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Spin Matrix theory describes near-BPS limits of $\mathcal{N}=4$ SYM theory, which enables us to probe finite $N$ effects like D-branes and black hole physics. In previous works, we have developed the spherical reduction and spin chain methods to construct Spin Matrix theory for various limits. In this paper, by considering a supercharge $\mathcal{Q}$ which is cubic in terms of the letters, we construct the Hamiltonian of the largest Spin Matrix theory of $\mathcal{N}=4$ SYM, called the PSU$(1,2|3)$ Spin Matrix theory, as $H = \{\mathcal{Q}, \mathcal{Q}^\dagger \}$. We show the resulting Hamiltonian is automatically positive definite and manifestly invariant under supersymmetry. The Hamiltonian is made of basic blocks which transform as supermultiplets. A novel feature of this Hamiltonian is its division into D-terms and F-terms that are separately invariant under PSU$(1,2|3)$ symmetry and positive definite. As all the other Spin Matrix theories arising from $\mathcal{N}=4$ SYM can be acquired by turning off certain letters in the theory, we consider our work as revealing the "Panorama" of Spin Matrix theory., 32+11 pages, 3 figures; v2: minor changes; v3: journal version with minor changes

We consider limits of $\mathcal{N} = 4$ super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic theories that describe the effective dynamics near the BPS bounds and upon quantization are known as Spin Matrix Theories. The near-BPS theories can be obtained by reducing $\mathcal{N}=4$ SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. In previous works we have considered various SU(1,1) and SU(1,2) types of subsectors in this limit. In the current work, we will construct the remaining Spin Matrix Theories defined near the $\frac{1}{8}$-BPS subsectors, which include the PSU(1,1|2) and SU(2|3) cases. We derive the Hamiltonians by applying the spherical reduction algorithm and show that they match with the spin chain result, coming from the loop corrections to the dilatation operator. In the PSU(1,1|2) case, we prove the positivity of the spectrum by constructing cubic supercharges using the enhanced PSU$(1|1)^2$ symmetry and show that they close to the interacting Hamiltonian. We finally analyse the symmetry structure of the sectors in view of an interpretation of the interactions in terms of fundamental blocks., 49 pages, 2 figures

We compute holographic complexity for the nonsupersymmetric Janus deformation of ${\mathrm{AdS}}_{5}$ according to the volume conjecture. The result is characterized by a power-law ultraviolet divergence. When a ball-shaped region located around the interface is considered, a subleading logarithmic divergent term and a finite part appear in the corresponding subregion volume complexity. Using two different prescriptions to regularize the divergences, we find that the coefficient of the logarithmic term is universal.

We investigate the complexity=volume proposal in the case of Janus AdS3 geometries, both at zero and finite temperature. The leading contribution coming from the Janus interface is a logarithmic divergence, whose coefficient is a function of the dilaton excursion. In the presence of the defect, complexity is no longer topological and becomes temperature-dependent. We also study the time evolution of the extremal volume for the time-dependent Janus BTZ black hole. This background is not dual to an interface but to a pair of entangled CFTs with different values of the couplings. At late times, when the equilibrium is restored, the couplings of the CFTs do not influence the complexity rate. On the contrary, the complexity rate for the out-of-equilibrium system is always smaller compared to the pure BTZ black hole background.

Computational complexity is a new quantum information concept that may play an important role in holography and in understanding the physics of the black hole interior. We consider quantum computational complexity for $n$ qubits using Nielsen's geometrical approach. We investigate a choice of penalties which, compared to previous definitions, increases in a more progressive way with the number of qubits simultaneously entangled by a given operation. This choice turns out to be free from singularities. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space., 41 pages, several figures; v2: journal version; v3: journal title

We consider limits of $\mathcal{N} = 4$ super-Yang-Mills (SYM) theory that approach BPS bounds. These limits result in non-relativistic near-BPS theories that describe the effective dynamics near the BPS bounds and upon quantization are known as Spin Matrix theories. The near-BPS theories can be obtained by reducing $\mathcal{N}=4$ SYM on a three-sphere and integrating out the fields that become non-dynamical in the limits. We perform the sphere reduction for the near-BPS limit with $\mathrm{SU}(1,2|2)$ symmetry, which has several new features compared to the previously considered cases with $\mathrm{SU}(1,1)$ symmetry, including a dynamical gauge field. We discover a new structure in the classical limit of the interaction term. We show that the interaction term is built from certain blocks that comprise an irreducible representation of the $\mathrm{SU}(1,2|2)$ algebra. Moreover, the full interaction term can be interpreted as a norm in the linear space of this representation, explaining its features including the positive definiteness. This means one can think of the interaction term as a distance squared from saturating the BPS bound. The $\mathrm{SU}(1,1|1)$ near-BPS theory, and its subcases, is seen to inherit these features. These observations point to a way to solve the strong coupling dynamics of these near-BPS theories., 54 pages, 4 figures. V2: minor comments added

We consider limits of $\mathcal{N}=4$ super Yang-Mills (SYM) theory that approach BPS bounds and for which an $SU(1,1)$ structure is preserved. The resulting near-BPS theories become non-relativistic, with a $U(1)$ symmetry emerging in the limit that implies the conservation of particle number. They are obtained by reducing $\mathcal{N}=4$ SYM on a three-sphere and subsequently integrating out fields that become non-dynamical as the bounds are approached. Upon quantization, and taking into account normal-ordering, they are consistent with taking the appropriate limits of the dilatation operator directly, thereby corresponding to Spin Matrix theories, found previously in the literature. In the particular case of the $SU(1,1|1)$ near-BPS/Spin Matrix theory, we find a superfield formulation that applies to the full interacting theory. Moreover, for all the theories we find tantalizingly simple semi-local formulations as theories living on a circle. Finally, we find positive-definite expressions for the interactions in the classical limit for all the theories, which can be used to explore their strong coupling limits. This paper will have a companion paper in which we explore BPS bounds for which a $SU(2,1)$ structure is preserved., Comment: 62 pages, 2 figures; v2: minor corrections

We analytically compute subsystem action complexity for a segment in the BTZ black hole background up to the finite term, and we find that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. This elegant structure does not survive to more complicated geometries: in the case of a two segments subregion in AdS$_3$, complexity has additional finite contributions. We give analytic results for the mutual action complexity of a two segments subregion., Comment: 29 pages, 7 figures, V2: minor changes, refs added

We compute the ultraviolet divergences of holographic subregion complexity for the left and right factors of the thermofield double state in warped AdS$_3$ black holes, both for the action and the volume conjectures. Besides the linear divergences, which are also present in the BTZ black hole, additional logarithmic divergences appear. For the action conjecture, these log divergences are not affected by the arbitrarity in the length scale associated with the counterterm needed to ensure reparameterization invariance. We find that the subregion action complexity obeys the superadditivity property for the thermofield double in warped AdS$_3$, independently from the action counterterm coefficient. We study the temperature dependence of subregion complexity at constant angular momentum and we find that it is correlated with the sign of the specific heat., Comment: 26 pages, 6 figures, V2: comments on regularization added

We study the Complexity=Volume conjecture for Warped AdS$_3$ black holes. We compute the spatial volume of the Einstein-Rosen bridge and we find that its growth rate is proportional to the Hawking temperature times the Bekenstein-Hawking entropy. This is consistent with expectations about computational complexity in the boundary theory., 18 pages, 3 figures, V2: refs added

Using the heat kernel method, we compute nonrelativistic trace anomalies for Schr\"odinger theories in flat spacetime, with a generic background gauge field for the particle number symmetry, both for a free scalar and a free fermion. The result is genuinely nonrelativistic, and it has no counterpart in the relativistic case. Contrary to the naive expectations, the anomaly is not gauge-invariant; this is similar to the non-gauge covariance of the non-abelian relativistic anomaly. We also show that, in the same background, the gravitational anomaly for a nonrelativistic scalar vanishes., Comment: 20 pages; V2 minor changes also in title, typos

The Complexity=Action conjecture is studied for black holes in Warped AdS$_3$ space, realized as solutions of Einstein gravity plus matter. The time dependence of the action of the Wheeler-DeWitt patch is investigated, both for the non-rotating and the rotating case. The asymptotic growth rate is found to be equal to the Hawking temperature times the Bekenstein-Hawking entropy; this is in agreement with a previous calculation done using the Complexity=Volume conjecture., Comment: 21 pages, 8 figures, V2:minor changes

We study the coupling of a 2+1 dimensional non-relativistic spin 1/2 fermion to a curved Newton-Cartan geometry, using null reduction from an extra-dimensional relativistic Dirac action in curved spacetime. We analyze Weyl invariance in detail: we show that at the classical level it is preserved in an arbitrary curved background, whereas at the quantum level it is broken by anomalies. We compute the trace anomaly using the Heat Kernel method and we show that the anomaly coefficients a, c are proportional to the relativistic ones for a Dirac fermion in 3+1 dimensions. As for the previously studied scalar case, these coefficents are proportional to 1/m, where m is the non-relativistic mass of the particle., Comment: 23 pages

Weyl consistency conditions are a powerful tool to study the irreversibility properties of the renormalization group. We apply this formalism to non-relativistic theories in 2 spatial dimensions with boost invariance and dynamical exponent z=2. Different possibilities are explored, depending on the structure of the gravitational background used as a source for the energy-momentum tensor., Comment: 22 pages; V2 refs. added

We classify the trace anomaly for parity-invariant non-relativistic Schr\"odinger theories in 2+1 dimensions coupled to background Newton-Cartan gravity. The general anomaly structure looks very different from the one in the z=2 Lifshitz theories. The type A content of the anomaly is remarkably identical to that of the relativistic 3+1 dimensional case, suggesting the conjecture that an a-theorem should exist also in the Newton-Cartan context. Erratum: due to an overcounting of the number of linearly-independent terms in the basis, the type A anomaly disappears if Frobenius condition is imposed. See appended erratum for details. This crucial mistake was pointed out to us in arXiv:1601.06795., Comment: 16 pages, V2:few equations corrected (final results unchanged), references added, typos, V3: erratum included

Questa tesi si focalizza sullo studio di due ambiti di ricerca principali: teorie di campo non relativistiche e complessità olografica. Nella prima parte riesaminiamo la classificazione generale dell'anomalia di traccia per teorie di campo 2+1 dimensionali accoppiate a una geometria di Newton-Cartan e revisioniamo anche il metodo dell'heat kernel, che è utilizzato per studiare azioni effettive a un loop e permette quindi di calcolare le anomalie per una data teoria. Applichiamo questa tecnica per estrarre i coefficienti esatti dei termini di curvatura dell'anomalia di traccia sia per uno scalare che per un fermione liberi e non relativistici, trovando una relazione con l'anomalia conforme della controparte relativistica in 3+1 dimensioni che suggerisce l'esistenza di una versione non relativistica del teorema a su cui commentiamo. Proseguiamo l'analisi dello scalare e del fermione libero e non relativistico con il metodo dell'heat kernel accendendo una sorgente per la massa delle particelle: in questa geometria, troviamo che non c'è anomalia gravitazionale, ma l'anomalia di traccia non è invariante di gauge. In seguito consideriamo un modello specifico che realizza un'estensione supersimmetrica N=2 del gruppo di Bargmann in 2+1 dimensioni con superpotenziale non nullo, ottenuto dalla riduzione dimensionale lungo una direzione nulla del modello di Wess-Zumino relativistico. Controlliamo che il superpotenziale è protetto contro le correzioni quantistiche come nella controparte relativistica, così trovando une versione non relativistica del teorema di non rinormalizzazione. Inoltre, abbia una forte evidenza che la teoria è esatta a un loop, a causa della struttura causale del propagatore non relativistico e a causa della conservazione della massa. Nella seconda parte della tesi revisioniamo le congetture olografiche proposte da Susskind per descrivere l'evoluzione temporale del ponte di Einstein-Rosen in teorie gravitazionali: la complessità=volume e la complessità=azione. Queste quantità possono essere usate come un modo per studiare dualità, e noi consideriamo sia il volume che l'azione di buchi neri in spazi AdS_3 warped, che costituiscono delle deformazioni non banali dell'usuale AdS_3, aventi isometrie al bordo non relativistiche. In particolare, calcoliamo analiticamente la dipendenza temporale della complessità trovando una derivata temporale asintotica proporzionale al prodotto della temperatura di Hawking e dell'entropia di Bekenstein-Hawking. In questo contesto, esistono estensioni delle proposte olografiche quando lo stato duale in teoria dei campi è misto, cioé consideriamo soltanto una sottoregione del bordo. Calcoliamo la struttura delle divergenze ultraviolette, il comportamento sub/super-additivo della complessità e la sua dipendenza dalla temperatura per buchi neri warped in 2+1 dimensioni, quando la sottoregione è uno dei bordi disconnessi dello spaziotempo. Infine, calcoliamo analiticamente la complessità=azione per una sottoregione data da un segmento nella geometria del buco nero BTZ, dimostrando che è uguale alla somma di un termine divergente direttamente proporzionale alla lunghezza della sottoregione, più un termine proporzionale all'entropia di entanglement. Mentre questo risultato suggerisce un forte legame tra complessità ed entropia di entanglement, dimostriamo che nel caso di due segmanti disgiunti nella geometria del BTZ ci sono dei termini finiti aggiuntivi: quindi, la precedente elegante struttura rimane vera solo per le parti divergenti. This thesis focuses on the investigation of two broad research areas: non-relativistic field theories and holographic complexity. In the first part we review the general classification of the trace anomaly for 2+1 dimensional field theories coupled to a Newton-Cartan background and we also review the heat kernel method, which is used to study one-loop effective actions and then allows to compute anomalies for a given theory. We apply this technique to extract the exact coefficients of the curvature terms of the trace anomaly for both a non-relativistic free scalar and a fermion, finding a relation with the conformal anomaly of the 3+1 dimensional relativistic counterpart which suggests the existence of a non-relativistic version of the a-theorem on which we comment. We continue the analysis of non-relativistic free scalar and fermion with the heat kernel method by turning on a source for the particle mass: on this background, we find that there is no gravitational anomaly, but the trace anomaly is not gauge invariant. We then consider a specific model realizing a N=2 supersymmetric extension of the Bargmann group in 2+1 dimensions with non-vanishing superpotential, obtained by null reduction of a relativistic Wess-Zumino model. We check that the superpotential is protected against quantum corrections as in the relativistic parent theory, thus finding a non-relativistic version of the non-renormalization theorem. Moreover, we find strong evidence that the theory is one-loop exact, due to the causal structure of the non-relativistic propagator together with mass conservation. In the second part of the thesis we review the holographic conjectures proposed by Susskind to describe the time-evolution of the Einstein-Rosen bridge in gravitational theories: the complexity=volume and complexity=action. These quantities may be used as a tool to investigate dualities, and we investigate both the volume and the action for black holes living in warped AdS_3 spacetime, which is a non-trivial modification of usual AdS_3 with non-relativistic boundary isometries. In particular, we analytically compute the time dependence of complexity finding an asymptotic growth rate proportional to the product of Hawking temperature and Bekenstein-Hawking entropy. In this context, there exist extensions of the holographic proposals when the dual state from the field theory side is mixed, i.e. we consider only a subregion on the boundary. We study the structure of UV divergences, the sub/super-additivity behaviour of complexity and its temperature dependence for warped black holes in 2+1 dimensions when the subregion is taken to be one of the two disconnected boundaries. Finally, we analytically compute the subregion action complexity for a general segment on the boundary in the BTZ black hole background, finding that it is equal to the sum of a linearly divergent term proportional to the size of the subregion and of a term proportional to the entanglement entropy. While this result suggests a strong relation of complexity with entanglement entropy, we find after investigating the case of two disjoint segments in the BTZ background that there are additional finite contributions: then the previous elegant structure holds only for the divergent parts.