# Joanna Karczmarek

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#### Supervision Enquiry

## Graduate Student Supervision

##### Doctoral Student Supervision (Jan 2008 - May 2021)

We study the non-commutative geometry associated with matrices of N quantum particles in matrix models. The earlier work established a surface embedded in flat ℝ³ from three Hermitian matrices. We construct coherent states corresponding to points in the emergent geometry and find that the original matrices determine not only the shape of the emergent surface, but also a unique Poisson structure. Through our construction, we can realize arbitrary non-commutative membranes embedded in ℝ³. We further conjecture an embedding operator that assigns, to any 2n+1 N-dimensional Hermitian matrices, a 2n-dimensional hypersurface in flat (2n+1)-dimensional Euclidean space. This corresponds to defining a fuzzy D(2n)-brane corresponding to N D0-branes. Points on the hypersurface correspond to zero eigenstates of the embedding operator, which have an interpretation as coherent states underlying the emergent non-commutative geometry. Using this correspondence, all physical properties of the emergent D(2n)-brane can be computed. Many studies have been carried out exploring the geometry emerging from the matrix configuration, but they have not always produced consistent results. We apply two types of point-probe methods, as well as the supergravity charge density formula to the generalized fuzzy sphere. Its tangled structure challenges the applicability of these probing methods. We propose to disentangle blocks of the generalized fuzzy sphere regarding the geometrical symmetry and retrieve the generalized fuzzy sphere as a thick two sphere with coherent layers consistently in three methods. The Yang-Mills (YM) matrix model with mass term representing a cutoff radius generates remarkable spherical solutions of the emergent universe, but it is unsolvable, unlike for matrix models dominated by a Gaussian potential. By coarse-graining the dimension of matrices, quantum gravity is reproduced by the Gaussian model at the fixed point of the dimensional-renormalization group flow. We approach the unsolvable YM model using the same dimensional-renormalization and discover a non-trivial fixed point after imposing spherical topology. This fixed point might lead to a new duality between quantum gravity and the massive YM model, and its existence also sets a density condition on the generalized fuzzy sphere.

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This thesis presents various examples of the application of quantum-mechanical methods to the understanding of the structure of space-time. It focuses on noncommutative geometry and the gauge/gravity duality as intermediaries between quantum mechanics and classical geometry. First, we numerically calculate entanglement entropy and mutual information for a massive free scalar field on commutative and noncommutative (fuzzy) spheres. To define a subregion with a well-defined boundary in the noncommutative geometry, we use the symbol map between elements of the noncommutative algebra and functions on the sphere. We show that the UV-divergent part of the entanglement entropy on a fuzzy sphere does not follow an area law. In agreement with holographic predictions, it is extensive for small (but fixed) regions. This is true even in the limit of small noncommutativity. Nonetheless, we find that mutual information (which is UV-finite) is the same in both theories. This suggests that nonlocality at short distances does not affect quantum correlations over large distances in a free field theory. Previous work has shown that a surface embedded in flat ℝ³ can be associated to any three Hermitian matrices. By constructing coherent states corresponding to points in the emergent geometry, we study this emergent surface when the matrices are large. We find that the original matrices determine not only shape of the emergent surface, but also a unique Poisson structure. We prove that commutators of matrix operators correspond to Poisson brackets. Through our construction, we can realize arbitrary noncommutative membranes.Finally, we use the gauge/gravity correspondence to translate the positivity of relative entropy on the boundary into constraints on allowable space-time metrics in the bulk. Using the Einstein equations, we interpret these constraints as energy conditions. For certain three-dimensional bulks, we obtain strict constraints coming from the positivity of relative entropy with a thermal reference state which turn out to be equivalent to a version of the weak energy condition near the boundary. In higher dimensions, we use the canonical energy formalism to obtain similar but weaker constraints.

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The holographic Ryu-Takayanagi formula for entanglement entropy connects the entanglement of a field theory to the geometry of a dual gravitational theory in a straightforward and universal way. The first part of this thesis applies this formula to study the entanglement entropy in strongly coupled noncommutative field theories. It is found that the ground state of these theories have substantial entanglement at the length scale of the noncommutativity. The entanglement entropy in a different perturbative regime is also computed, where in contrast it is found that noncommutative interactions do not induce long range entanglement in the ground state to leading order in perturbations theory. The second part of this thesis explores some general consequences of this holographic formula for the entanglement entropy. Identities involving entanglement entropies are related to nontrivial geometric constraints on gravitational duals. In particular, the strong subadditivity of entanglement entropy is used to show that dual three dimensional asymptotically anti-de Sitter gravitational states must obey an averaged null energy condition. Finally, this holographic formula allows us at least in principle to express the entanglement entropy of a region in a holographic field theory in terms of the one-point functions in that theory. This is explored in the context of a two dimensional conformal field theory where explicit calculations are possible. Our results in this case allow us to extend a recent proposal that the entanglement entropy of states near the vacuum of conformal theories can be understood by propagation in an auxiliary de Sitter space.

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The study of solutions to open string field theory remains very much a work in progress, even for the bosonic string. In this dissertation I consider in detail two of these solutions involving marginal deformations of the original boundary conformal field theory. The first is a previously unknown solution in which two D-branes are translated before tachyon condensation occurs. This solution is studied in the level truncation scheme, in a sector which is larger than the universal subspace, but still less than the whole string Fock space due to several symmetries of the theory which take on a different content in the presence of two D-branes. This solution brings us a step closer to a full understanding of the relationship between the magnitude of a marginal deformation in BCFT and the strength of the corresponding marginal operator in OSFT. The other solution I study was first written down formally by Kiermaier and Okawa, and involves the renormalization of an exactly marginal operator. I consider the same solution with a more general renormalization scheme and find a set of sufficient restrictions for the solution’s validity. While this proceeds much as in the original work on this solution, I find some freedom in the solution as well as additional algebraic structure for renormalization schemes. I also present a collection of procedures written in Maple which define and manipulate wedge states with insertions, as well as computing correlation functions for such states provided that all inserted operators are sufficiently simple. Using this code I am able to calculate the tachyon profile of this solution for the time-symmetric rolling tachyon at 6th order in λ and describe its properties in comparison to previously known rolling tachyon profiles. I find the same unwanted oscillations that were seen in previous work on the time-asymmetric rolling tachyon.

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The holographic principle connects theories with gravity to lower dimensional theories without gravity. Notably, the AdS/CFT correspondence — the first concrete realization of the holographic principle — provides a one to one map between string theory in Anti de Sitter space, and a strongly coupled, large N, SU(N) super Yang-Mills gauge theory in one less dimension.In this thesis, within the context of holographic field theories, I improve on the current understanding of the map between gravity (bulk) and gauge theory (boundary) degrees of freedom. Furthermore, I explore some of the applications of the AdS/CFT correspondence to the study of strongly coupled field theories.I study the map between bulk and boundary degrees of freedom mainly by trying to determine what is the gravity dual of a subset of the boundary field theory. In the process of doing so I show how extremal surfaces, entanglement entropy, hyperbolic black holes, and boson stars are fundamental tools in this quest.Next, I explore a few examples of direct applications of the correspondence as a model building device. I discuss how AdS/CFT can be used to construct quasi realistic strongly coupled physical systems ranging from relativistic fluids to plasmas and high temperature superconductors. Finally, I compare some of the results obtained in this thesis with known standard field theory results.

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##### Master's Student Supervision (2010 - 2020)

Over the years, deep insights into string theory and supersymmetric gauge theorieshave come from studying geometry emerging from matrix models. In this thesis,I study the ℕ = 1* and ℕ = 2* theories from which an elliptic curve withmodular parameter τ is known to emerge, alongside an elliptic function called thegeneralized resolvent into which the physics is encoded. This is indicative of thecommon origin of the two theories in ℕ = 4 SYM. The ℕ = 1* Dijkgraaf-Vafamatrix model is intrinsically holomorphic with parameter space corresponding tothe upper-half plane ℍ. The Dijkgraaf-Vafa matrix model ’t Hooft coupling S(τ)has been previously shown to be holomorphic on ℍ and quasi-modular with respectto SL(2,ℤ). The allowed ℕ = 2* coupling is constrained to a Hermitianslice through the enlarged moduli space of the holomorphic ℕ = 1* model.After explicitly constructing the map from the elliptic curve to the eigenvalueplane, I argue that the ℕ = 1* coupling S(τ) encodes data reminiscent of ℕ = 2*.A collection of extrema (saddle-points) of S(τ) behave curiously like the quantumcritical points of ℕ = 2* theory. For the first critical point, the match is exact. Thiscollection of points lie on the line of degeneration which behaves in a sense, like aboundary at infinityI also show explicitly that the emergent elliptic curve along with the generalizedresolvent allow one to recover exact eigenvalue densities. At weak coupling, mymethod reproduces the inverse square root of ℕ = 2* as well as the Wigner semicirclein ℕ = 1*. At strong coupling in ℕ = 1*, I provide encouraging evidenceof the parabolic density arising in the neighborhood of the line of degeneration.To my knowledge, the parabolic density has only been observed asymptotically. Itis interesting to see evidence that it may be exactly encoded in the other form ofemergent geometry: the elliptic curve with the generalized resolvent.

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Over the years, deep insights into string theory and supersymmetric gauge theorieshave come from studying geometry emerging from matrix models. In this thesis,I study the ℕ = 1* and ℕ = 2* theories from which an elliptic curve withmodular parameter τ is known to emerge, alongside an elliptic function called thegeneralized resolvent into which the physics is encoded. This is indicative of thecommon origin of the two theories in ℕ = 4 SYM. The ℕ = 1* Dijkgraaf-Vafamatrix model is intrinsically holomorphic with parameter space corresponding tothe upper-half plane ℍ. The Dijkgraaf-Vafa matrix model ’t Hooft coupling S(τ)has been previously shown to be holomorphic on ℍ and quasi-modular with respectto SL(2,ℤ). The allowed ℕ = 2* coupling is constrained to a Hermitianslice through the enlarged moduli space of the holomorphic ℕ = 1* model.After explicitly constructing the map from the elliptic curve to the eigenvalueplane, I argue that the ℕ = 1* coupling S(τ) encodes data reminiscent of ℕ = 2*.A collection of extrema (saddle-points) of S(τ) behave curiously like the quantumcritical points of ℕ = 2* theory. For the first critical point, the match is exact. Thiscollection of points lie on the line of degeneration which behaves in a sense, like aboundary at infinityI also show explicitly that the emergent elliptic curve along with the generalizedresolvent allow one to recover exact eigenvalue densities. At weak coupling, mymethod reproduces the inverse square root of ℕ = 2* as well as the Wigner semicirclein ℕ = 1*. At strong coupling in ℕ = 1*, I provide encouraging evidenceof the parabolic density arising in the neighborhood of the line of degeneration.To my knowledge, the parabolic density has only been observed asymptotically. Itis interesting to see evidence that it may be exactly encoded in the other form ofemergent geometry: the elliptic curve with the generalized resolvent.

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A noncommutative geometric configuration of D1-branes is obtained by numeric methods. This configuration is a static BPS solution to the non-Abelian Born-Infeld action, which is dual to an Abelian BPS D3-brane solution with magnetic charges. These monopoles correspond to emergent D1-branes that are attached to the D3-brane's surface and span a transverse direction. In the non-Abelian geometry, there is a topology change from a single noncommutative two-sphere of infinite radius at one end into two isolated two-spheres at infinity, with asymptotically vanishing radii. Recent method for constructing an Abelian surface from a non-Abelian geometry is used on the D1-brane configuration to compare it to the Abelian D3-brane configuration. The D3-brane and the D1-brane pictures are expected to converge for large N, yet surprisingly good agreement is found for N only reaching as high as 6.

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The goal of this thesis is to study non-singlet sectors of the c=1 matrix model in order to determine their connection with string theory in two space-time dimensions. It is well understood that the singlet sector of this matrix model is related to the closed string sector of the string theory. The adjoint sector has been connected to long strings stretching from an FZZT brane at infinity. The goal is to find a sector which includes a large number of these long strings which condense to form an FZZT brane in line with a proposal due to Gaiotto. Sectors corresponding to symmetric and antisymmetric tensor products of the fundamental and antifundamental representation of the U(N) symmetry of the matrix model are studied. Partition functions of the matrix model with a harmonic oscillator potential as well as bases for the Hilbert spaces are found for all these sectors. The first two representations are studied using a collective field approach with the inverted oscillator potential applicable to the c=1 model and are found to contain large numbers of long strings which form a free bosonic gas in the large N limit and so do not condense. On the other hand, in the third sector the degrees of freedom are fermionic and a shift in the ground state energy is found in the harmonic oscillator potential, which points to the existence of a Fermi sea and the condensation of the long strings. A collective field that can be used to study this sector is proposed and the technical difficulties presented by the study of this sector are discussed, but its study is left for future work.

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To derive the phase transition of large N gauge theories, the calculation of the partition function is restricted to the singlet sector. In this thesis, we examined whether the singlet constraint of gauge theory at the deconfinement phase was a physical consequence. We found changing the integration variables from group elements to eigenvalues explains the discrepancy of the free energy between imposing and relaxing the single constraint. Gauge theory cannot simply relax the singlet constraint at high temperature without adding a normalization factor.

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