N=2⋆\mathcal{N}=2^\star from Topological Amplitudes in String Theory

In this paper, we explicitly construct string theory backgrounds that realise the so-called $\mathcal N=2^\star$ gauge theory. We prove the consistency of our models by calculating their partition function and obtaining the correct gauge theory spectrum. We further provide arguments in favour of the universality of our construction which covers a wide class of models all of which engineer the same gauge theory. We reproduce the corresponding Nekrasov partition function once the $\Omega$-deformation is included and the appropriate field theory limit taken. This is achieved by calculating the topological amplitudes $F_g$ in the string models. In addition to heterotic and type II constructions, we also realise the mass deformation in type I theory, thus leading to a natural way of uplifting the result to the instanton sector.Comment: 27 page

Instanton Corrections for m and Omega

In this paper, we study instanton corrections in the N=2* gauge theory by using its description in string theory as a freely-acting orbifold. The latter is used to compute, using the worldsheet, the deformation of the Yang-Mills action. In addition, we calculate the deformed instanton partition function, thus extending the results to the non-perturbative sector of the gauge theory. As we point out, the structure of the deformation is extremely similar to the Omega-deformation, therefore confirming the universality of the construction. Finally, we comment on the realisation of the mass deformation using physical vertex operators by exploiting the equivalence between Scherk-Schwarz deformations and freely-acting orbifolds

Topological Amplitudes and the String Effective Action

In this work, we study a class of higher derivative couplings in the string effective action arising at the junction of topological string theory and supersymmetric gauge theories in the $\Omega$-background. They generalise a series of gravitational couplings involving gravitons and graviphotons, which reproduces the topological string theory partition function. The latter reduces, in the field theory limit, to the partition function of the gauge theory in the $\Omega$-background when one if its parameters, say $\epsilon_+$, is set to zero. This suggests the existence of a one-parameter extension called the refined topological string. The couplings considered in this work involve an additional vector multiplet and are evaluated, perturbatively and non-perturbatively, at the string level. In the field theory limit, they correctly reproduce the partition function of the gauge theory in a general $\Omega$-background. Hence, these couplings provide new perspectives toward a worldsheet definition of the refined topological string.Comment: Ph.D. dissertation, 12 figure

Non-Perturbative Nekrasov Partition Function from String Theory

We calculate gauge instanton corrections to a class of higher derivative string effective couplings introduced in [1]. We work in Type I string theory compactified on K3xT2 and realise gauge instantons in terms of D5-branes wrapping the internal space. In the field theory limit we reproduce the deformed ADHM action on a general {\Omega}-background from which one can compute the non-perturbative gauge theory partition function using localisation. This is a non-perturbative extension of [1] and provides further evidence for our proposal of a string theory realisation of the {\Omega}-background.Comment: 23 page

Probing the Moduli Dependence of Refined Topological Amplitudes

With the aim of providing a worldsheet description of the refined topological string, we continue the study of a particular class of higher derivative couplings $F_{g,n}$ in the type II string effective action compactified on a Calabi-Yau threefold. We analyse first order differential equations in the anti-holomorphic moduli of the theory, which relate the $F_{g,n}$ to other component couplings. From the point of view of the topological theory, these equations describe the contribution of non-physical states to twisted correlation functions and encode an obstruction for interpreting the $F_{g,n}$ as the free energy of the refined topological string theory. We investigate possibilities of lifting this obstruction by formulating conditions on the moduli dependence under which the differential equations simplify and take the form of generalised holomorphic anomaly equations. We further test this approach against explicit calculations in the dual heterotic theory.Comment: 30 page

Amplitudes Topologiques et l'Action Effective de la Théorie des Cordes

In this thesis, we study a class of higher derivative couplings in the string effective action arising at the junction of topological string theory and supersymmetric gauge theories in the Omega-background. They generalise a series of gravitational couplings involving gravitons and graviphotons, which reproduces the topological string theory partition function. The latter reduces, in the field theory limit, to the partition function of the gauge theory in the Omega-background when one if its parameters, say epsilon_+, is set to zero. This suggests the existence of a one-parameter extension called the refined topological string. The couplings considered in this work involve an additional vector multiplet and are evaluated, perturbatively and non-perturbatively, at the string level. In the field theory limit, they correctly reproduce the partition function of the gauge theory in a general Omega-background. Hence, these couplings provide new perspectives toward a worldsheet definition of the refined topological string.Cette thèse est dédiée à l'étude d'une classe de couplages dans l'action effective de la théorie des cordes qui se trouvent au croisement entre la théorie des cordes topologique et les théories de jauge supersymétriques. Ces couplages généralisent un ensemble de couplages gravitationnels qui calculent la fonction de partition de la théorie des cordes topologique. Dans la limite de théorie des champs, ces derniers reproduisent la fonction de partition de la théorie de jauge dans le fond Oméga lorsque l'un des paramètres de ce dernier, epsilon_+ , est égal à zéro. Cela suggère naturellement l'existence d'une généralisation dénommée la corde topologique raffinée. Les couplages étudiés dans ce manuscrit sont caractérisés par un multiplet vectoriel supplémentaire et sont calculés, en théorie des cordes, aux niveaux perturbatif et non-perturbatif. De plus, leur limite de théorie des champs donne la fonction de partition de la théorie des champs dans un fond Oméga général. Ainsi, ces couplages ouvrent de nouvelles perspectives pour la définition, au niveau de la surface d'univers, de la théorie des cordes topologiques raffinée

Ω versus graviphoton

In this work, I study the deformation of the topological string by Ω¯, the complex conjugate of the Ω-deformation. Namely, I identify Ω¯ in terms of a physical state in the string spectrum and verify that the deformed Yang-Mills and ADHM actions are reproduced. This completes the study initiated in [1]where we show that Ω¯ decouples from the one-loop topological amplitudes in heterotic string theory. Simi-larly to the N=2*deformation, I show that the quadratic terms in the effective action play a crucial role in obtaining the correct realisation of the full Ω-deformation. Finally, I comment on the differences between the graviphoton and the Ω-deformation in general and discuss possible Ω¯ remnants at the boundary of the string moduli space