Algebraic-geometrical formulation of two-dimensional quantum gravity

We find a volume form on moduli space of double punctured Riemann surfaces whose integral satisfies the Painlev\'e I recursion relations of the genus expansion of the specific heat of 2D gravity. This allows us to express the asymptotic expansion of the specific heat as an integral on an infinite dimensional moduli space in the spirit of Friedan-Shenker approach. We outline a conjectural derivation of such recursion relations using the Duistermaat-Heckman theorem.Comment: 10 pages, Latex fil

RG Flow Irreversibility, C-Theorem and Topological Nature of 4D N=2 SYM

We determine the exact beta function and a RG flow Lyapunov function for N=2 SYM with gauge group SU(n). It turns out that the classical discriminants of the Seiberg-Witten curves determine the RG potential. The radial irreversibility of the RG flow in the SU(2) case and the non-perturbative identity relating the $u$-modulus and the superconformal anomaly, indicate the existence of a four dimensional analogue of the c-theorem for N=2 SYM which we formulate for the full SU(n) theory. Our investigation provides further evidence of the essentially topological nature of the theory.Comment: 9 pages, LaTeX file. Discussion on WDVV and integrability. References added. Version published in PR

Mass Deformations of Super Yang-Mills Theories in D= 2+1, and Super-Membranes: A Note

Mass deformations of supersymmetric Yang-Mills theories in three spacetime dimensions are considered. The gluons of the theories are made massive by the inclusion of a non-local gauge and Poincare invariant mass term due to Alexanian and Nair, while the matter fields are given standard Gaussian mass-terms. It is shown that the dimensional reduction of such mass deformed gauge theories defined on $R^3$ or $R\times T^2$ produces matrix quantum mechanics with massive spectra. In particular, all known massive matrix quantum mechanical models obtained by the deformations of dimensional reductions of minimal super Yang-Mills theories in diverse dimensions are shown also to arise from the dimensional reductions of appropriate massive Yang-Mills theories in three spacetime dimensions. Explicit formulae for the gauge theory actions are provided.Comment: 20 Page

Taming open/closed string duality with a Losev trick

A target space string field theory formulation for open and closed B-model is provided by giving a Batalin-Vilkovisky quantization of the holomorphic Chern-Simons theory with off-shell gravity background. The target space expression for the coefficients of the holomorphic anomaly equation for open strings are obtained. Furthermore, open/closed string duality is proved from a judicious integration over the open string fields. In particular, by restriction to the case of independence on continuous open moduli, the shift formulas of [7] are reproduced and shown therefore to encode the data of a closed string dual.Comment: 22 pages, no figures; v.2 Refs. and a comment added

Nonperturbative Relations in N=2 SUSY Yang-Mills and WDVV equation

We find the nonperturbative relation between $\langle {\rm tr} \phi^2 \rangle$, $\langle {\rm tr} \phi^3\rangle$ the prepotential ${\cal F}$ and the vevs $\langle \phi_i\rangle$ in $N=2$ supersymmetric Yang-Mills theories with gauge group $SU(3)$. Nonlinear differential equations for ${\cal F}$ including the Witten -- Dijkgraaf -- Verlinde -- Verlinde equation are obtained. This indicates that $N=2$ SYM theories are essentially topological field theories and that should be seen as low-energy limit of some topological string theory. Furthermore, we construct relevant modular invariant quantities, derive canonical relations between the periods and investigate the structure of the beta function by giving its explicit form in the moduli coordinates. In doing this we discuss the uniformization problem for the quantum moduli space. The method we propose can be generalized to $N=2$ supersymmetric Yang-Mills theories with higher rank gauge groups.Comment: 12 pages, LaTex. Expanded version. New results, corrections, references and acknowledgements adde

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

Irregular singularities in Liouville theory

Motivated by problems arising in the study of N=2 supersymmetric gauge theories we introduce and study irregular singularities in two-dimensional conformal field theory, here Liouville theory. Irregular singularities are associated to representations of the Virasoro algebra in which a subset of the annihilation part of the algebra act diagonally. In this paper we define natural bases for the space of conformal blocks in the presence of irregular singularities, describe how to calculate their series expansions, and how such conformal blocks can be constructed by some delicate limiting procedure from ordinary conformal blocks. This leads us to a proposal for the structure functions appearing in the decomposition of physical correlation functions with irregular singularities into conformal blocks. Taken together, we get a precise prediction for the partition functions of some Argyres-Douglas type theories on the four-sphere.Comment: 84 pages, 6 figure

Polynomial Structure of the (Open) Topological String Partition Function

In this paper we show that the polynomial structure of the topological string partition function found by Yamaguchi and Yau for the quintic holds for an arbitrary Calabi-Yau manifold with any number of moduli. Furthermore, we generalize these results to the open topological string partition function as discussed recently by Walcher and reproduce his results for the real quintic.Comment: 15 page

Non-holomorphic terms in N=2 SUSY Wilsonian actions and RG equation

In this paper we first investigate the Ansatz of one of the present authors for K(\Psi,\bar\Psi), the adimensional modular invariant non-holomorphic correction to the Wilsonian effective Lagrangian of an N=2 globally supersymmetric gauge theory. The renormalisation group beta-function of the theory crucially allows us to express K(\Psi,\bar\Psi) in a form that easily generalises to the case in which the theory is coupled to N_F hypermultiplets in the fundamental representation of the gauge group. This function satisfies an equation which should be viewed as a fully non-perturbative ``non-chiral superconformal Ward identity". We also determine its renormalisation group equation. Furthermore, as a first step towards checking the validity of this Ansatz, we compute the contribution to K(\Psi,\bar\Psi) from instantons of winding number k=1 and k=2. As a by-product of our analysis we check a non-renormalisation theorem for N_F=4.Comment: 39 pages, LaTex file, no figure

Partonic description of a supersymmetric p-brane

We consider supersymmetric extensions of a recently proposed partonic description of a bosonic p-brane which reformulates the Nambu-Goto action as an interacting multi-particle action with Filippov-Lie algebra gauge symmetry. We construct a worldline supersymmetric action by postulating, among others, a p-form fermion. Demanding a local worldline supersymmetry rather than the full worldvolume supersymmetry, we circumvent a known no-go theorem against the construction of a Ramond-Neveu-Schwarz supersymmetric action for a p-brane of p>1. We also derive a spacetime supersymmetric Green-Schwarz extension from the preexisting kappa-symmetric action.Comment: 1+16 pages, no figure; References added and Concluding section expanded. Final version to appear in JHE