Braiding and fusion properties of the Neveu-Schwarz super-conformal blocks

We construct, generalizing appropriately the method applied by J. Teschner in the case of the Virasoro conformal blocks, the braiding and fusion matrices of the Neveu-Schwarz super-conformal blocks. Their properties allow for an explicit verification of the bootstrap equation in the NS sector of the N=1 supersymmetric Liouville field theory.Comment: 41 pages, 3 eps figure

Noncommutative Solitons: Moduli Spaces, Quantization, Finite Theta Effects and Stability

We find the N-soliton solution at infinite theta, as well as the metric on the moduli space corresponding to spatial displacements of the solitons. We use a perturbative expansion to incorporate the leading 1/theta corrections, and find an effective short range attraction between solitons. We study the stability of various solutions. We discuss the finite theta corrections to scattering, and find metastable orbits. Upon quantization of the two-soliton moduli space, for any finite theta, we find an s-wave bound state.Comment: Second revision: Discussions of translation zero-modes in section 4 and scales in section 5 improved; web addresses of movies changed. First revision: Section 6 is rewritten (thanks to M. Headrick for pointing out a mistake in the original version); some references and acknowledgements added. 21 pages, JHEP style, Hypertex, 1 figure, 3 MPEG's at: http://www.physto.se/~unge/traj1.mpg http://www.physto.se/~unge/traj2.mpg http://www.physto.se/~unge/traj3.mp

Ground state energy of the modified Nambu-Goto string

We calculate, using zeta function regularization method, semiclassical energy of the Nambu-Goto string supplemented with the boundary, Gauss-Bonnet term in the action and discuss the tachyonic ground state problem.Comment: 10 pages, LaTeX, 2 figure

Classical conformal blocks from TBA for the elliptic Calogero-Moser system

The so-called Poghossian identities connecting the toric and spherical blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain expressions for the classical 4-point block on the sphere. The main motivation for this line of research is the longstanding open problem of uniformization of the 4-punctured Riemann sphere, where the 4-point classical block plays a crucial role. It is found that the obtained representation for certain 4-point classical blocks implies the relation between the accessory parameter of the Fuchsian uniformization of the 4-punctured sphere and the eCMY functional. Additionally, a relation between the 4-point classical block and the $N_f=4$, ${\sf SU(2)}$ twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the $N_f=4$, ${\sf SU(2)}$ supersymmetric gauge theory from the classical block.Comment: 25 pages, no figures, latex+JHEP3, published versio

Recursive representation of the torus 1-point conformal block

The recursive relation for the 1-point conformal block on a torus is derived and used to prove the identities between conformal blocks recently conjectured by R. Poghossian. As an illustration of the efficiency of the recurrence method the modular invariance of the 1-point Liouville correlation function is numerically analyzed.Comment: 14 pages, 1 eps figure, misprints corrected and a reference adde

Liouville theory and uniformization of four-punctured sphere

Few years ago Zamolodchikov and Zamolodchikov proposed an expression for the 4-point classical Liouville action in terms of the 3-point actions and the classical conformal block. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on n-punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.Comment: 17 pages, no figure

Airy structures for semisimple Lie algebras

We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $\mathbb C$, and to some extent also over $\mathbb R$, up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $\mathfrak{sl}_2$, $\mathfrak{sp}_4$ and $\mathfrak{sp}_{10}$. Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a number of interesting properties of these Airy structures and constructed many examples. Techniques used to derive our results may be described, broadly speaking, as an application of representation theory in semiclassical analysis.Comment: Some references were adde

Uniformization, Calogero-Moser/Heun duality and Sutherland/bubbling pants

Inspired by the work of Alday, Gaiotto and Tachikawa (AGT), we saw the revival of Poincar{\'{e}}'s uniformization problem and Fuchsian equations obtained thereof. Three distinguished aspects are possessed by Fuchsian equations. First, they are available via imposing a classical Liouville limit on level-two null-vector conditions. Second, they fall into some A_1-type integrable systems. Third, the stress-tensor present there (in terms of the Q-form) manifests itself as a kind of one-dimensional "curve". Thereby, a contact with the recently proposed Nekrasov-Shatashvili limit was soon made on the one hand, whilst the seemingly mysterious derivation of Seiberg-Witten prepotentials from integrable models become resolved on the other hand. Moreover, AGT conjecture can just be regarded as a quantum version of the previous Poincar{\'{e}}'s approach. Equipped with these observations, we examined relations between spheric and toric (classical) conformal blocks via Calogero-Moser/Heun duality. Besides, as Sutherland model is also obtainable from Calogero-Moser by pinching tori at one point, we tried to understand its eigenstates from the viewpoint of toric diagrams with possibly many surface operators (toric branes) inserted. A picture called "bubbling pants" then emerged and reproduced well-known results of the non-critical self-dual c=1 string theory under a "blown-down" limit.Comment: 17 pages, 4 figures; v2: corrections and references added; v3: Section 2.4.1 newly added thanks to JHEP referee advice. That classical four-point spheric conformal blocks reproducing known SW prepotentials is demonstrated via more examples, to appear in JHEP; v4: TexStyle changed onl