THE STRING DIFFERENCE EQUATION OF THE D=1 MATRIX MODEL AND W1+∞ SYMMETRY OF THE KP HIERARCHY

1992 ◽  
Vol 07 (20) ◽  
pp. 4791-4802 ◽  
Author(s):  
M.A. AWADA ◽  
S.J. SIN
Keyword(s):  
Free Energy ◽  
String Theory ◽  
Difference Equation ◽  
Matrix Model ◽  
Zero Mode ◽  
Legendre Transformation ◽  
Kp Hierarchy ◽  
Kdv Hierarchy ◽  
The Difference ◽  
Special Case

We give a connection between the D=1 matrix model and the generalized KP hierarchy. First, we find a difference equation satisfied by F, the Legendre transformation of the free energy of the D=1 matrix model on a circle of radius R. Then we show that it is a special case of the difference equation of the generalized KP hierarchy with its zero mode identified with the scaling variable of the D=1 string theory. We propose that the massive D=1 matrix model is described by the generalized KP hierarchy, which implies the manifest integrability of D=1 string theory. We also show that the (generalized) KP hierarchy has an underlying W1+∞ symmetry. By reduction, we prove that the generalized KdV hierarchy has a subalgebra of the above symmetry which again forms a W1+∞. We argue that there are no W constraints in D=1 string theory, which is in contrast to D<1 theories, where there are W1+∞ constraints.

scholarly journals 1/N expansion of circular Wilson loop in $$ \mathcal{N} $$ = 2 superconformal SU(N) × SU(N) quiver

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
M. Beccaria ◽  
A.A. Tseytlin
Keyword(s):  
Free Energy ◽  
Gauge Theory ◽  
String Theory ◽  
Strong Coupling ◽  
Matrix Model ◽  
Wilson Loop ◽  
Numerical Study ◽  
Expectation Value ◽  
Non Gaussian ◽  
Localization Approach

Abstract Localization approach to $$ \mathcal{N} $$ N = 2 superconformal SU(N) × SU(N) quiver theory leads to a non-Gaussian two-matrix model representation for the expectation value of BPS circular SU(N) Wilson loop $$ \left\langle \mathcal{W}\right\rangle $$ W . We study the subleading 1/N2 term in the large N expansion of $$ \left\langle \mathcal{W}\right\rangle $$ W at weak and strong coupling. We concentrate on the case of the symmetric quiver with equal gauge couplings which is equivalent to the ℤ2 orbifold of the SU(2N) $$ \mathcal{N} $$ N = 4 SYM theory. This orbifold gauge theory should be dual to type IIB superstring in AdS5 × (S5/ℤ2). We present a string theory argument suggesting that the 1/N2 term in $$ \left\langle \mathcal{W}\right\rangle $$ W in the orbifold theory should have the same strong-coupling asymptotics λ3/2 as in the $$ \mathcal{N} $$ N = 4 SYM case. We support this prediction on the gauge theory side by a numerical study of the localization matrix model. We also find a relation between the 1/N2 term in the Wilson loop expectation value and the derivative of the free energy of the orbifold gauge theory on 4-sphere.

scholarly journals Integrability of type 0A matrix model in the presence of D-brane

2019 ◽  
Vol 34 (03n04) ◽  
pp. 1950020
Author(s):  
Chandrima Paul
Keyword(s):  
Free Energy ◽  
String Theory ◽  
Partition Function ◽  
Matrix Model ◽  
Canonical Ensemble ◽  
Fredholm Determinant ◽  
Grand Canonical Ensemble ◽  
Integrable Structure ◽  
The Matrix ◽  
Grand Canonical

We consider type 0A matrix model in the presence of spacelike D-brane which is localized in matter direction at any arbitrary point. In string theory, the boundary state, which in matrix model corresponds to the Laplace transform of the macroscopic loop operator, is known to obey the operator constraints corresponding to open string boundary condition. When we analyze MQM as well as the respective collective field theory and compare it with dual string theory, it appears that consistency of the theory requires a condition equivalent to a constraint on the matter part that needed to be imposed in the matrix model. We identified this condition and observed that this results in constraining the macroscopic loop operator so that it projects the Hilbert space generated by the operator to its physical sector at the point of insertion while keeping the bulk matrix model unaffected, thereby describing a situation parallel to string theory. We analyzed the theory with uncompactified time and have shown explicitly that the matrix model predictions are in good agreement with the relevant string theory. Next, we considered the theory with compactified time, analyzed MQM on a circle in the presence of D-brane. We evaluated the partition function along with the constrained macroscopic loop operator in the grand canonical ensemble and showed the free energy corresponds to that of a deformed Fermi surface. We have also shown that the path integral in the presence of D-brane can be expressed as the Fredholm determinant. We have studied the fermionic scattering in a semiclassical regime. Finally, we considered the compactified theory in the presence of the D-brane with tachyonic background. We evaluated the free energy in the grand canonical ensemble. We have shown the integrable structure of the respective partition function and it corresponds to the tau function of Toda hierarchy. We have also analyzed the dispersionless limit.

scholarly journals MATRIX MODELS AND ONE-DIMENSIONAL OPEN STRING THEORY

1993 ◽  
Vol 08 (20) ◽  
pp. 3599-3614 ◽  
Author(s):  
JOSEPH A. MINAHAN
Keyword(s):  
Free Energy ◽  
String Theory ◽  
Matrix Model ◽  
Scaling Limit ◽  
Open String ◽  
Model Potential ◽  
Boundary Field ◽  
Random Matrix Model ◽  
Open Strings ◽  
The Matrix

We propose a random matrix model as a representation for D = 1 open strings. We show that the model with one flavor of boundary fields is equivalent to N fermions with spin in a central potential that also includes a long-range ferromagnetic interaction between the fermions that falls off as 1/(rij)2. We also generalize this theory to contain an arbitrary number of flavors. For an appropriate choice of the matrix model potential the ground state of the system can be found. Using this potential, we calculate the free energy in the double scaling limit and show that the free energy expansion has the expected form for a theory of open and closed strings if the boundary field mass and couplings have a logarithmic divergence. We then examine the critical properties of this theory and show that the length of the boundary around a hole remains finite, even near the critical point. We also argue that unlike critical string theory or a D = 0 theory, the open string coupling constant is a free parameter.

Synthesis and characterization of a series of 1-methyl-4-[2-aryl-1-diazenyl]piperazines and a series of ethyl 4-[2-aryl-1-diazenyl]-1-piperazinecarboxylates

2004 ◽  
Vol 82 (8) ◽  
pp. 1294-1303 ◽  
Author(s):  
Vanessa Renée Little ◽  
Keith Vaughan
Keyword(s):  
Free Energy ◽  
Chemical Shifts ◽  
Linear Free Energy Relationship ◽  
Piperazine Ring ◽  
Linear Free Energy ◽  
In Series ◽  
Methylene Groups ◽  
Diazonium Coupling ◽  
The Difference

1-Methylpiperazine was coupled with a series of diazonium salts to afford the 1-methyl-4-[2-aryl-1-diazenyl]piperazines (2), a new series of triazenes, which have been characterized by 1H and 13C NMR spectroscopy, IR spectroscopy, and elemental analysis. Assignment of the chemical shifts to specific protons and carbons in the piperazine ring was facilitated by comparison with the chemical shifts in the model compounds piperazine and 1-methylpiperazine and by a HETCOR experiment with the p-tolyl derivative (2i). A DEPT experiment with 1-methylpiperazine (6) was necessary to distinguish the methyl and methylene groups in 6, and a HETCOR spectrum of 6 enabled the correlation of proton and carbon chemical shifts. Line broadening of the signals from the ring methylene protons is attributed to restricted rotation around the N2-N3 bond of the triazene moiety in 2. The second series of triazenes, the ethyl 4-[2-phenyl-1-diazenyl]-1-piperazinecarboxylates (3), have been prepared by similar diazonium coupling to ethyl 1-piperazinecarboxylate and were similarly characterized. The chemical shifts of the piperazine ring protons are much closer together in series 3 than in series 2, resulting in distortion of the multiplets for these methylenes. It was noticed that the difference between these chemical shifts in 3 exhibited a linear free energy relationship with the Hammett substituent constants for the substituents in the aryl ring. Key words: triazene, piperazine, diazonium coupling, NMR, HETCOR, linear free energy relationship.

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals Boundary conditions in topological AdS4/CFT3

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Pietro Benetti Genolini ◽  
Matan Grinberg ◽  
Paul Richmond
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Boundary Conditions ◽  
Smooth Solution ◽  
Three Dimensional ◽  
Field Theories ◽  
Legendre Transformation ◽  
Generating Functional ◽  
Holographic Duals

Abstract We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

scholarly journals Detection of New Translocation in Infant Twins with Concordant ALL and Discordant Outcome

2020 ◽  
Vol 13 (1) ◽  
pp. 9-14
Author(s):  
Golamreza Bahoush ◽  
Maryam Vafapour ◽  
Roxana Kariminejad
Keyword(s):  
Gene Rearrangement ◽  
Cytogenetic Study ◽  
Lymphoblastic Leukemia ◽  
Monozygotic Twins ◽  
Protective Role ◽  
Genetic Studies ◽  
Cytogenetic Testing ◽  
The Difference ◽  
Special Case ◽  
Mll Gene Rearrangement

About 2–5% of acute lymphoblastic leukemia (ALL) cases in pediatric patients are infants with an unfavorable prognosis because of high relapse probability. Early detection of the disease is, therefore, very important. Despite the fact that leukemia in twins occurs rarely, more attention has been paid to it in genetic studies. In the present study, through cytogenetic testing, a special case of concordant ALL in monozygotic twins was presented with different outcomes. In spite of an acceptable initial consequence to medical treatment in twins, in another brother (Twin B), early relapse was observed. In the cytogenetic study, both twins expressed t (4; 11) (q21; q23) while twin A expressed t (2; 7) (p10; q10). No cases have previously reported this mutation. Whether this translocation has a protective role for leukemia with mixed-lineage leukemia (MLL) gene rearrangement is still unclear. The difference in the translocation identified in the identical twins is also subject to further investigations.

scholarly journals An experimental and theoretical investigation of diffusion in a two-phase alloy

1948 ◽  
Vol 192 (1031) ◽  
pp. 575-592 ◽  
Keyword(s):  
Free Energy ◽  
Single Phase ◽  
Phase State ◽  
Theoretical Treatment ◽  
Research Association ◽  
Essential Difference ◽  
Two Phase ◽  
Two Factors ◽  
Order Of Magnitude ◽  
The Difference

The present paper describes an investigation of diffusion in the solid state. Previous experimental work has been confined to the case in which the free energy of a mixture is a minimum for the single-phase state, and diffusion decreases local differences of concentration. This may be called ‘diffusion downhill’. However, it is possible for the free energy to be a minimum for the two-phase state; diffusion may then increase differences of concentration; and so may be called ‘diffusion uphill’. Becker (1937) has proposed a simple theoretical treatment of these two types of diffusion in a binary alloy. The present paper describes an experimental test of this theory, using the unusual properties of the alloy Cu 4 FeNi 3 . This alloy is single phase above 800° C and two-phase at lower temperatures, both the phases being face-centred cubic; the essential difference between the two phases is their content of copper. On dissociating from one phase into two the alloy develops a series of intermediate structures showing striking X-ray patterns which are very sensitive to changes of structure. It was found possible to utilize these results for a quantitative study of diffusion ‘uphill’ and ‘downhill’ in the alloy. The experimental results, which can be expressed very simply, are in fair agreement with conclusions drawn from Becker’s theory. It was found that Fick’s equation, dc / dt = D d2c / dx2 , can, within the limits of error, be applied in all cases, with the modification that c denotes the difference of the measured copper concentration from its equilibrium value. The theory postulates that D is the product of two factors, of which one is D 0f the coefficient of diffusion that would be measured if the alloy were an ideal solid solution. The theory is able to calculate D/D 0 , if only in first approximation, and the experiments confirm this calculation. It was found that in most cases the speed of diffusion—‘uphill’ or ‘downhill’—has the order of magnitude of D 0 . * Now with British Electrical Research Association.

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