## Activities and role of the various Research Units

The research topics will include:

- Integrability in the AdS/CFT correspondence (BO,FI,PG,PR); Study of quantum behavior of solitons in gauge theories; confinement and dynamical symmetry breaking;
- AdS/CFT correspondence and D-branes dynamics and thermodynamics (BO,FI,PG,PI);
- Applied Holography: finite temperature gauge theories and AdS/CondMat (FI,PG,PI);
- Supersymmetric and cusped Wilson loops: localization and integrability (BO,FI,PR);
- Non-perturbative aspects in gauge theories: solitons, confinement and dynamical symmetry breaking (PI);
- Study of quantum behavior of solitons in gauge theories (PI,TS);
- Vacuum structure in supersymmetric gauge theories gauge theories, integrable systems and localization (TS);
- Mathematical structures in QFT: sigma models and topological strings (FI,PG,PR,TS);
- EApplication of matrix model techniques to gauge and string theories (FI,PG,PR);
- String field theories and non-local theories (PG);
- Worldline approaches to quantum field theories (BO);
- Glueball and meson propagators in large-N QCD, large-N QCD S-matrix, Twistorial Topological Field Theories and Twistorial Topological String Theories for large-N confining asymptotically-free gauge theories, such as large-N QCD (PI - INFN Roma1).

## Research activities of the various Research Units

### Sezione di BO

We have quantized a class of higher spin fields on (A)dS spaces using a worldline formulation based on the so called O(N) spinning particles, and computed the first few Seeley-DeWitt coefficients of the corresponding one-loop effective action, i.e the diverging terms in D=4. [F. Bastianelli, R. Bonezzi]We completed the study, from a worldline viewpoint, of quantum differential forms on Kaehler backgrounds treating the general case of (p,q) forms with and without gauge symmetries. [F. Bastianelli, R. Bonezzi, C. Iazeolla]

We studied the structure of the low energy limit of the one-loop photon-graviton amplitudes induced by massive scalars and spinors to search for KLT-type of relations, where effectively two photons merge into a graviton, and finding a new example of a KLT-like factorization in field theory at the loop level. [F. Bastianelli]

We applied first quantized methods to represent effective actions for non-commutative quantum field theories. These methods showed to be quite flexible and may be employed in the future to treat interactions with higher spin backgrounds. [R. Bonezzi]

We have been investigating the interplay between conformal invariance and electric-magnetic duality in electromagnetism. [R. Bonezzi]

We continued our investigation on a possible statistical origin for gravity, analysing in particular the derivation of field equations from an extremum condition for a suitable entropy functional. [A. Pesci]

The Unruh effect for the rate of emission and absorption of neutral massive Majorana particles, the most plausible constituents of the Dark Matter, has been investigated. The corresponding Bogolyubov coefficients in a Rindler space have been calculated and the consistency with the Fermi-Dirac statistics actually verified. [R. Soldati]

A formulation of higher gauge theory, whose symmetry is encoded in a semistrict Lie 2-algebra v and which we call semistrict has been worked out by viewing v as a 2-term L infinity algebra. Fields are v-valued and gauge transformations are special Aut(v)-valued maps organized as a strict 2-group and acting on them. Using the BV quantization method in the AKSZ geometrical version, a 3-dimensional semistrict higher BF gauge theory and a 4-dimensional higher Chern-Simons theory have been written down. An analysis of observables has been carried out. [E. Soncini, R. Zucchini]

We derived the dispersion relations of all the excitations over the GKP vacuum beyond the scalar sector. In particular, we payed particular attention to the strong coupling limit where they should become relativistic (free) particles, but with Lorentz-violating interaction. In this respect, we have determined the two-particle S-matrix for the gluons at any coupling, and then computed the 'slight' relativistic violation (at strong coupling). Moreover, the scattering is clearly elastic and integrable (factorisable) and thus we have put forward the computation of its basic building blocks, i.e. the two-particle amplitudes for all the excitations (scalars, gluons and gauginos) over the GKP vacuum. As the scholarly understanding is more and more stressing, this scattering theory is directly related to the exact knowledge of the scattering amplitudes in 4D N=4 SYM, and as a consequence acquires a particular importance. [D. Fioravanti]

In the one dimension less correspondence, AdS4/CFT3, we have developed the TBA approach for the low energy string theory which decouples at strong coupling (starting from the asymptotic Bethe Ansatz), which is the analogue of the O(6) NLSM. This model is a CP3 bosonic system coupled, via a U(1) gauge field, to a massless Dirac fermion with Thirring interaction. In this way, we discovered a new class of system of crossed form. [D. Fioravanti]

### Sezione di CS

Integrability in N=4 SYM (M.Rossi, G.Infusino)[In collaboration with:

D.Fioravanti (Dipartimento di Fisica, Universita' di Bologna and INFN, Sez.

di Bologna)

D. Bombardelli (universita' di Porto, Portogallo]

In collaboration with D. Fioravanti, M.Rossi completed a first project - paper 1a) - concerning twist operators in the sl(2) sector of planar N=4 SYM in a special limit in which the relevant dynamics should be give by integral equations of the O(6) non-linear sigma model.

Going into details, we compute anomalous dimensions by means of a nonlinear integral equation which is equivalent to the asymptotic Bethe Ansatz equations. We expand in powers of ln S, where S is the semiclassical spin parameter. At the leading order ln S we can re-confirm the O(6) non-linear sigma model description by Alday and Maldacena. By computing the first finite size correction 1/ln S, we find agreement with available string theory results at one loop and we give predictions for all loops result.

A second project - preprint 1b) - collaboration between D. Fioravanti, G.Infusino and M.Rossi, concerns the study of reciprocity and

functional relations in anomalous dimensions of twist operators in N=4 SYM.

In particular we performed precision computations of corrections to the anomalous dimension going like the inverse power of the Lorentz spin.

Other three projects are in progress.

The first concerns the study of excitations on the GKP vacuum. In particular we are interested in determining scattering phases between excitations and dispersion relations of the latters.

This project is carried on by M.Rossi in collaboration with D. Fioravanti and his PhD student S. Piscaglia.

The second project is aimed at studying a set of TBA equations with boundary.

Following the seminal paper of Correa, Maldacena, Sever this set of integral equations

can be used to determine by means of the AdS/CFT correspondence the

potential between coloured particles in a supersymmetric theory.

This project is carried on by M. Rossi in collaboration with D.Bombardelli and D.Fioravanti.

The third project involves G. Infusino and concerns numerical solutions of sl(n) Bethe equations.

### Sezione di FI

Supersymmetric Wilson loops [FIRENZE-PARMA]:Recently, it was shown that supersymmetric Wilson loop in N=4 falls into two families: (I) the loop is an orbit of some conformal transformation of the space- time; (II) an arbitrary contour in the subspace where local super-algebra

generator is a pure spinor. We have investigated systematically the first class. We have constructed the most general loop in this family and we have studied its quantum properties at weak and strong coupling. [Cardinali, Griguolo, Seminara]

Correlators of Wilson loops: [FIRENZE-PARMA]:

We have studied the correlator of two "antiparallel" Hopf fibers in N=4 SYM theory. The analysis has been performed at

strong coupling and a connected classical string surface linking the two fibers was constructed. This system is interesting since it can be considered a small deformation of the ordinary antiparallel lines giving the static quark-antiquark potential. Indeed, the correct potential is reproduced at weak and strong coupling, as the fibres approach one another. [Griguolo, Mori, Nieri, Seminara]

Potential and cusped loops in ABJM: [FIRENZE-PARMA]:

We constructed a generalized cusped Wilson loop operator in N = 6 super Chern-Simons-matter theories which is locally invariant under half of the supercharges. It depends on two parameters and interpolates smoothly between the 1/2 BPS line or circle and a pair of antiparallel lines, representing a natural generalization of the quark-antiquark potential in ABJ(M) theories. For particular choices of the parameters we obtain 1/6 BPS configurations that, mapped on S^2 by a conformal transformation, realize a three-dimensional analogue of the wedge DGRT Wilson loop of N = 4. The cusp couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bifundamental representation of the U(N)xU(M) gauge group and its expectation value is expressed as the holonomy of a suitable superconnection. We discussed the definition of these observables in terms of traces and the role of the boundary conditions of fermions along the loop. We performed a complete two-loop analysis, obtaining an explicit result for the generalized cusp at the second non-trivial order, from which we read off the interaction potential between heavy 1/2 BPS particles in the ABJ(M) model. Our results open the possibility to explore in the three-dimensional case the connection between localization properties and integrability, recently advocated in D = 4.

[Griguolo, Marmiroli, Martelloni, Seminara]

Supersymmetric Wilson loops in ABJ(M) theories: [FIRENZE-PARMA]:

We presented two new families of Wilson loop operators in N= 6 supersymmetric Chern-Simons theory. The first one is defined for an arbitrary contour on the three dimensional space and it resembles the Zarembo's construction in N=4 SYM. The second one involves arbitrary curves on the two dimensional sphere. In both cases one can add certain scalar and fermionic couplings to the Wilson loop so it preserves at least two supercharges. Some previously known loops, notably the 1/2 BPS circle, belong to this class, but we point out more special cases which were not known before. They could provide further tests of the gauge/gravity correspondence in the ABJ(M) case and interesting observables, exactly computable by localization. [V. Cardinali, L. Griguolo, G. Martelloni, D. Seminara]

Unbalanced superconductor [PISA-FIRENZE]:

The occurrence of superconductive (or superfluid) phases where two fermionic species with different chemical potential condense is a very interesting possibility relevant both in condensed matter and in finite density QCD contexts. We plan to investigate this system from an holographic point of view. [Bigazzi, Cotrone, Musso Pinzani-Fokeeva, Seminara]

Snyder Geometry and Quantum Field Theory [FIRENZE]:

We find that, in presence of the Snyder geometry, the notion of translational invariance needs to be modified, allowing a momentum dependence of this symmetry. This step is necessary to build the maximally localized states and the Feynman rules of the corresponding quantum field theory. [Valtancoli]

Simple consequences of minimal position uncertainty [FIRENZE]

We consider a simple deformation of the usual commutation relations in quantum mechanics which induces a minimal length. We have studied how this change affects the spectrum of the harmonic oscillator.[Valtancoli]

### Sezione di PD

Title: Mellin amplitudes and AdS/CFT (PADOVA)"Mellin amplitudes" are a very convenient framework in which correlators of conformal theories on the boundary of an AdS space can be related to weakly coupled theories in the bulk and a conjecture was proposed that in suitable asymptotic limits they may provide a holographic description of the S-matrix in the flat-space limit of AdS. These topics are under investigation both at a basic and at a "practical" levels. [Bassetto]

Title: New non-BCS superconductivity (PADOVA)

We analyze the general framework of our new proposal of a non-BCS mechanism of superconductivity based on spin-vortex attraction, for the high Tc superconductors, within a spin-charge gauge approach to the t-J model. In particular we extend our previous treatment to the more realistic t-t'-J model and discuss the influence of the formation of charge pairs without condensation on the spectral weight of the hole, obtaining results in agreement with data of photoemission and tunneling experiments.

[Marchetti]

Title: Measure for string theory (PADOVA)

Belavin and Knizhnik conjectured an expression for g=4 bosonic measure in string theory which has been proved in the framework of the vector-valued Teichmueller modular forms introduced in collaboration with Roberto Volpato. It turns out that for g>3 the bosonic measure is expressed in terms of such forms and it has an elegant geometrical

interpretation as generating the quadrics in P^{g-1} characterizing the Riemann surface.

It has been shown that there is an infinite class of partition functions with world-sheet metric, space-time coordinates and first order systems, that correspond to volume forms on the moduli space of Riemann surfaces and are free of singularities at the Deligne-Mumford boundary. Such partition functions are derived from the mapping of the Mumford forms to non-factorized scalar forms on M_g we introduced in a previous work.[Matone]

Title: Quantum Hamilton-Jacobi theory (PADOVA)

It has been investigated the quantum version of the dispersion relations which follow from the quantum relativistic Hamilton-Jacobi equation. Furthermore, in collaboration

with Alon Faraggi, it has been shown that if space is compact, then the time parametrization does not exist in the framework of Quantum Hamilton-Jacobi theory. [Matone]

Title: Supersymmetry and localization (PADOVA)

The problem of finding low-energy effective action in three-dimensional supersymmetric

field theories was solved. Superfield methods for studying partition functions in two- and

three-dimensional field theories with localization techniques are developed in collaboration with D. Sorokin. [Samsonov]

### Sezione di PG

a) Holographic three point functionsThe agreement between string theory and field theory is demonstrated in the leading order

by providing the first calculation of the correlator of three two-impurity BMN states with

all non-zero momenta. The calculation is performed in two completely independent ways:

in field theory by using the large-N perturbative expansion, up to the terms subleading in

finite-size, and in string theory by using the Dobashi-Yoneya 3-string vertex in the leading order of the Penrose expansion. The two results come out to be completely identical.

We also showed that the fully dynamical three-point correlation functions of BMN operators are identical at the tree level in the planar limit of perturbative field theory and, on the string theory side, calculated by means of

the Dobashi-Yoneya three string vertex in the Penrose limit. We presented a one-loop calculation of the same quantity both on the field-theory and string-theory side, where a complete identity between the two results is demonstrated.

We conjectured a generalization of a formula for an SU(2) threepoint correlation function at weak coupling based on integrability techniques. to the SO(6) sector, thus including all possible single-trace scalar operators in N = 4 super Yang-Mills, and prove, by direct comparison to a perturbative SO(6) calculations, that our generalization is valid.

b) D-brane models

We considered a holographic model of dynamical symmetry breaking in 2+1-dimenisons, where a parallel D7-anti-D7 brane pair fuse into a single object. We showed that the current-current correlation functions can be computed analytically and exhibit the low momentum structure that is expected when global symmetries are spontaneously broken. We also find that these correlation functions have poles attributable to infinite towers of vector mesons with equally spaced masses.

c) Finite temperature D-brane probes

We applyed a new description of thermal fundamental string probes to the study of finite

temperature Wilson loops in the context of the AdS/CFT correspondence. Previously this

problem has been considered using extremal probes even though the background is at finite

temperature. As a result of our analysis we found a new term in the potential between static quarks in the symmetric representation which for suffciently small temperatures is the leading correction to the Coulomb force potential. We also find an order 1/N correction to the onset of the Debye screening of the quarks. These effects arise from including the thermal excitations of the string probe and demanding the probe to be in thermodynamic equilibrium with the Anti-de Sitter black hole background.

### Sezione di PI

Our research activity has been focused on the study of strongly interacting quantum field theories, at finite temperature and density, using the holographic correspondence as a tool. I have investigated models of quark-gluon plasmas in 3+1 dimensions (studying D3-D7 systems) as well as effective (bottom-up) holographic models for unbalanced superconductors in 2+1 dimensions. I have studied both static (thermodynamics, phase diagrams) and real-time dynamical issues (hydrodynamic transport coefficients and jet quenching in the quark-gluon plasma case; conductivities and spintronic effects in the 2+1 dimensional model).[F. Bigazzi]

In 2012 I collaborated with Muneto Nitta (and Walter Vinci) in the context of High density QCD. We studied the stability and the properties of the vortices arising in color-superconductors which are believed to be realized in the core of neutron stars. At high densities and low temperatures like the ones of the core of these stars, quarks are the relevant degrees of freedom and condense in Cooper pairs. There vortices can appear which are both superconducting and superfluid. We found the most stable vortex solution and studied the colorful boojum standing where these vortices reach the interface where the color-superconducting phase changes into the npe-phase. There they connect to the superconducting and superfluid vortices constituted by condensed protons and neutrons respectively.

In order to study the vortex lattice in this system, we started a systematic study of vortex lattices with numerical simulation performed with the finite element method by the FreeFem++ package. To simplify the calculation we started from simulations of ungauged systems, namely multi-components BECs. The first result we are publishing is relative to a study of the vortex lattice in a coherently coupled two-components BEC, where we find many interesting phenomena like multi-dimer molecules and partner changing. We are also ending a study on three-component BEC, which we believe is the first step to understand the properties of vortices in high density QCD, because of the similarity of the condensates, when the gauge coupling is neglected in the latter. Simulations of vortices in gauged multi-components systems will be our next subject of interest.

[M. Cipriani, W. Vinci]

New types of confinement phase appear as the singular SCFT appearing as infrared-fixed-points in N=2 supersymmetric QCD's are deformed by small adjoint scalar mass term. Based on some remarkable developments on these singular SCFT "of higest criticalities" - where physics is characterized by strongly-interacting monopoles and dyons of non-Abelian variety, we investigated which type of confinement phase can arise and how confinement and dynamical symmetry breaking (DSB) related to each other. We examined various models based on SU(N), SO(N) and USp(2N) gauge groups and with several choices of the number of flavors, finding that DSB and confinement are related to each other but in a subtler way than expected in a straightforward dual superconductor picture of quark confinement.

[S.Giacomelli, K. Konishi]

Non-Abelian vortices have orientational zeromodes arising from unbroken color-flavor diagonal symmetries, which can be excited along the vortex length and time (vortex worldsheet). When some or all of teh exact global symmetry is weakly gauged interesting phenomena (supersymmetry breaking or weak Higgs mechanism in the worldsheet. The study of this calss of phenomena has been initiated in a work with collaboration with M. Nitta.

[W. Vinci, K. Konishi]

Many of SCFT found in N=2 supersymmetric theories have exactly marginal coupling constant. At "infinite coupling" these SCFT allow (Argyres-Seiberg) dual description in terms of weak SU(2) gauge interactions connecting two, separate SCFT's. This phenomena have been extended to some singular SCFT's appearing as infrared-fixed-points of various N=2 gauge theories, by Gaiotto, Seiberg and Tachikawa.

I extended the GST analysis to much wider class of gauge theories, finding new, dual description of singular SCFT's in SO(N), USp(2N) and other theories.

[S. Giacomelli]

During 2012 my research has mainly focused on the study of nonperturbative aspects of supersymmetric field theories in four dimensions. In particular I have analyzed the implications of the recent progress in the understanding of superconformal N=2 field theories for confinement and chiral symmetry breaking in the softly broken version of these models. I have also worked on the construction and analysis of non lagrangian superconformal field theories and on the M-theory realization of N=1 supersymmetric field theories, trying to generalize the construction of N=2 theories proposed by Gaiotto.

[S. Giacomelli]

In 2015 we outlined a proposal for a candidate [1] string solution of large-N QCD, satisfying the following two fundamental properties,
contrary to all the previous proposal based either on old-fashioned string theories or on Gauge/Gravity duality:

1) A large-N spectrum of mesons and glueballs, linear in the masses squared, and remarkably compatible [2]
both with experiments. i.e the PDG (2015), and lattice gauge-theory computation at large-N.

2) A large-N S-matrix compatible with asymptotic freedom [1], i.e. that leads to asymptotically-free correlation functions working out the LSZ
formulae the other way around.

The rationale behind the aforementioned proposal is the existence of a Twistorial Topological String Theory [1] dual at large-N
to a Topological Field Theory underlying large-N QCD [3], and recent theoretical developments summarized in [3].

[M. Bochicchio - INFN Roma1]

[1] M. Bochicchio, An asymptotic solution of large-N QCD, for the glueball and meson spectrum and the collinear S-matrix.
In: Proceedings of the XVIth Conference on Hadron Spectroscopy (HADRON 2015), AIP Conference Series (2015), in print.

[2] M. Bochicchio, Glueball and Meson spectrum in large-N QCD.
In: Proceedings of LIGHT CONE 2015, FBS (2016), available "Online First" DOI 10.1007/s00601-016-1100-6, in print.

[3] M. Bochicchio, Yang-Mills mass gap at large-N, non-commutative YM theory, topological quantum field theory and hyperfiniteness,
IJMPD 24(06) (2015). arXiv:1202.4476 [hep-th].

### Sezione di PR

The activity of the PI14 group of Parma has been focussed in 2012 on the following topics1) Pure spinor superstrings (A. Camobreco, M. Bonini)

The action for the pure spinor superstring in the coset description of AdS_4 x CP^3 superspace has been constructed, using the variables which solve the pure spinor condition. As a test of the consistency of the approach, the background field method has been exploited to verify the absence of central charge at the second order in the expansion and to show the one-loop finiteness of the effective action.

2) Colloidal particles as dynamical systems (G. Cicuta, E. Onofri)

A system of active colloidal particles driven by harmonic potentials to oscillate about the vertices of a regular polygon, with hydrodynamic coupling between all particles, have been described by a piecewise linear model which exhibits various patterns of synchronization. Analytical solutions have been obtained for this class of dynamical systems. Depending only on the number of particles, the synchronization has been observed into states in which nearest neighbors oscillate in in-phase, antiphase, or phase-locked (time-shifted) trajectories.

3) AdS/CFT correspondence: correlators of Hopf Wilson loops at strong coupling (L. Griguolo, S. Mori)

It has been performed a study at quantum level correlators of supersymmetric Wilson loops with contours lying on Hopf fibers of S3. In N=4 SYM theory the strong coupling analysis is performed using the AdS/CFT correspondence and a connected classical string surface, linking two different fibers, has been found: it describes oppositely oriented fibers with the same scalar coupling and depends on an angular parameter, interpolating between a non-BPS configuration and a BPS one. The system represents a deformation of the ordinary antiparallel lines giving the static quark-antiquark potential, that is reproduced, at weak and strong coupling, as the fibers approach one another.

4) Killing spinors and supersymmetric Wilson loops in N=4 SYM: the impure case (L. Griguolo)

A general class of supersymmetric Wilson loops operator in N = 4 super Yang-Mills theory, obtained as orbits of conformal transformations, has been studied. These loops are the natural generalization of the familiar circular Wilson-Maldacena operator and their supersymmetric properties are encoded into a Killing spinor that is not pure. A systematic analysis of their scalar couplings and of the preserved supercharges, modulo the action of the global symmetry group, both in the compact and in the non-compact case has been presented. The quantum behavior of their expectation value, in the simplest case of the Lissajous contours, has been examined: explicit computations at weak-coupling, through Feynman diagrams expansion, and at strong-coupling, by means of AdS/CFT correspondence, suggest the possibility of an exact evaluation

5) The generalized cusp in ABJ(M) N = 6 Super Chern-Simons theories (L. Griguolo, D. Marmiroli)

A generalized cusped Wilson loop operator in N = 6 super Chern-Simons-matter theories which is locally invariant under half of the supercharges has been constructed, depending on two parameters and interpolating smoothly between the 1/2 BPS line or circle and a pair of antiparallel lines. The cusp couples, in addition to the gauge and scalar fields of the theory, also to the fermions in the bifundamental representation of the U(N)xU(M) gauge group and its expectation value is expressed as the holonomy of a suitable superconnection. The definition of these observables in terms of traces and the role of the boundary conditions of fermions along the loop is carefully discussed. The complete two-loop analysis has been presented, obtaining an explicit result for the generalized cusp at the second non-trivial order, from which we read off the interaction potential between heavy 1/2 BPS particles in the ABJ(M) model. These results open the possibility to explore in the three-dimensional case the connection between localization properties and integrability, recently advocated in D = 4.

6) Supersymmetric loops in ABJ(M) theories (L. Griguolo)

Two new families of Wilson loop operators in N= 6 supersymmetric Chern-Simons theory have been obtained. The first one is defined for an arbitrary contour on the three dimensional space and it resembles the Zarembo's construction in N=4 SYM. The second one involves arbitrary curves on the two dimensional sphere. In both cases certain scalar and fermionic couplings can be added to the Wilson loop so it preserves at least two supercharges. Some previously known loops, notably the 1/2 BPS circle, belong to this class, but more special cases, which were not known before, have been presented. They could provide further tests of the gauge/gravity correspondence in the ABJ(M) case and interesting observables, exactly computable by localization

### Sezione di TN

Theories defined on fractional space times are described by a continuous geometrywith constant non-integer Hausdorff and spectral dimension. The interest in such theories is due to the possibility that the "perceived" dimension of a theory can be different at different scales. In spite of the lack of translational invariance of these theories, it is possible to find representations of the Poincare' group, which are therefore the free fractional fields, and quantum states of definite mass and spin can be consistently defined in fractional spacetimes.

### Sezione di TS

We studied the correspondence between four dimensional N=2 supersymmetric gauge theoriesand two-dimensional conformal field theories. We analyzed gauge theories on resolved toric singularities and the corresponding 2d CFTs by extending the original AGT correspondence to these cases. In this context we found a correspondence between gauge theories on ALE spaces

and supersymmetric or more in general parafermionic Liouville theory.

We studied non-Lagrangian sectors in supersymmetric gauge theories obtained by compactifying the six dimensional A_1 (2,0) theory on Riemann surfaces with irregular punctures. These are naturally associated to Hitchin systems with wild ramification whose spectral curves provide the relevant Seiberg-Witten geometries. We computed the prepotential of these gauge theories from the corresponding irregular conformal blocks on the Riemann surfaces via a generalization of the coherent state construction to the case of higher order singularities.

We studied also: differential and twistor geometry of quantum projective spaces.

Instantons and monopoles on quantum projective spaces and their moduli spaces.

Higgs fields on quantum projective spaces; dimensional reduction.

Holomorphic structures and equivariant bundles in noncommutative geometry.

Generalized TKNN equations and noncommutative tori.

Other subjects investigated: study of nonabelian Lie algebroid extensions (obstruction and classification), related spectral sequences.

Noether-Lefschetz loci of ample hypersurfaces in torics 3-folds.

Stacky resolutions of moduli spaces of instantons and instanton counting for open toric surfaces

## List of the main international collaborations related to the proposal:

- R. Poghossian, Yerevan Institute of Physic, Yerevan, Armenia
- S. Giacomelli, Bruxelles University (ULB), Belgium
- C. Burgess, Perimeter Institute and McMaster University, Canada
- G. W. Semenoff, University of British Columbia, Canada
- J. Alfaro, Pontificia Univ. Catolica de Chile, Santiago de Chile, Chile
- J. Evslin, Beijing University, China
- T. Harmark, Niels Bohr Institute, Denmark
- N. Obers, Niels Bohr Institute, Denmark
- M. Dubois-Violette, Université Paris Sud, France
- V. Fateev, CNRS-Montpellier, France
- M. Goodsell, Ecole Polytechnique, Paris, France
- V. Roubtsov, LAREMA, Université d'Angers, Angers, France
- R. Santachiara, LPT Orsay, Parigi, France
- G. Akemann, Universitat Bielefeld, Germany
- D. Arean, Munich University, Germany
- K. Dutta, DESY, Germany
- S. Krippendorf, Bonn University, Germany
- C. Mayrhofer, Heidelberg University, Germany
- A. Ringwald, DESY Hamburg, Germany
- A. Westphal, DESY Hamburg, Germany
- A. Maharana, Harish-Chandra Research Institute, Allahbad, India
- S. Bolognesi, Hebrew University, Israel
- S. Abdussalam, ICTP, Trieste, Italy
- L. Aparicio, ICTP, Trieste, Italy
- D. Musso, ICTP, Trieste, Italy
- R. Valandro, ICTP, Trieste, Italy
- M. Nitta, Keio University, Japan
- K. Ohashi, Osaka City University, Japan
- M. Cambiaso, UNAM, Mexico DF, Mexico
- O. Corradini, Universidad Autónoma de Chiapas, Mexico
- C. Schubert, Universidad Michoacana, Morelia, Mexico
- D. Bombardelli, University of Porto, Portugal
- A. A. Andrianov, St. Petersburg State Univ., Russia
- G. Calcagni, Instituto de Estructura de la Materia, CSIC Madrid, Spain
- J. Mas, Santiago de Compostela University, Spain
- A. Ramallo, Santiago de Compostela University, Spain
- J. Tarrio, Barcelona University, Spain
- D. Young, Nordita, Sweden
- A. S. Cattaneo, Institut für Mathematik, Universität Zürich, Switzerland
- E. Latini, Institut für Mathematik, Universitädt Zürich, Switzerland
- D. Mayerson, Amsterdam University, The Netherlands
- N. Pinzani Fokeeva, Amsterdam University, The Netherlands
- W. D. van Suijlekom, Radboud University, Nijmegen, The Netherlands
- B. Acharya, ICTP and King's College, London, UK
- A. Bissi, Mathematical Institute, University of Oxford, UK
- J. Conlon, Oxford University, UK
- A. Mazumdar, Lancaster University, UK
- F. Quevedo, ICTP Director and Cambridge University, UK
- S. Pasquetti, University of Surrey, UK
- R. Szabo, Heriot-Watt University, Edinburgh, UK
- R. Allahverdi, University of New Mexico, USA
- S. De Alwis, University of Colorado, USA
- D.E. Diaconescu, Rutgers, USA
- B. Dutta, Texas A&M, USA
- A. Grassi, University of Pennsylvania, Philadelphia, USA
- D. Klevers, UPenn, Philadelphia, USA
- P. Kumar, Yale University, USA
- P. Longhi, Rutgers University, Piscataway, NJ, USA
- F. Luef, MIT, Boston, USA
- K. Maruyoshi, Caltech, USA
- K. Sinha, Texas A&M, USA
- A. Waldron, University of California at Davis, USA