119225597
ABJM Wilson loops in the Fermi gas approach
The matrix model of ABJM theory can be formulated in terms of a Fermi gas in an external potential. We show that, in this formalism, vevs of Wilson loops correspond to averages of operators in the statistical-mechanical problem. This makes it possible to calculate these vevs at all orders in 1/N, up to exponentially small corrections, and for arbitrary Chern-Simons coupling, by using the WKB expansion. We present explicit results for the vevs of 1/6 and the 1/2 BPS Wilson loops, at any winding number, in terms of Airy functions. Our expressions are shown to reproduce the low genus results obtained previously in the 't Hooft expansion.
119256812
Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops
We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the $${\mathcal N=4}$$ supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure $${\mathcal N=2}$$ and the $${\mathcal N=2^*}$$ supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional $${\mathcal N=2}$$ superconformal gauge theory is treated similarly.
7447447
Wilson Loops in N=4 Supersymmetric Yang--Mills Theory
Perturbative computations of the expectation value of the Wilson loop in N=4 supersymmetric Yang-Mills theory are reported. For the two special cases of a circular loop and a pair of anti-parallel lines, it is shown that the sum of an infinite class of ladder-like planar diagrams, when extrapolated to strong coupling, produces an expectation value characteristic of the results of the AdS/CFT correspondence, $ \sim\exp((constant)\sqrt{g^2N})$. For the case of the circular loop, the sum is obtained analytically for all values of the coupling. In this case, the constant factor in front of $\sqrt{g^2N}$ also agrees with the supergravity results. We speculate that the sum of diagrams without internal vertices is exact and support this conjecture by showing that the leading corrections to the ladder diagrams cancel identically in four dimensions. We also show that, for arbitrary smooth loops, the ultraviolet divergences cancel to order $g^4N^2$.
7417188
An exact prediction of
We propose that the expectation value of a circular BPS-Wilson loop in N=4 supersymmetric Yang--Mills can be calculated exactly, to all orders in a 1/N expansion and to all orders in g{sup 2}N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in {alpha}' and to all orders in g{sub s}. We then compare this result with string theory. We find that the gauge theory calculation, for large g{sup 2}N and to all orders in the 1/N{sup 2} expansion, does agree with the leading string theory calculation, to all orders in g{sub s} and to lowest order in {alpha}'. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.
15963486
Exact results for Wilson loops in superconformal Chern-Simons theories with matter
We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any $$\mathcal{N}= 2 $$ theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.
118581984
The exact superconformal R-symmetry extremizes Z
A bstractThe three sphere partition function, Z , of three dimensional theories with four supercharges and an R-symmetry is computed using localization, resulting in a matrix integral over the Cartan of the gauge group. There is a family of couplings to the curved background, parameterized by a choice of R-charge, such that supersymmetry is preserved; Z is a function of those parameters. The magnitude of the result is shown to be extremized for the superconformal R-charge of the infrared conformal field theory, in the absence of mixing of the R-symmetry with accidental symmetries. This exactly determines the IR superconformal R-charge.
118564204
Notes on SUSY gauge theories on three-sphere
We extend theformulaforpartitionfunctions of $$\mathcal{N}=2 $$ superconformalgauge theories on S3 obtained recently by Kapustin, Willett and Yaakov, to incorporate matter fields with arbitrary R-charge assignments. We use the result to check that the self-mirror property of $$\mathcal{N}=4 $$ SQED with two electron hypermultiplets is preserved under a certain mass deformation which breaks the supersymmetry to $$\mathcal{N}=2 $$.
16987793
N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
We construct three dimensional Chern-Simons-matter theories with gauge groups U(N) × U(N) and SU(N) × SU(N) which have explicit = 6 superconformal symmetry. Using brane constructions we argue that the U(N) × U(N) theory at level k describes the low energy limit of N M2-branes probing a C4/Zk singularity. At large N the theory is then dual to M-theory on AdS4 × S7/Zk. The theory also has a 't Hooft limit (of large N with a fixed ratio N/k) which is dual to type IIA string theory on AdS4 × CP3. For k = 1 the theory is conjectured to describe N M2-branes in flat space, although our construction realizes explicitly only six of the eight supersymmetries. We give some evidence for this conjecture, which is similar to the evidence for mirror symmetry in d = 3 gauge theories. When the gauge group is SU(2) × SU(2) our theory has extra symmetries and becomes identical to the Bagger-Lambert theory.
2229881
Wilson loops in 3-dimensional N=6 supersymmetric Chern-Simons Theory and their string theory duals
archiveprefix: arXiv primaryclass: hep-th reportnumber: HU-EP-08-20 slaccitation: %%CITATION = ARXIV:0809.2787;%%
16908302
Supersymmetric Wilson loops in N = 6 super-ChernSimons-matter theory
Abstract We study supersymmetric Wilson loop operators in ABJM theory from both sides of the AdS 4 / CFT 3 correspondence. We first construct some supersymmetric Wilson loops. The perturbative computations are performed in the field theory side at the first two orders. A fundamental string solution ending on a circular loop is also studied.
7462160
Wilson loops in superconformal Chern-Simons theory and fundamental strings in Anti-de Sitter supergravity dual
We study Wilson loop operators in three-dimensional, = 6 superconformal Chern-Simons theory dual to IIA superstring theory on AdS4 ? 3. Novelty of Wilson loop operators in this theory is that, for a given contour, there are two linear combinations of Wilson loop transforming oppositely under time-reversal transformation. We show that one combination is holographically dual to IIA fundamental string, while orthogonal combination is set to zero. We gather supporting evidences from detailed comparative study of generalized time-reversal transformations in both D2-brane worldvolume and ABJM theories. We then classify supersymmetric Wilson loops and find at most 1/6 supersymmetry. We next study Wilson loop expectation value in planar perturbation theory. For circular Wilson loop, we find features remarkably parallel to circular Wilson loop in = 4 super Yang-Mills theory in four dimensions. First, all odd loop diagrams vanish identically and even loops contribute nontrivial contributions. Second, quantum corrected gauge and scalar propagators take the same form as those of = 4 super Yang-Mills theory. Combining these results, we propose that expectation value of circular Wilson loop is given by Wilson loop expectation value in pure Chern-Simons theory times zero-dimensional Gaussian matrix model whose variance is specified by an interpolating function of `t Hooft coupling. We suggest the function interpolates smoothly between weak and strong coupling regime, offering new test ground of the AdS/CFT correspondence.
118833950
From Weak to Strong Coupling in ABJM Theory
The partition function of $${\mathcal{N}=6}$$ supersymmetric Chern–Simons-matter theory (known as ABJM theory) on $${\mathbb{S}^3}$$ , as well as certain Wilson loop observables, are captured by a zero dimensional super-matrix model. This super–matrix model is closely related to a matrix model describing topological Chern–Simons theory on a lens space. We explore further these recent observations and extract more exact results in ABJM theory from the matrix model. In particular we calculate the planar free energy, which matches at strong coupling the classical IIA supergravity action on $${{\rm AdS}_4\times\mathbb{C}\mathbb{P}^3}$$ and gives the correct N3/2 scaling for the number of degrees of freedom of the M2 brane theory. Furthermore we find contributions coming from world-sheet instanton corrections in $${\mathbb{C}\mathbb{P}^3}$$ . We also calculate non-planar corrections, both to the free energy and to the Wilson loop expectation values. This matrix model appears also in the study of topological strings on a toric Calabi–Yau manifold, and an intriguing connection arises between the space of couplings of the planar ABJM theory and the moduli space of this Calabi–Yau. In particular it suggests that, in addition to the usual perturbative and strong coupling (AdS) expansions, a third natural expansion locus is the line where one of the two ’t Hooft couplings vanishes and the other is finite. This is the conifold locus of the Calabi–Yau, and leads to an expansion around topological Chern–Simons theory. We present some explicit results for the partition function and Wilson loop observables around this locus.
59322523
Nonperturbative aspects of ABJM theory
A bstractUsing the matrix model which calculates the exact free energy of ABJM theory on $ {\mathbb{S}^3} $ we study non-perturbative effects in the large N expansion of this model, i.e., in the genus expansion of type IIA string theory on AdS4 × $ \mathbb{C}{\mathbb{P}^3} $. We propose a general prescription to extract spacetime instanton actions from general matrix models, in terms of period integrals of the spectral curve, and we use it to determine them explicitly in the ABJM matrix model, as exact functions of the ’t Hooft coupling. We confirm numerically that these instantons control the asymptotic growth of the genus expansion. Furthermore, we find that the dominant instanton action at strong coupling determined in this way exactly matches the action of an Euclidean D2-brane instanton wrapping $ \mathbb{R}{\mathbb{P}^3} $.
119331715
Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes
We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the ‘number’ of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.
6313413
Holomorphic anomaly in gauge theories and matrix models
We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be calculated to all genus as quasimodular functions of Γ(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.
509033
Holomorphic anomaly and matrix models
The genus g free energies of matrix models can be promoted to modular invariant, non-holomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these non-holomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf-Vafa conjecture relating matrix models to type B topological strings on certain local Calabi-Yau threefolds.
119114923
Integrability of the holomorphic anomaly equations
We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the 1/N2 expansion of matrix models with more then one cut.
53492847
Exact results in ABJM theory from topological strings
Recently, Kapustin, Willett and Yaakov have found, by using localization techniques, that vacuum expectation values of Wilson loops in ABJM theory can be calculated with a matrix model. We show that this matrix model is closely related to Chern-Simons theory on a lens space with a gauge supergroup. This theory has a topological string large N dual, and this makes possible to solve the matrix model exactly in the large N expansion. In particular, we find the exact expression for the vacuum expectation value of a 1/6 BPS Wilson loop in the ABJM theory, as a function of the 't Hooft parameters, and in the planar limit. This expression gives an exact interpolating function between the weak and the strong coupling regimes. The behavior at strong coupling is in precise agreement with the prediction of the AdS string dual. We also give explicit results for the 1/2 BPS Wilson loop recently constructed by Drukker and Trancanelli.
10253968
Higher genus correlators for the Hermitian matrix model with multiple cuts
Abstract An iterative scheme is set up for solving the loop equation of the hermitian one-matrix model with a multi-cut structure. Explicit results are presented for genus one for an arbitrary but finite number of cuts. Due to the complicated form of the boundary conditions, the loop correlators now contain elliptic integrals. This demonstrates the existence of new universality classes for the hermitian matrix model. The two-cut solution is investigated in more detail, including the double scaling limit. It is shown that in special cases it differs from the known continuum solution with one cut.
119116841
Summing up all genus free energy of ABJM matrix model
The localization technique allows us to compute the free energy of the U(N)k × U(N)−k Chern-Simons-matter theory dual to type IIA strings on AdS4 × CP3 from weak to strong ’t Hooft coupling λ = N/k at finite N, as demonstrated by Drukker, Mariño, and Putrov. In this note we study further the free energy at large ’t Hooft coupling with the aim of testing AdS/CFT at the quantum gravity level and, in particular, sum up allthe1/N corrections, apart from the worldsheet instanton contributions. The all genus partition function takes a remarkably simple form — the Airy function, $ {\text{Ai}}\left( {{{\left( {{{{\pi {k^2}}} \left/ {{\sqrt {2} }} \right.}} \right)}^{{{2} \left/ {3} \right.}}}{\lambda_{\text{ren}}}} \right) $, with the renormalized ’t Hooft coupling λren.
15441660
Multimatrix models and tri-Sasaki Einstein spaces
Localization methods reduce the path integrals in N{>=}2 supersymmetric Chern-Simons gauge theories on S{sup 3} to multimatrix integrals. A recent evaluation of such a two-matrix integral for the N=6 superconformal U(N)xU(N) Aharony-Bergman-Jafferis-Maldacena theory produced detailed agreement with the AdS/CFT correspondence, explaining, in particular, the N{sup 3/2} scaling of the free energy. We study a class of p-matrix integrals describing N=3 superconformal U(N){sup p} Chern-Simons gauge theories. We present a simple method that allows us to evaluate the eigenvalue densities and the free energies in the large N limit keeping the Chern-Simons levels k{sub i} fixed. The dual M-theory backgrounds are AdS{sub 4}xY, where Y are seven-dimensional tri-Sasaki Einstein spaces specified by the k{sub i}. The gravitational free energy scales inversely with the square root of the volume of Y. We find a general formula for the p-matrix free energies that agrees with the available results for volumes of the tri-Sasaki Einstein spaces Y, thus providing a thorough test of the corresponding AdS{sub 4}/CFT{sub 3} dualities. This formula is consistent with the Seiberg duality conjectured for Chern-Simons gauge theories.
55759997
ABJM theory as a Fermi gas
The partition function on the three-sphere of many supersymmetric Chern-Simons-matter theories reduces, by localization, to a matrix model. We develop a new method to study these models in the M-theory limit, but at all orders in the 1/N expansion. The method is based on reformulating the matrix model as the partition function of an ideal Fermi gas with a non-trivial, one-particle quantum Hamiltonian. This new approach leads to a completely elementary derivation of the N^{3/2} behavior for ABJM theory and N=3 quiver Chern-Simons-matter theories. In addition, the full series of 1/N corrections to the original matrix integral can be simply determined by a next-to-leading calculation in the WKB or semiclassical expansion of the quantum gas, and we show that, for several quiver Chern-Simons-matter theories, it is given by an Airy function. This generalizes a recent result of Fuji, Hirano and Moriyama for ABJM theory. It turns out that the semiclassical expansion of the Fermi gas corresponds to a strong coupling expansion in type IIA theory, and it is dual to the genus expansion. This allows us to calculate explicitly non-perturbative effects due to D2-brane instantons in the AdS background.
14124024
Fractional M2-branes
We consider two generalizations of the = 6 superconformal Chern-Simons-matter theories with gauge group U(N) × U(N). The first generalization is to = 6 superconformal U(M) × U(N) theories, and the second to = 5 superconformal O(2M) × USp(2N) and O(2M+1) × USp(2N) theories. These theories are conjectured to describe M2-branes probing C4/Zk in the unitary case, and C4/k in the orthogonal/symplectic case, together with a discrete flux, which can be interpreted as |M−N| fractional M2-branes localized at the orbifold singularity. The classical theories with these gauge groups have been constructed before; in this paper we focus on some quantum aspects of these theories, and on a detailed description of their M theory and type IIA string theory duals.
197472145
Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories
In these lectures, I give a pedagogical presentation of some of the recent progress in supersymmetric Chern–Simons-matter theories, coming from the use of localization and matrix model techniques. The goal is to provide a simple derivation of the exact interpolating function for the free energy of ABJM theory on the three-sphere, which implies in particular the N3/2 behavior at strong coupling. I explain in detail part of the background needed to understand this derivation, like holographic renormalization, localization of path integrals and large N techniques in matrix models.
211258262
A supermatrix model for N = 6 super Chern-Simons-matter theory
We construct the Wilson loop operator of N = 6 super Chern-Simons-matter which is invariant under half of the supercharges of the theory and is dual to the simplest macroscopic open string in AdS4 ×CP3. The Wilson loop couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bi-fundamental representation of the U(N) × U(M) gauge group. These ingredients are naturally combined into a superconnection whose holonomy gives the Wilson loop, which can be defined for any representation of the supergroup U(N |M). Explicit expressions for loops supported along an infinite straight line and along a circle are presented. Using the localization calculation of Kapustin et al. we show that the circular loop is computed by a supermatrix model and discuss the connection to pure Chern-Simons theory with supergroup U(N |M).
119190806
1/2-BPS Wilson loops and vortices in ABJM model
We explore the low-energy dynamics of 1/2-BPS heavy particles coupled to the ABJM model via the Higgsing of M2-branes, with focus on physical understanding of the recently discovered 1/2-BPS Wilson loop operators. The low-energy theory of 1/2-BPS heavy particles turns out to have the U(N|N) supergauge symmetry, which explains the novel structure of the 1/2-BPS Wilson loop operator as a holonomy of a U(N|N) superconnection. We show that the supersymmetric transformation of the Wilson loop operator can be identified as a fermionic supergauge transformation, which leads to their invariance under half of the supersymmetry. We also argue that 1/2-BPS Wilson loop operators appear as 1/2-BPS vortices with vorticity 1/k. Such a vortex can be naturally interpreted as a membrane wrapping the $$ {\mathbb{Z}_t} $$ cycle once, or type IIA fundamental string.
115884066
Supergroups and Their Representations
About a decade ago a new kind of symmetry principle appeared in physics, namely supersymmetry. The novel feature of this symmetry is that it operates between bosons and fermions which have different space-time (or spin and statistics) properties. The generators of supersymmetry transformations form a Lie superalgebra whose even subalgebra is an ordinary Lie algebra and the odd generators, which mix bosons and fermions, close under anti-commutation to the even part.
56086058
Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories
We study various aspects of the matrix models calculating free energies and Wilson loop observables in supersymmetric Chern-Simons-matter theories on the three-sphere. We first develop techniques to extract strong coupling results directly from the spectral curve describing the large N master field. We show that the strong coupling limit of the gauge theory corresponds to the so-called tropical limit of the spectral curve. In this limit, the curve degenerates to a planar graph, and matrix model calculations reduce to elementary line integrals along the graph. As an important physical application of these tropical techniques, we study $ \mathcal{N} = 3 $ theories with fundamental matter, both in the quenched and in the unquenched regimes. We calculate the exact spectral curve in the Veneziano limit, and we evaluate the planar free energy and Wilson loop observables at strong coupling by using tropical geometry. The results are in agreement with the predictions of the AdS duals involving tri-Sasakian manifolds.
14162048
Topological expansion for the 1-hermitian matrix model correlation functions
We rewrite the loop equations of the hermitian matrix model, in a way which involves no derivative with respect to the potential, we compute all the correlation functions, to all orders in the topological 1/N2 expansion, as residues on an hyperelliptical curve. Those residues, can be represented as Feynman graphs of a cubic field theory on the curve.
11353852
Local Mirror Symmetry at Higher Genus
We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa and the local mirror map, the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa.
15702640
Exact Results for Topological Strings on Resolved Y p,q Singularities
We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y p,q manifolds and give rise via M-theory compactification to SU(p) gauge theories on $${\mathbb{R}^4\times S^1}$$ . As an application we present a detailed study of the local $${\mathbb{F}_2}$$ case and compute open and closed genus zero Gromov-Witten invariants of the $${\mathbb{C}^3/\mathbb{Z}_4}$$ orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A1 Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y p,q geometries.
14503397
Open string amplitudes and large order behavior in topological string theory
We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open string amplitudes beyond the disk, and in particular we write down the annulus amplitude in terms of theta functions on a Riemann surface. We test these ideas on local curves and local surfaces, providing in this way generating functionals for open Gromov-Witten invariants in the spirit of mirror symmetry. In the case of local curves, we study the open string sector near the critical point which leads to 2d gravity, and we show that toric D-branes become FZZT branes in a double-scaling limit. We use this connection to compute non-perturbative instanton effects due to D-branes that control the large order behavior of topological string theory on these backgrounds.
18733586
Remodeling the B-Model
We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi–Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of Ap fibrations, where the amplitudes compute the ’t Hooft expansion of vevs of Wilson loops in Chern-Simons theory on lens spaces. We also use our formalism to predict the disk amplitude for the orbifold $${{mathbb {C}}^3 /{mathbb{Z}}_3}$$ .
15279378
Gravitational corrections in supersymmetric gauge theory and matrix models
Gravitational corrections in N = 1 and N = 2 supersymmetric gauge theories are obtained from topological string amplitudes. We show how they are recovered in matrix model computations. This provides a test of the proposal by Dijkgraaf and Vafa beyond the planar limit. Both, matrix model and topological string theory, are used to check a conjecture of Nekrasov concerning these gravitational couplings in Seiberg-Witten theory. Our analysis is performed for those gauge theories which are related to the cubic matrix model, i.e. pure SU(2) Seiberg-Witten theory and N = 2 U(N) SYM broken to N = 1 via a cubic superpotential. We outline the computation of the topological amplitudes for the local Calabi-Yau manifolds which are relevant for these two cases.
62281211
Handbook of elliptic integrals for engineers and scientists
Download HANDBOOK OF ELLIPTIC INTEGRALS FOR ENGINEERS AND SCIENTISTS PDF. Consequently, it is of interest to approximate complete elliptic integrals by simple and accurate algebraic formulas over the As such, it should be of interest to practitioners in the water engineering community. Handbook of elliptic integrals for ResearchGate is the professional network for scientists and researchers. We also introduce an integral operator on the set of means and investigate its properties. Handbook of Elliptic Integrals for Engineers and Scientists. Springer.
120940283
Behavior of Reduced Density Matrices of the Ideal Fermi Gas
whereas for T=;60, this well-known behavior is shown to be valid only when r is smaller than Ar and the asymptotic behavior is found to be J..r- 4r- 2 (1+cos(2k 0r))exp(-ar/J..r), where a is a constant. This is a sum of terms which shows exponential decay and oscillatory exponential decay, respectively. An explicit form of the three-particle distribution function p<3) (r12 , r 23, r 31 ) of the ideal Fermi gas is also given and its behaviors are discussed.
118957135
Statistical Mechanics: Algorithms and Computations
1. Monte Carlo Methods 2. Hard Disks and Spheres 3. Density Matrices and Path Integrals 4. The Bose Gas 5. Order and Disorder in Spin Systems 6. Entropic Forces 7. Dynamic Monte-Carlo Methods
27443761
Off-diagonal long-range order, cycle probabilities, and condensate fraction in the ideal Bose gas.
We discuss the relationship between the cycle probabilities in the path-integral representation of the ideal Bose gas, off-diagonal long-range order, and Bose-Einstein condensation. Starting from the Landsberg recursion relation for the canonic partition function, we use elementary considerations to show that in a box of size L3 the sum of the cycle probabilities of length k>>L2 equals the off-diagonal long-range order parameter in the thermodynamic limit. For arbitrary systems of ideal bosons, the integer derivative of the cycle probabilities is related to the probability of condensing k bosons. We use this relation to derive the precise form of the pik in the thermodynamic limit. We also determine the function pik for arbitrary systems. Furthermore, we use the cycle probabilities to compute the probability distribution of the maximum-length cycles both at T=0, where the ideal Bose gas reduces to the study of random permutations, and at finite temperature. We close with comments on the cycle probabilities in interacting Bose gases.
119807660
Distribution functions in physics: Fundamentals
This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.
121577136
Semiclassical approximations for nuclear hamiltonians. I. Spin-independent potentials
Abstract A systematic semiclassical expansion procedure for physical quantities in nuclei, based on the Thomas-Fermi approximation to the Hartree-Fock equations and constructed in a previous work, is extended here to the realistic case where the effective one-body Hamiltonian for nucleons contains spin-dependent terms. Spin-induced corrections to the kinetic energy density and surface energy of the nucleus, and expressions for various spindependent densities inside the nucleus are computed up to fourth order in for several nuclear Hamiltonians.
119176900
Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant
A bstractWe show that the ABJM theory, which is an $ \mathcal{N} = {6} $ superconformal U(N) × U(N) Chern-Simons gauge theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility of probing the quantum aspects of M-theory and testing the AdS4/CFT3 duality at the quantum level. Here we calculate the free energy, and confirm the N3/2 scaling in the M-theory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion. The method can be easily generalized to the calculations of BPS operators and to other theories that reduce to matrix models.
14603774
On the Moduli Space of Elliptic Maxwell-Chern-Simons Theories
We analyze the moduli space of the low-energy limit of 3-dimensional N = 3 MaxwellChern-Simons theories described by circular quiver diagrams, as for 4-dimensional elliptic models. We define the theories by using D3-NS5-(k,1)5-brane systems with an arbitrary number of fivebranes. The supersymmetry is expected to be enhanced to N =4i n the low-energy limit. We show that the Higgs branch, in which all bifundamental scalar fields develop vacuum expectation values, is an abelian orbifold of C 4 . We confirm that the same geometry is obtained as an M-theory dual of the brane system. We also consider theories realized by introducing more than two kinds of fivebranes, and obtain nontoric fourfolds as moduli spaces. Recently, there has been great interest in 3-dimensional superconformal field theories as theories for describing multiple M2-branes in various backgrounds. This was triggered by the proposal of a new class of 3-dimensional theories by Bagger and Lambert, 1)–3) and Gusstavson. 4),5) The model (BLG model) possesses N(d=3) =8 superconformal symmetry and is based on Lie 3-algebra. The action of the BLG model includes the structure constant f abc d of a Lie 3-algebra, which determines the form of the interactions, and a metric h ab , which appears in the coefficients of the kinetic terms. These tensors must satisfy certain conditions required by the supersymmetry invariance of the action. If these tensors satisfy the conditions, we can write down the action of a BLG model. The constraint imposed on the structure constant is called a fundamental identity. It was soon realized that the identity is very restrictive, 6) and it was proved that if we assume that the metric is positive definite and the algebra is finite dimensional, there is only one nontrivial Lie 3-algebra, 7),8) which is called an A4 algebra. The BLG model based on the A4 algebra is a SU(2)×SU(2) Chern-Simons theory with levels k and −k for each SU(2) factor. Analysis of this model showed that it describes a pair of M2-branes in certain orbifold backgrounds. 9)–11) As a theory for an arbitrary number of M2-branes, a model based on an algebra with a Lorenzian metric was proposed in Refs. 12)– 14). Because of the indefinite metric, the model includes unwanted ghost modes. Although the ghost modes can be removed by treating them as background fields satisfying classical equations of motion, 14),15) or by gauging certain symmetries and fixing them, 16),17) this procedure breaks the conformal invariance, and the theory becomes D2-brane theory 14),16),18) by the mechanism proposed in Ref. 19) unless the
14764800
A simple class of N = 3 gauge/gravity duals
We find the gravity duals to an infinite series of = 3 Chern-Simons quiver theories. They are AdS4 × M7 vacua of M-theory, with M7 in a certain class of 3-Sasaki-Einstein manifolds obtained by a quotient construction. The field theories can be engineered from a brane configuration; their geometry is summarized by a ``hyperKahler toric fan'' that can be read off easily from the relative angles of the branes. The singularity at the tip of the cone over M7 is generically not an orbifold. The simplest new manifolds we consider can be written as the biquotient U(1)\U(3)/U(1). We also comment on the relation between our theories and four-dimensional = 1 theories with the same quiver.
14361074
Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
Abstract We study the vacuum structure and don spectrum of N =2 supersymmetric gauge theory in four dimensions, with gauge group SU(2). The theory turns out to have remarkably rich and physical properties which can nonetheless be described precisely; exact formulas can be obtained, for instance, for electron and dyon masses and the metric on the moduli space of vacua. The description involves a version of Olive-Montonen electric-magnetic duality. The “strongly coupled” vacuum turns out to be a weakly coupled theory of monopoles, and with a suitable perturbation confinement is described by monopole condensation.
8428666
An efficient algorithm for computing the Baker–Campbell–Hausdorff series and some of its applications
We provide a new algorithm for generating the Baker–Campbell–Hausdorff (BCH) series Z=log(eXeY) in an arbitrary generalized Hall basis of the free Lie algebra L(X,Y) generated by X and Y. It is based on the close relationship of L(X,Y) with a Lie algebraic structure of labeled rooted trees. With this algorithm, the computation of the BCH series up to degree of 20 [111 013 independent elements in L(X,Y)] takes less than 15min on a personal computer and requires 1.5Gbytes of memory. We also address the issue of the convergence of the series, providing an optimal convergence domain when X and Y are real or complex matrices.
199433938
Asymptotic ~-expansions of stationary quantum states
We discuss the mathematical framework for uniform asymptotic expansions in the parameter h of non-relativistic quantum mechanics. Quantum observables of interest are taken in an algebra of pseudo-differential operators defined with the help of the Wigner transformation. We define a class of asymptotic quantum states as certain linear functionals on those operators. These functionals are microlocal, and we study their supports in phase space in analogy with wave front set theory; they are also shown to be covariant under the « metaplectic representation » of the affine symplectic transformations in phase space. In this asymptotic framework we can formulate the eigenstate problem for the most general observable. This problem is formally solved by quadratures for onedimensional systems : a Bohr-Sommerfeld formula correct to all orders in h is the obtained for the discrete spectrum. RESUME. Nous formulons un cadre mathématique pour développer la mécanique quantique non-relativiste en series asymptotiques uniformes dans le paramètre h. II faut se restreindre a des observables quantiques appartenant a une certaine algèbre d’opérateurs pseudo-différentiels, définie a l’aide de la transformation de Wigner. Nous définissons alors une classe d’états quantiques asymptotiques comme fonctionnelles linéaires sur ces observables. Ces fonctionnelles sont microlocales, et nous étudions leurs supports dans l’espace de phase en analogie avec la théorie des fronts d’onde ; nous montrons aussi qu’elles sont covariantes par la « représen(*) Detache du C. N. R. S. Annales de l’Institut Henri Poincaré Section A Vol. XXVI, n° 4 1977.
14221396
Supergravity on AdS 5 / 4 × Hopf Fibrations and Conformal Field Theories
We obtain three and four dimensional conformal field theories with less than maximal supersymmetry by using their supergravity duals. These supergravity
13576595
Knot Invariants and Topological Strings
We find further evidence for the conjecture relating large N Chern-Simons theory on S^3 with topological string on the resolved conifold geometry by showing that the Wilson loop observable of a simple knot on S^3 (for any representation) agrees to all orders in N with the corresponding quantity on the topological string side. For a general knot, we find a reformulation of the knot invariant in terms of new integral invariants, which capture the spectrum (and spin) of M2 branes ending on M5 branes embedded in the resolved conifold geometry. We also find an intriguing link between knot invariants and superpotential terms generated by worldsheet instantons in N=1 supersymmetric theories in 4 dimensions.
123673530
Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc
In this paper, we present foundational material towards the development of a rigorous enumerative theory of stable maps with Lagrangian boundary conditions, ie stable maps from bordered Riemann surfaces to a symplectic manifold, such that the boundary maps to a Lagrangian submanifold. Our main application is to a situation where our proposed theory leads to a well-defined algebro-geometric computation very similar to well-known localization techniques in Gromov–Witten theory. In particular, our computation of the invariants for multiple covers of a generic disc bounding a special Lagrangian submanifold in a Calabi–Yau threefold agrees completely with the original predictions of Ooguri and Vafa based on string duality. Our proposed invariants depend more generally on a discrete parameter which came to light in the work of Aganagic, Klemm, and Vafa which was also based on duality, and our more general calculations agree with theirs up to sign.
59362887
Interacting fermions and $ \mathcal{N} $ = 2 Chern-Simons-matter theories
A bstractThe partition function on the three-sphere of $ \mathcal{N} $ = 3 Chern-Simons-matter theories can be formulated in terms of an ideal Fermi gas. In this paper we show that, in theories with $ \mathcal{N} $ = 2 supersymmetry, the partition function corresponds to a gas of interacting fermions in one dimension. The large N limit is the thermodynamic limit of the gas and it can be analyzed with the Hartree and Thomas-Fermi approximations, which lead to the known large N solutions of these models. We use this interacting fermion picture to analyze in detail $ \mathcal{N} $ = 2 theories with one single node. In the case of theories with no long-range forces we incorporate exchange effects and argue that the partition function is given by an Airy function, as in $ \mathcal{N} $ = 3 theories. For the theory with g adjoint superfields and long-range forces, the Thomas-Fermi approximation leads to an integral equation which determines the large N, strongly coupled R-charge.
5784033
Topological Strings and (Almost) Modular Forms
The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space H3(X). We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local Calabi-Yau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP2 and IP1  ×  IP1. As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold $${{\mathbb {C}^3} / {\mathbb {Z}_3}}$$.