality maps proposed from different standpoints are consistent with each other and with our definite spacetime picture for the AdS holography in the plane-wave limit., Comment: corrected typos, 57 pages

The recent proposal on the correspondence between the ${\cal N}=4$ super Yang-Mills theory and string theory in the Penrose limit of the AdS$_5\times$S$^5$ geometry involves a few puzzles from the viewpoint of holographic principle, especially in connection with the interpretation of times. To resolve these puzzles, we propose to interpret the PP-wave strings on the basis of tunneling null geodesics connecting boundaries of the AdS geometry. Our approach predicts a direct and systematic identification of the S-matrix of Euclidean string theory in the bulk with the short-distance structure of correlation functions of super Yang-Mills theory on the AdS boundary, as an extension of the ordinary relation in supergravity-CFT correspondence. Holography requires an infinite number of contact terms for interaction vertices of string field theory and constrains their forms in a way consistent with supersymmetry., Comment: Section 3 is revised, and a note added explaining the status of our prediction

It is reviewed how space-time supersymmetry is realized nonlinearly in open superstring theory without making the GSO projection. We show that the world-sheet string dualities, viz. dualities of open-closed strings and of open-open strings, play crucial roles for the existence of 10 dimensional N=2 supersymmetry in a spontaneously broken phase. We also speculate on a possible mechanism of the restoration of supersymmetry from the viewpoint of world-sheet dynamics., 12 pages, talk presented at the 10th International Symposium on String Theory (July 3-7, Fukuoka, Japan, 2001) and a part of lectures given at the Summer Conference on Strings (July 16-27, Beijing, China, 2001)

The notion of a space-time uncertainty principle in string theory is clarified and further developed. The motivation and the derivation of the principle are first reviewed in a reasonably self-contained way. It is then shown that the nonperturbative (Borel summed) high-energy and high-momentum transfer behavior of string scattering is consistent with the space-time uncertainty principle. It is also shown that, in consequence of the principle, string theories in 10 dimensions generically exhibit a characteristic length scale which is equal to the well-known 11 dimensional Planck length $g_s^{1/3}\ell_s$ of M-theory as the scale at which stringy effects take over the effects of classical supergravity, even without involving D-branes directly. The meanings of the space-time uncertainty relation in connection with D-branes and black holes are discussed and reinterpreted. Finally, we present a novel interpretation of the Schild-gauge action for strings from a viewpoint of noncommutative geometry, which conforms to the space-time uncertainty relation by manifestly exhibiting a noncommutativity of quantized string coordinates dominantly between space and time. We also discuss the consistency of the space-time uncertainty relation with S and T dualities., Comment: 45 pages, 1 figure, remaining typos corrected, further typos of published version are corrected

Orientable open string theories containing both bosons and fermions without the GSO projection are expected to have the 10 dimensional N=2(A) space-time supersymmetry in a spontaneously broken phase. We study the low-energy theorem for the nonlinearly realized N=2 supersymmetry using the effective action for an unstable D9-brane. It is explicitly confirmed that the 4-fermion open string amplitudes without the GSO projection obey the low-energy theorem derived from the nonlinear N=2 supersymmetry. An intimate connection between the existence of the hidden supersymmetry and the open-open string (s-t) duality is pointed out., Comment: 18 pages, no figure, further typos are corrected

The large N behavior of Matrix theory is discussed on the basis of the previously proposed generalized conformal symmetry. The concept of `oblique' AdS/CFT correspondence, in which the conformal symmetry involves both the space-time coordinates and the string coupling constant, is proposed. Based on the explicit predictions for two-point correlators, possible implications for the Matrix-theory conjecture are discussed., Comment: LaTeX, 10 pages, 2 figures, written version of the talk presented at Strings'99

Motivated by the space-time uncertainty principle, we establish a conformal symmetry in the dynamics of D-particles. The conformal symmetry, combined with the supersymmetric non-renormalization theorem, uniquely determines the classical form of the effective action for a probe D-particle in the background of a heavy D-particle source, previously constructed by Becker-Becker-Polchinski-Tseytlin. Our results strengthen the conjecture proposed by Maldacena on the correspondence, in the case of D-particles, between the supergravity and the supersymmetric Yang-Mills matrix models in the large $N$-limit, the latter being the boundary conformal field theory of the former in the classical D-particle background in the near horizon limit., Comment: 18 pages, no figures, final version to be published in Nucl. Phys. B, typos are corrected and a few sentences are added

We discuss a T-duality transformation for the c=1/2 matrix model for the purpose of studying duality transformations in a possible toy example of nonperturbative frameworks of string theory. Our approach is to first investigate the scaling limit of the Schwinger-Dyson equations and the stochastic Hamiltonian in terms of the dual variables and then compare the results with those using the original spin variables. It is shown that the c=1/2 model in the scaling limit is T-duality symmetric in the sphere approximation. The duality symmetry is however violated when the higher-genus effects are taken into account, owing to the existence of global Z_2 vector fields corresponding to nontrivial homology cycles. Some universal properties of the stochastic Hamiltonians which play an important role in discussing the scaling limit and have been discussed in a previous work by the last two authors are refined in both the original and dual formulations. We also report a number of new explicit results for various amplitudes containing macroscopic loop operators., Comment: RevTex, 46 pages, 5 eps figures

We present a new class of matrix models which are manifestly symmetric under the T-duality transformation of the target space. The models may serve as a nonperturbative regularization for the T-duality symmetry in continuum string theory. In particular, it now becomes possible to extract winding modes explicitly in terms of extended matrix variables., Comment: REVTeX, 15 pages, no figures

We argue from two complementary viewpoints of Holography that the 2-point correlation functions of 1/2-BPS multi-trace operators in the large-N (planar) limit are nothing but the (Wick-rotated) S-matrix elements of c=1 matrix model. On the bulk side, we consider an Euclideanized version of the so-called bubbling geometries and show that the corresponding droplets reach the conformal boundary. Then the scattering matrix of fluctuations of the droplets gives directly the two-point correlators through the GKPW prescription. On the Yang-Mills side, we show that the two-point correlators of holomorphic and anti-holomorphic operators are essentially equivalent with the transformation functions between asymptotic in- and out-states of c=1 matrix model. Extension to non-planar case is also discussed., 28 pages, 3 figures, corrected typos, version to appear in JHEP

We study the expectation values of Wilson-loop operators with the insertionsof local operators Z^J and Zbar^J with large R-charge J from the bulk viewpoint of AdS/CFT correspondence. Classical solutions of strings attached to such deformed Wilson loops at the conformal boundary are constructed and are applied to the computation of Wilson-loop expectation values. We argue that in order to have such solutions for general insertions at finite positions in the base spacetime of the gauge theory, it is crucial to interpret the holographic correspondence in the semi-classical picture as a tunneling phenomenon, as has been previously established for holographic computations of correlators of BMN operators. This also requires to use the Euclideanized AdS background and Euclidean super Yang-Mills theory., 16 pages, 5 figures, version to be published in JHEP, no change from the previous version, only extraneous figure file is removed

On the basis of new integral representation proposed by Fairlie and Martin, we develop a manifestly super-gauge invariant formulation of the Neveu-Schwarz model. We show that the Fairlie-Martin representation is invariant under wider trafisformations (super-Mobius transformations) than the usual Mobius transformations. The corresponding algebra (super­ Mobius algebra) is a closed subalgebra of the super-gauge algebra. The operator formalism is reconstructed as a representation of the super-Mobius algebra. As a generalization of our formulation, we introduce closed-string emission vertices for the Neveu-Schwarz open-string which are manifestly covariant under the super-gauge transformations as well as the confotmal transformations. Finally we discuss briefly the general representation theory of .the s~per­ Mobius algebra. Unfortunately, the representation appeared in the Neveu-Schwarz model is essentially unique as the representation of the full super-gauge algebra. However, two types of_ the mass spectrum are possible.

We discuss the topological properties of the Yang–Mills fields from a unified point of view of the Wu–Yang global formulation. In four‐dimensional Euclidean space, the gauge type is characterized by the second Chern class. The multi‐instanton solution recently found by ’t Hooft and generalized by Jackiw, Nohl, and Rebbi is discussed from this point of view. We also discuss the generalization of the topology of instantons and monopoles in higher dimensional spaces and the relation among them.

Collective motions in internal space are investigated with emphasis on their relevance to the large- N limit. If the internal symmetry is maximal in the sense that the orbit space of the internal symmetry transformations effectively covers the field space in the large- N limit, the internal collective motions are the dominant modes of quantum fluctuations. Conditions for the existence of the large- N master field are discussed from this point of view. Several soluble systems are treated within this formulation and the importance of possible higher hidden symmetries for realistic models is emphasized.