The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsi... more The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional N = 2 gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in SU(2) theory with Nf = 4 fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with 5 regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.
We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory o... more We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory of the moduli spaces of torsion free sheaves. In the spirit of the BPS/CFT correspondence the partition function and the expectation values of the chiral, BPS protected observables are given by the matrix elements and more generally by the correlation functions in some q-deformed conformal field theory in two dimensions. We show that the coupling of the gauge theory to the bi-fundamental matter hypermultiplet inserts a particular vertex operator in this theory. In this way we get a generalization of the main result of \\cite{CO} to $K$-theory. The theory of interpolating Macdonald polynomials is an important tool in our construction.
Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twis... more Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twisted masses down to = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU (2) XXX spin chain which is mapped to the two dimensional U (N) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model which are related to the four dimensional theory compactified on T2. A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as cotangent bundles to (partial) flag varieties and the ring of quantum integrals of motion of various spin chains. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models a...
Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twis... more Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twisted masses down to = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU (2) XXX spin chain which is mapped to the two dimensional U (N) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model which are related to the four dimensional theory compactified on T2. A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as cotangent bundles to (partial) flag varieties and the ring of quantum integrals of motion of various spin chains. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models a...
Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are ... more Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become essentially equivalent to the large N ordinary gauge theories ...
Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are ... more Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become essentially equivalent to the large N ordinary gauge theories ...
We propose an approximate wavefunction of the bound state of $N$ $D0$-branes. Its spread grows as... more We propose an approximate wavefunction of the bound state of $N$ $D0$-branes. Its spread grows as $N^{1\over 3}$ per particle, i.e. it saturates the Polchinski's bound.
We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dime... more We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dimension interacting via the pair-wise potentials $U_I(q) = (\nu^2/4R^2)\sin^2(q/2R)$ and the one of the particles with the pair potential $U_{II}(q) = \nu^2/q^2$ and the external potential $U_{ext} = \omega^2 q^2/2$. The natural relation between the frequency $\omega$ and the radius $R$ is: $\omega R = 1$.
The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsi... more The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional N = 2 gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in SU(2) theory with Nf = 4 fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with 5 regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.
We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory o... more We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory of the moduli spaces of torsion free sheaves. In the spirit of the BPS/CFT correspondence the partition function and the expectation values of the chiral, BPS protected observables are given by the matrix elements and more generally by the correlation functions in some q-deformed conformal field theory in two dimensions. We show that the coupling of the gauge theory to the bi-fundamental matter hypermultiplet inserts a particular vertex operator in this theory. In this way we get a generalization of the main result of \\cite{CO} to $K$-theory. The theory of interpolating Macdonald polynomials is an important tool in our construction.
Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twis... more Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twisted masses down to = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU (2) XXX spin chain which is mapped to the two dimensional U (N) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model which are related to the four dimensional theory compactified on T2. A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as cotangent bundles to (partial) flag varieties and the ring of quantum integrals of motion of various spin chains. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models a...
Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twis... more Supersymmetric vacua of two dimensional = 4 gauge theories with matter, softly broken by the twisted masses down to = 2, are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. Examples include: the Heisenberg SU (2) XXX spin chain which is mapped to the two dimensional U (N) theory with fundamental hypermultiplets, the XXZ spin chain which is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model which are related to the four dimensional theory compactified on T2. A consequence of our correspondence is the isomorphism of the quantum cohomology ring of various quiver varieties, such as cotangent bundles to (partial) flag varieties and the ring of quantum integrals of motion of various spin chains. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schrödinger models a...
Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are ... more Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become essentially equivalent to the large N ordinary gauge theories ...
Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are ... more Recently there has been a revival of interest in noncommutative gauge theories86,14,21. They are interesting examples of nonlocal field theories which in the certain limit (of large noncommutativity) become essentially equivalent to the large N ordinary gauge theories ...
We propose an approximate wavefunction of the bound state of $N$ $D0$-branes. Its spread grows as... more We propose an approximate wavefunction of the bound state of $N$ $D0$-branes. Its spread grows as $N^{1\over 3}$ per particle, i.e. it saturates the Polchinski's bound.
We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dime... more We point out a map between the dynamics of a non-relativistic system of $N$ particles in one dimension interacting via the pair-wise potentials $U_I(q) = (\nu^2/4R^2)\sin^2(q/2R)$ and the one of the particles with the pair potential $U_{II}(q) = \nu^2/q^2$ and the external potential $U_{ext} = \omega^2 q^2/2$. The natural relation between the frequency $\omega$ and the radius $R$ is: $\omega R = 1$.
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Papers by Nikita Nekrasov