Cardy Formulae for SUSY Theories in d=4 and d=6

We consider supersymmetric theories on a space with compact space-like slices. One can count BPS representations weighted by (-1)^F, or, equivalently, study supersymmetric partition functions by compactifying the time direction. A special case of this general construction corresponds to the counting of short representations of the superconformal group. We show that in four-dimensional N=1 theories the "high temperature" asymptotics of such counting problems is fixed by the anomalies of the theory. Notably, the combination a-c of the trace anomalies plays a crucial role. We also propose similar formulae for six-dimensional (1,0) theories.Comment: 33 pages; added reference

(Non-)Decoupled Supersymmetric Field Theories

We study some consequences of coupling supersymmetric theories to (super)gravity. To linear order, the couplings are determined by the energy-momentum supermultiplet. At higher orders, the couplings are determined by contact terms in correlation functions of the energy-momentum supermultiplet. We focus on the couplings of one particular field in the supergravity multiplet, the auxiliary field $M$. We discuss its linear and quadratic (seagull) couplings in various supersymmetric theories. In analogy to the local renormalization group formalism, we provide a prescription for how to fix the quadratic couplings. They generally arise at two-loops in perturbation theory. We check our prescription by explicitly computing these couplings in several examples such as mass-deformed ${\cal N}$ = 4 and in the Coulomb phase of some theories. These couplings affect the Lagrangians of rigid supersymmetric theories in curved space. In addition, our analysis leads to a transparent derivation of the phenomenon known as Anomaly Mediation. In contrast to previous approaches, we obtain both the gaugino and scalar masses of Anomaly Mediation by relying just on classical, minimal supergravity and a manifestly local and supersymmetric Wilsonian point of view. Our discussion naturally incorporates the connection between Anomaly Mediation and supersymmetric $AdS_4$ Lagrangians. This note can be read without prior familiarity with Anomaly Mediated Supersymmetry Breaking (AMSB).Comment: 35 pages, 3 figure

Programmable interactions with biomimetic DNA linkers at fluid membranes and interfaces

At the heart of the structured architecture and complex dynamics of biological systems are specific and timely interactions operated by biomolecules. In many instances, biomolecular agents are spatially confined to flexible lipid membranes where, among other functions, they control cell adhesion, motility and tissue formation. Besides being central to several biological processes, \emph{multivalent interactions} mediated by reactive linkers confined to deformable substrates underpin the design of synthetic-biological platforms and advanced biomimetic materials. Here we review recent advances on the experimental study and theoretical modelling of a heterogeneous class of biomimetic systems in which synthetic linkers mediate multivalent interactions between fluid and deformable colloidal units, including lipid vesicles and emulsion droplets. Linkers are often prepared from synthetic DNA nanostructures, enabling full programmability of the thermodynamic and kinetic properties of their mutual interactions. The coupling of the statistical effects of multivalent interactions with substrate fluidity and deformability gives rise to a rich emerging phenomenology that, in the context of self-assembled soft materials, has been shown to produce exotic phase behaviour, stimuli-responsiveness, and kinetic programmability of the self-assembly process. Applications to (synthetic) biology will also be reviewed.Comment: 63 pages, revie

Quantum Electrodynamics in d=3 from the epsilon-expansion

We study Quantum Electrodynamics in d=3 (QED_3) coupled to N_f flavors of fermions. The theory flows to an IR fixed point for N_f larger than some critical number N_f^c. For N_f<= N_f^c, chiral-symmetry breaking is believed to take place. In analogy with the Wilson-Fisher description of the critical O(N) models in d=3, we make use of the existence of a perturbative fixed point in d=4-2epsilon to study the three-dimensional conformal theory. We compute in perturbation theory the IR dimensions of fermion bilinear and quadrilinear operators. For small N_f, a quadrilinear operator can become relevant in the IR and destabilize the fixed point. Therefore, the epsilon-expansion can be used to estimate N_f^c. An interesting novelty compared to the O(N) models is that the theory in d=3 has an enhanced symmetry due to the structure of 3d spinors. We identify the operators in d=4-2epsilon that correspond to the additional conserved currents at d=3 and compute their infrared dimensions.Comment: 6 pages, 3 figures. v2: references added, minor correction

Scalar QED in AdS

We consider scalar QED with $N_f$ flavors in AdS$_D$. For $D<4$ the theory is strongly-coupled in the IR. We use the spin 1 spectral representation to compute and efficiently resum the bubble diagram in AdS, in order to obtain the exact propagator of the photon at large $N_f$. We then apply this result to compute the boundary four-point function of the charged operators at leading order in $1/N_f$ and exactly in the coupling, both in the Coulomb and in the Higgs phase. In the first case a conserved current is exchanged in the four-point function, while in the second case the current is absent and there is a pattern of double-trace scaling dimension analogous to a resonance in flat space. We also consider the BCFT data associated to the critical point with bulk conformal symmetry separating the two phases. Both in ordinary perturbation theory and at large $N_f$, in integer dimension $D= 3$ an IR divergence breaks the conformal symmetry on the boundary by inducing a boundary RG flow in a current-current operator.Comment: 40 pages, 7 figure

Summer pruning in table grape

This paper reviews cultural practices to improve fruit quality in table grape during vegetative and reproductive seasons. Summer pruning in table grape (Vitis vinifera L.) has more effects than winter pruning, above all with regard to plant productivity and final number of bunches for harvesting. Thinning is one of the most cultural technique and it consists in the elimination of vegetative or reproductive organs in excess. Other summer canopy management techniques include leaf removal, fruit shoots positioning, shoot trimming and girdling

The Casimir Energy in Curved Space and its Supersymmetric Counterpart

We study $d$-dimensional Conformal Field Theories (CFTs) on the cylinder, $S^{d-1}\times \mathbb{R}$, and its deformations. In $d=2$ the Casimir energy (i.e. the vacuum energy) is universal and is related to the central charge $c$. In $d=4$ the vacuum energy depends on the regularization scheme and has no intrinsic value. We show that this property extends to infinitesimally deformed cylinders and support this conclusion with a holographic check. However, for $\mathcal{N}=1$ supersymmetric CFTs, a natural analog of the Casimir energy turns out to be scheme independent and thus intrinsic. We give two proofs of this result. We compute the Casimir energy for such theories by reducing to a problem in supersymmetric quantum mechanics. For the round cylinder the vacuum energy is proportional to $a+3c$. We also compute the dependence of the Casimir energy on the squashing parameter of the cylinder. Finally, we revisit the problem of supersymmetric regularization of the path integral on Hopf surfaces.Comment: 53 pages; v2: minor changes, references added, version published in JHE

Metodi non perturbativi in teorie di gauge supersimmetriche

Questa tesi si propone di studiare alcuni dei metodi di indagine non perturbativa che sono disponibili nelle teorie di gauge supersimmetriche, in particolare: 1. i risultati esatti per le teorie con N = 1 supersimmetria 2. la soluzione di Seiberg e Witten della teoria con supersimmetria N = 2 (per soluzione si intende la determinazione della dinamica a basse energie) La parte originale della tesi mostra come questi due approcci non perturbativi siano tra loro consistenti nell’ambito di una classe di teorie per cui entrambi sono applicabili per opportuni valori dei parametri. Seguiremo una linea usata precedentemente da Gorsky, Vainshtein e Yung generalizzando il loro risultato dal caso di un flavor a quello di due e tre flavor. Il nostro studio porta a deter- minare la struttura dei vuoti e la rottura della simmetria di flavor da un’analisi semiclassica che fa uso dei metodi non perturbativi N = 1 e pertanto fornisce una verifica degli analoghi risultati per la regione di strong-cupling, che sfrut- tano la soluzione di Seiberg-Witten. Nella prima parte si descrive la definizione delle teorie di gauge supersimmetriche. Successivamente si passa ad analizzare piu` in dettaglio la versione supersimmetrica della QCD (SQCD): partendo dal lavoro del 1984 di Affleck, Dine e Seiberg sul superpotenziale esatto per il caso Nc > Nf, si discutono alcuni dei notevoli passi avanti degli anni ’90, dovuti principalmente a Seiberg, sullo spazio dei moduli quantistico della SQCD. Il terzo capitolo riguarda la soluzione di Seiberg-Witten della SQCD con supersimmetria estesa N = 2. I due fondamentali lavori del ’94 hanno rivolu- zionato la comprensione delle teorie di gauge: hanno permesso di vedere per la prima volta i monopoli magnetici come gradi di libert`a quantistici della teoria in regime di strong-coupling e hanno mostrato analiticamente che tali cariche magnetiche condensano dando luogo al confinamento. Nel capitolo finale viene esposto il lavoro di Gorsky, Vainshtein e Yung e i risultati originali sulla sua generalizzazione. La classe di teorie che prendiamo in considerazione sono teorie N = 1 che oltre ai quark hanno un campo di materia nella rappresentazione aggiunta del gruppo di gauge. Dopo aver discusso la struttura dello spazio dei moduli classico e le simme- trie globali della teoria, si passa ad analizzare i casi di Nf = 1, 2, 3. L’analisi procede secondo i seguenti punti: • il campo nell’aggiunta viene disaccoppiato, producendo una versione della SQCD modificata da un termine di interazione quartico nei campi dei quark • sfruttando i risultati della SQCD N = 1, si deduce il superpotenziale esatto della teoria quantistica e si determinano le equazioni per i VEV degli squark • mediante l’anomalia di Konishi si ricava il VEV U = ⟨TrΦ2⟩ del campo nell’aggiunta in termini di quelli degli squark • risolvendo le equazioni cos`ı ottenute si ricavano i punti corrispondenti ai vuoti sul piano U e si confrontano i risultati con quelli della teoria N =

Inhomogeneous Phase of the Chiral Gross-Neveu Model

There is substantial evidence that the ground state of the 2D chiral Gross-Neveu model, in the presence of a U(1) fermion number chemical potential mu and in the large N limit, is given by a "chiral spiral" phase, namely an inhomogeneous phase with a chiral condensate having a spatially periodic phase. We show that the chiral spiral configuration persists at finite N and T = 0 for any mu &gt; 0. Our analysis is based on nonabelian bosonization, that relates the model to a U(N)(1) Wess-Zumino-Witten model deformed by current-current interactions. In this description, the appearance of the inhomogeneous phase is surprisingly simple. We also rederive the phase diagram of the large N chiral Gross-Neveu model via a direct diagrammatic computation, finding agreement with previous results in the literature