Integrable Models and Geometry of Target Spaces from the Partition Function of N=(2,2) theories on S^2

In this thesis we analyze the exact partition function for N=(2,2) supersymmetric theories on the sphere S^2. Especially, its connection to geometry of target spaces of a gauged linear sigma model under consideration is investigated. First of all, such a model has different phases corresponding to different target manifolds as one varies the Fayet-Iliopoulos parameters. It is demonstrated how a single partition function includes information about geometries of all these target manifolds and which operation corresponds to crossing a wall between phases. For a fixed phase we show how one can extract from the partition function the I-function, a central object of Givental's formalism developed to study mirror symmetry. It is in some sense a more fundamental object than the exact Kahler potential, since it is holomorphic in the coordinates of the moduli space (in a very vague sense it is a square root of it), and the main advantage is that one can derive it from the partition function in a more effective way. Both these quantities contain genus zero Gromov--Witten invariants of the target manifold. For manifolds where mirror construction is not known (this happens typically for targets of non-abelian gauged linear sigma models), this method turns out to be the only available one for obtaining these invariants. All discussed features are illustrated on numerous examples throughout the text. Further, we establish a way for obtaining the effective twisted superpotential based on studying the asymptotic behavior of the partition function for large radius of the sphere. Consequently, it allows for connecting the gauged linear sigma model with a quantum integrable system by applying the Gauge/Bethe correspondence of Nekrasov and Shatashvili. The dominant class of examples we study are ''ADHM models``, i.e. gauged linear sigma models with target manifold the moduli space of instantons (on C^2 or C^2/Gamma). For the case of a unitary gauge group we were able to identify the related integrable system, which turned out to be the Intermediate Long Wave system describing hydrodynamics of two layers of liquids in a channel. It has two interesting limits, the Korteweg--deVries integrable system (limit of shallow water with respect to the wavelength) and Benjamin--Ono integrable system (deep water limit). Another integrable model that naturally enters the scene is the (spin) Calogero--Sutherland model. We examine relations among energy eigenvalues of the latter, the spectrum of integrals of motion for Benjamin--Ono and expectation values of chiral correlators in the ADHM model

Goldstone bosons on celestial sphere and conformal soft theorems

In this paper, we study celestial amplitudes of Goldstone bosons and conformal soft theorems. Motivated by the success of soft bootstrap in momentum space and the important role of the soft limit behavior of tree-level amplitudes, our goal is to extend some of the methods to the celestial sphere. The crucial ingredient of the calculation is the Mellin transformation which transforms four-dimensional scattering amplitudes to correlation functions of primary operators in the celestial CFT. The soft behavior of the amplitude is then translated to the singularities of the correlator. Only for amplitudes in "UV completed theories" (with sufficiently good high energy behavior) the Mellin integration can be properly performed, in all other cases, the celestial amplitude is only defined in a distributional sense with delta functions. We provide many examples of celestial amplitudes in UV-completed models including linear sigma models and Z-theory, which is a certain completion of the SU(N) non-linear sigma model. We also comment on the BCFW-like and soft recursion relations for celestial amplitudes and the extension of soft bootstrap ideas. 45 pages of main text + 6 appendicess and 6 figuresComment: 45 pages of the main text, 6 appendices and 6 figure

Six-dimensional supersymmetric gauge theories, quantum cohomology of instanton moduli spaces and gl(N) Quantum Intermediate Long Wave Hydrodynamics

We show that the exact partition function of U(N) six-dimensional gauge theory with eight supercharges on \u21022 7 S 2 provides the quantization of the integrable system of hydrodynamic type known as gl(N) periodic Intermediate Long Wave (ILW). We characterize this system as the hydrodynamic limit of elliptic Calogero-Moser integrable system. We compute the Bethe equations from the effective gauged linear sigma model on S 2 with target space the ADHM instanton moduli space, whose mirror computes the Yang-Yang function of gl(N) ILW. The quantum Hamiltonians are given by the local chiral ring observables of the six-dimensional gauge theory. As particular cases, these provide the gl(N) Benjamin-Ono and Korteweg-de Vries quantum Hamiltonians. In the four dimensional limit, we identify the local chiral ring observables with the conserved charges of Heisenberg plus W N algebrae, thus providing a gauge theoretical proof of AGT correspondence. \ua9 2014 The Author(s)

The stringy instanton partition function

We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A 1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants

Extended DBI and its generalizations from graded soft theorems

We analyze a theory known as extended DBI, which interpolates between DBI and the $U(N)\times U(N)/U(N)$ non-linear sigma model and represents a nontrivial example of theories with mixed power counting. We discuss symmetries of the action and their geometrical origin; the special case of SU(2) extended DBI theory is treated in great detail. The revealed symmetries lead to a new type of graded soft theorem that allows us to prove on-shell constructibility of the tree-level S-matrix. It turns out that the on-shell constructibility of the full extended DBI remains valid, even if its DBI sub-theory is modified in such a way to preserve its own on-shell constructibility. We thus propose a slight generalization of the DBI sub-theory, which we call 2-scale DBI theory. Gluing it back to the rest of the extended DBI theory gives a new set of on-shell reconstructible theories -- the 2-scale extended DBI theory and its descendants. The uniqueness of the parent theory is confirmed by the bottom-up approach that uses on-shell amplitude methods exclusively.Comment: 52 pages, 6 figures, 3 appendices. v2: minor changes, matches accepted version in JHE

On special limit of non-supersymmetric effective actions of type II string theory

Abstract In this paper we first address four point functions of string amplitudes in both type IIA and IIB string theories. Making use of non-BPS scattering amplitudes, we explore not only several Bianchi identities that hold in both transverse and world volume directions of the brane, but also we reveal various new couplings. These couplings can just be found by taking into account the mixed pull-back and Taylor couplings where their all order alpha-prime higher derivative corrections have been derived as well. For the first time, we also explore the complete form of a six point non-BPS amplitude, involving three open string tachyons, a scalar field and a Ramond–Ramond closed string in both IIA, IIB. In a special limit of the amplitude and using the proper expansion we obtain an infinite number of bulk singularities that are being constructed in the effective field theory. Finally, using new couplings we construct all the other massless and tachyon singularities in type IIA, IIB string theories. All higher derivative corrections to these new couplings to all orders in $$\alpha '$$ α′ and new restricted Bianchi identities have also been obtained

Quantum cohomology and quantum hydrodynamics from supersymmetric quiver gauge theories

We study the connection between N=2 supersymmetric gauge theories, quantum cohomology and quantum integrable systems of hydrodynamic type. We consider gauge theories on ALE spaces of A and D -type and discuss how they describe the quantum cohomology of the corresponding Nakajima\u2019s quiver varieties. We also discuss how the exact evaluation of local BPS observables in the gauge theory can be used to calculate the spectrum of quantum Hamiltonians of spin Calogero integrable systems and spin Intermediate Long Wave hydrodynamics. This is explicitly obtained by a Bethe Ansatz Equation provided by the quiver gauge theory in terms of its adjacency matrix. \ua9 2015 Elsevier B.V

Vortex Partition Functions, Wall Crossing and Equivariant Gromov-Witten Invariants

In this paper we identify the problem of equivariant vortex counting in a (2,2) supersymmetric two dimensional quiver gauged linear sigma model with that of computing the equivariant Gromov\u2013Witten invariants of the GIT quotient target space determined by the quiver. We provide new contour integral formulae for the I and J-functions encoding the equivariant quantum cohomology of the target space. Its chamber structure is shown to be encoded in the analytical properties of the integrand. This is explained both via general arguments and by checking several key cases. We show how several results in equivariant Gromov\u2013Witten theory follow just by deforming the integration contour. In particular, we apply our formalism to compute Gromov\u2013Witten invariants of the C3/Zn orbifold, of the Uhlembeck (partial) compactification of the moduli space of instantons on C2, and of An and Dn singularities both in the orbifold and resolved phases. Moreover, we analyse dualities of quantum cohomology rings of holomorphic vector bundles over Grassmannians, which are relevant to BPS Wilson loop algebrae