arXiv subjects

A. Marshakov

Publications and source records attributed to A. Marshakov.

At least 37 records · Page 2Linked to original sources

arXiv2009

Strong versus Weak Coupling Confinement in N=2 Supersymmetric QCD

We consider N=2 supersymmetric QCD with the gauge group SU(N_c)=SU(N+1) and N_f number of quark matter multiplets, being perturbed by a small mass term for the adjoint matter, so that its Coulomb branch shrinks to a number of isolated vacua. We discuss the vacuum where r=N quarks develop VEV's for N_f\geq 2N=2N_c-2 (in particular, we focus on the N_f= 2N and N_f= 2N+1 cases). In the equal quark mass limit at large masses this vacuum stays at weak coupling, the low-energy theory has U(N) gauge symmetry and one observes the non-Abelian confinement of monopoles. As we reduce the average quark mass and enter the strong coupling regime the quark condensate transforms into the condensate of dyons. We show that the low energy description in the strongly-coupled domain for the original theory is given by U(N) dual gauge theory of N_f\geq 2N light non-Abelian dyons, where the condensed dyons still cause the confinement of monopoles, and not of the quarks, as can be thought by naive duality.

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arXiv2009

Zamolodchikov asymptotic formula and instanton expansion in N=2 SUSY N_f=2N_c QCD

The AGT relations allow one to convert the Zamolodchikov asymptotic formula for the conformal block into the instanton expansion of the Seiberg-Witten prepotential for theory with two colors and four fundamental flavors. This provides an explicit formula for the instanton corrections in this model, resolving in this way an old problem in Seiberg-Witten theory. The answer is especially elegant for vanishing matter masses, then the bare charge of gauge theory 16q_0 = 16e^{iπτ_0} plays the role of a branch point on the spectral torus. The exact prepotential at this point is F a^2\log q with q\neq q_0, unlike the case of another conformal theory, with massless adjoint field. Instead, 16q_0 = θ_{10}^4/θ_{00}^4(q) = 16q(1+O(q)), i.e. the Zamolodchikov asymptotics gives rise, in particular, to an exact non-perturbative beta-function so that the effective coupling differs from the bare charge by infinite number of instantonic corrections.

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arXiv2009

On non-conformal limit of the AGT relations

The Seiberg-Witten prepotentials for N=2 SUSY gauge theories with N_f<2N_c fundamental multiplets are obtained from conformal N_f=2N_c theory by decoupling 2N_c-N_f multiplets of heavy matter. This procedure can be lifted to the level of Nekrasov functions with arbitrary background parameters epsilon_1 and epsilon_2. The AGT relations imply that similar limit exists for conformal blocks (or, for generic N_c>2, for the blocks in conformal theories with W_{N_c} chiral algebra). We consider the limit of the four-point function explicitly in the Virasoro case of N_c=2, by bringing the dimensions of external states to infinity. The calculation is performed entirely in terms of representation theory for the Virasoro algebra and reproduces the answers conjectured in arXiv:0908.0307 with the help of the brane-compactification analysis and computer simulations. In this limit, the conformal block involving four external primaries, corresponding to the theory with vanishing beta-function, turns either into a 2-point or 3-point function, with certain coherent rather than primary external states.

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arXiv2009

First Order String Theory and the Kodaira-Spencer Equations. II

The first-order bosonic string theory, perturbed by primary operator, corresponding to the deformation of target-space complex structure is considered. We compute the correlation functions in this theory and study their divergencies. It is found, that consistency of these correlation functions with the world-sheet conformal invariance requires the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. This statement is checked explicitly for the three-point and four-point correlators, containing one probe operator. We discuss the origin of these divergences and their relation with beta-functions or effective action and polyvertex structures in BRST approach.

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arXiv2009

First Order String Theory and the Kodaira-Spencer Equations. I

We consider first-order bosonic string theory, perturbed by the primary operator, corresponding to deformation of the target-space complex structure. We compute the effective action in this theory and find that its consistency with the world-sheet conformal invariance requires necessarily the Kodaira-Spencer equations to be satisfied by target-space Beltrami differentials. We discuss the symmetries of the theory and its reformulation in terms of the vielbein background fields.

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arXiv2009

On two-dimensional quantum gravity and quasiclassical integrable hierarchies

The main results for the two-dimensional quantum gravity, conjectured from the matrix model or integrable approach, are presented in the form to be compared with the world-sheet or Liouville approach. In spherical limit the integrable side for minimal string theories is completely formulated using simple manipulations with two polynomials, based on residue formulas from quasiclassical hierarchies. Explicit computations for particular models are performed and certain delicate issues of nontrivial relations among them are discussed. They concern the connections between different theories, obtained as expansions of basically the same stringy solution to dispersionless KP hierarchy in different backgrounds, characterized by nonvanishing background values of different times, being the simplest known example of change of the quantum numbers of physical observables, when moving to a different point in the moduli space of the theory.

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arXiv2008

Non Abelian gauge theories, prepotentials and Abelian differentials

I discuss particular solutions of the integrable systems, starting from well-known dispersionless KdV and Toda hierarchies, which define in most straightforward way the generating functions for the Gromov-Witten classes in terms of the rational complex curve. On the ``mirror'' side these generating functions can be identified with the simplest prepotentials of complex manifolds, and I present few more exactly calculable examples of them. For the higher genus curves, corresponding in this context to the non Abelian gauge theories via the topological gauge/string duality, similar solutions are constructed using extended basis of Abelian differentials, generally with extra singularities at the branching points of the curve.

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arXiv2008

Seiberg-Witten Theory and Extended Toda Hierarchy

The quasiclassical solution to the extended Toda chain hierarchy, corresponding to the deformation of the simplest Seiberg-Witten theory by all descendants of the dual topological string model, is constructed explicitly in terms of the complex curve and generating differential. The first derivatives of prepotential or quasiclassical tau-function over the extra times, extending the Toda chain, are expressed through the multiple integrals of the Seiberg-Witten one-form. We derive the corresponding quasiclassical Virasoro constraints, discuss the functional formulation of the problem and propose generalization of the extended Toda hierarchy to the nonabelian theory.

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arXiv2007

On Microscopic Origin of Integrability in Seiberg-Witten Theory

We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of hep-th/0612019, as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of the spectral curve.

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arXiv2007

WDVV equations for 6d Seiberg-Witten theory and bi-elliptic curves

We present a generic derivation of the WDVV equations for 6d Seiberg-Witten theory, and extend it to the families of bi-elliptic spectral curves. We find that the elliptization of the naive perturbative and nonperturbative 6d systems roughly "doubles" the number of moduli describing the system.

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arXiv2006

WDVV Equations as Functional Relations

We discuss the associativity or WDVV equations and demonstrate that they can be rewritten as certain functional relations between the {\it second} derivatives of a single function, similar to the dispersionless Hirota equations. The properties of these functional relations are further discussed.

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arXiv2006

Matrix Models, Complex Geometry and Integrable Systems. I

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of the quasiclassical integrable systems, solved by construction of tau-functions or prepotentials. The complex curves and tau-functions of one- and two- matrix models are discussed in detail.

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arXiv2006

Matrix Models, Complex Geometry and Integrable Systems. II

We consider certain examples of applications of the general methods, based on geometry and integrability of matrix models, described in hep-th/0601212. In particular, the nonlinear differential equations, satisfied by quasiclassical tau-functions are investigated. We also discuss a similar quasiclassical geometric picture, arising in the context of multidimensional supersymmetric gauge theories and the AdS/CFT correspondence.

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arXiv2005

Matrix Model and Stationary Problem in Toda Chain

We analyze the stationary problem for the Toda chain, and show that arising geometric data exactly correspond to the multi-support solutions of one-matrix model with a polynomial potential. For the first nontrivial examples the Hamiltonians and symplectic forms are calculated explicitly, and the consistency checks are performed. The corresponding quantum problem is formulated and some its properties and perspectives are discussed.

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arXiv2005

Complex Geometry of Matrix Models

The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by quasiclassical or generalized Whitham hierarchies and are directly related to the superpotentials of four-dimensional N=1 SUSY gauge theories. We study the derivatives of tau-functions for these solutions, associated with the families of Riemann surfaces (with possible double points), and relations for these derivatives imposed by complex geometry, including the WDVV equations. We also find the free energy in subleading order of the 't Hooft expansion and prove that it satisfies certain determinant relations.

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arXiv2004

Non-Abelian Vortices in N=1* Gauge Theory

We consider the N=1* supersymmetric SU(2) gauge theory and demonstrate that the Z_2 vortices in this theory acquire orientational zero modes, associated with the rotation of magnetic flux inside SU(2) group, and turn into the non-Abelian strings, when the masses of all chiral fields become equal. These non-Abelian strings are not BPS-saturated. We study the effective theory on the string world sheet and show that it is given by two-dimensional non-supersymmetric O(3) sigma model. The confined 't Hooft-Polyakov monopole is seen as a junction of the Z_2-string and anti-string, and as a kink in the effective world sheet sigma model. We calculate its mass and show that besides the four-dimensional confinement of monopoles, they are also confined in the two-dimensional theory: the monopoles stick to anti-monopoles to form the meson-like configurations on the strings they are attached to.

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arXiv2004

Quasiclassical Geometry and Integrability of AdS/CFT Correspondence

We discuss the quasiclassical geometry and integrable systems related to the gauge/string duality. The analysis of quasiclassical solutions to the Bethe anzatz equations arising in the context of the AdS/CFT correspondence is performed, compare to stationary phase equations for the matrix integrals. We demonstrate how the underlying geometry is related to the integrable sigma-models of dual string theory, and investigate some details of this correspondence.

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arXiv2004

Classical/quantum integrability in AdS/CFT

We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N=4 super Yang-Mills and the energy of their dual semiclassical string states in AdS(5) X S(5). The anomalous dimensions can be computed using a set of Bethe equations, which for ``long'' operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU(2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

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