# Dr Andrea Prinsloo

## About

### Biography

- 2018-present: Teaching Fellow B in the Department of Mathematics at the University of Surrey
- 2015-2018: Teaching Fellow A in the Department of Mathematics at the University of Surrey
- 2015-2018: Associate Lecturer at the Open University
- 2012-2015: Research Fellow in the Department of Mathematics at the University of Surrey
- 2010-2012: NRF Innovation Postdoctoral Fellow at the National Institute for Theoretical Physics of South Africa and University of Cape Town
- 2010-2012: Assistant Lecturer in the Department of Mathematics and Applied Mathematics at the University of Cape Town
- 2007-2009: PhD in Applied Mathematics at the University of Cape Town
- 2005-2006: MSc in Physics at the University of the Witwatersrand (with distinction)
- 2004: BSc (Hons) in Theoretical Physics at the University of Cape Town (with distinction)
- 2001-2003: BSc majoring in Applied Mathematics, Mathematics and Physics at the University of Cape Town (with a triple distinction).

### University roles and responsibilities

- Personal Tutor of nine mathematics undergraduate students in 2022/23
- Departmental Coordinator of the Mathematics Student Mentoring Scheme, run by the University of Surrey Mathematics Society (MathSoc) since 2019
- Information Asset Owner, jointly coordinating (with Dr M. Wolf) the Department of Mathematics student data policy since 2018
- Organiser of the Amazing Maths seminars for undergraduate mathematics students since 2017
- University of Surrey Mathematics Society (MathSoc) Liaison since 2017
- Organiser of the Fields, Strings and Geometry Group seminars since 2012

### Affiliations and memberships

## Research

### Research interests

I am a member of the Fields, Strings and Geometry Group in the Department of Mathematics. My research focuses on gauge-gravity dualities, superstring and supergravity theories, D and M-branes, and quantum integrable systems and underlying algebraic structures. My current research interests include:

- Integrable spin-chains and scattering matrices in AdS/CFT correspondences
- Hopf superalgebras and quantum group symmetries
- Supergravity geometries from back-reacting brane configurations
- D and M-brane giant gravitons with non-trivial topologies in string theory and M-theory.

### Conference organising

- Local co-organiser of the 16th South-East Mathematical Physics Seminars meeting on 28 Apr 2022 at the University of Surrey, funded by the London Mathematical Society.
- Local co-organiser of the 11th South-East Mathematical Physics Seminars meeting on 28 Mar 2018 at the University of Surrey, funded by the London Mathematical Society.
- Local co-organiser of the 5th South-East Mathematical Physics Seminars meeting on 6 Nov 2015 at the University of Surrey, funded by the London Mathematical Society.

## Supervision

### Postgraduate research supervision

Co-supervisor (with Dr J. Gutowski) of MPhil student, Mr Mirco Di Gioia, since January 2020.

#### Project supervision

- Supervisor of a BSc Project (year 3) on
*Time-independent Perturbation Theory and its Applications to the Hydrogen Atom*in 2021/22. - Supervisor of a BSc Project (year 3) on
*Special and General Relativity*in 2020/21 - Co-supervisor (with Dr A. Torrielli) of an MMath Project on the
*Nested Algebraic Bethe Ansatz for su(3) spin-chains and beyond*in 2020/21 - Supervisor of a Literature Review (year 3) on
*Variational Methods in Physics*in Semester 1 of 2019/20 - Supervisor of a Literature Review (year 3) on
*The Stark Effect and Spectral Line Splitting in the Hydrogen Atom*in Semester 2 of 2018/19.

## Teaching

#### Semester 1 modules in 2022/23

- Mathematics I: Pure Mathematics (EEE1031)
- Groups and Rings (MAT2048)
- Advanced Techniques in Mathematics (MATM045).

#### Semester 2 modules in 2022/23

- Vector Calculus (MAT1005)
- Introduction to Complex Analysis (MAT2052)
- Representation Theory (MATM035).

#### Semester 1 modules in 2021/22

- Mathematics I: Pure Mathematics (EEE1031)
- Groups and Rings (MAT2048)
- Advanced Techniques in Mathematics (MATM045).

#### Semester 2 modules in 2021/22

- Vector Calculus (MAT1005)
- Introduction to Complex Analysis (MAT2052)
- General Topology (MATM042).

## Publications

We study some of the properties of dual giant gravitons - D2-branes wrapped on an S2⊂AdS4 - in type IIA string theory on AdS4×CP3. In particular we confirm that the spectrum of small fluctuations about the giant is both real and independent of the size of the graviton. We also extend previously developed techniques for attaching open strings to giants to this D2-brane giant and focus on two particular limits of the resulting string sigma model: In the pp-wave limit we quantize the string and compute the spectrum of bosonic excitations while in the semiclassical limit, we read off the fast string Polyakov action and comment on the comparison to the Landau-Lifshitz action for the dual open spin chain.

We construct the giant graviton on AdS4xCP3 out of a four-brane embedded in and moving on the complex projective space. This configuration is dual to the totally anti-symmetric Schur polynomial operator chiR(A1B1) in the 2+1-dimensional, N=6 super Chern-Simons ABJM theory. We demonstrate that this BPS solution of the D4-brane action is energetically degenerate with the point graviton solution and initiate a study of its spectrum of small fluctuations. Although the full computation of this spectrum proves to be analytically intractable, by perturbing around a 'small' giant graviton, we find good evidence for a dependence of the spectrum on the size, alpha0, of the giant. This is a direct result of the changing shape of the worldvolume as it grows in size.

In this article we consider gauge theories with a U(N)×U(N) gauge group. We provide, for the first time, a complete set of operators built from scalar fields that are in the bi-fundamental of the two groups. Our operators diagonalize the two point function of the free field theory at all orders in 1/N. We then use this basis to investigate non-planar anomalous dimensions in the ABJM theory. We show that the dilatation operator reduces to a set of decoupled harmonic oscillators, signaling integrability in a large N but non-planar limit of the theory.

We construct a new NS5-brane solution in AdS4×CP3 with S2×S3 topology. This solution belongs to the general class of non-Einstein N11 metrics to which T11 belongs, and carries a non-vanishing D0-brane charge. In eleven dimensions it gives rise to a squashed S2×S3 M5-brane giant graviton which is now of the N10 type. The energies of both solutions satisfy the BPS bound E = kQ/2, indicating supersymmetric configurations, where Q is interpreted as D0-brane charge for the NS5-brane and as angular momentum for the dynamically stable M5-brane giant graviton. The ground state is degenerate with a spherical D2 or M2-brane, rather than with a point-like object. Moreover, while the charge of the spherical 2-brane can be arbitrary, the charge of the S2×S3 5-brane is bounded by N/2, with N the rank of the ABJM gauge group, a manifestation of the stringy exclusion principle. A microscopic description, suitable for the study of the finite ’t Hooft coupling region, is provided in terms of spherical D2 or M2-branes expanding into fuzzy 3-spheres.

Following on from recent studies of string theory on a one-parameter family of integrable deformations of AdS5×S5 proposed by Lunin and Maldacena, we carry out a systematic analysis of the high temperature properties of type IIB strings on the associated pp-wave geometries. In particular, through the computation of the thermal partition function and free energy we find that not only does the theory exhibit a Hagedorn transition in both the (J,0,0) and (J,J,J) class of pp-waves, but that the Hagedorn temperature is insensitive to the deformation suggesting an interesting universality in the high temperature behaviour of the pp-wave string theory. We comment also on the implications of this universality on the confinement/deconfinement transition in the dual N=1 Leigh-Strassler deformation of N=4 Yang-Mills theory.

We implement Mikhailov's holomorphic curve construction to explore various properties of giant gravitons in type IIB string theory on AdS5×T11. By coloring the D-brane worldvolume, we are able to show how, in the string theory, the giant graviton factorizes at its maximal size into two dibaryons - topologically stable D-branes wrapping non-contractible cycles in the T11. This is related to the structure of the symmetry group of the emergent Klebanov-Witten gauge theory being a product - SU(N)×SU(N) instead of the canonical SU(N). Finally, we complete this study with a systematic and detailed construction of the spectrum of small fluctuations about the giant graviton configuration. Curiously, we find that the fluctuation spectrum depends on the size of the giant. The similarity of the operator structures in the Klebanov-Witten and ABJM theories leads us to believe that the D4-brane giant graviton in type IIA string theory on AdS4×CP3 factorizes into two CP2 dibaryons in a qualitatively similar way.

A 3-parameter generalization of the Lunin-Maldacena background has recently been constructed by Frolov. This gamma_i-deformed background is non-supersymmetric. We consider strings in this gamma_i-deformed R ____times S^5 background rotating in three orthogonal planes (the 3-spin sector) in a fast motion limit, in which the total angular momentum J is assumed to be large. We show that there exists a consistent transformation which takes the undeformed equations of motion into the gamma_i-deformed equations of motion. This transformation is used to construct a Lax pair for the bosonic part of the gamma_i-deformed theory in the fast motion limit. This implies the integrability of the bosonic part of the gamma_i-deformed string sigma model in the fast motion limit.

We study integrable open boundary conditions for d(2,1;____alpha)^2 and psu(1,1|2)^2 spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type IIB open superstrings ending on D-branes in the AdS_3 x S^3 x S^3 x S^1 and AdS_3 x S^3 x T^4 supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1 and D5-brane maximal giant gravitons.

We report on the discovery of at least 18 pulsation modes in β Pictoris from 697 high-dispersion spectra obtained over a two-week period. These are seen as absorption subfeatures moving across the spectral line profiles, indicating that they are of relatively high spherical harmonic order. They do not resemble the features seen in the Ca H and K lines generally attributed to infalling bodies. We use the correlation function between an unbroadened synthetic spectrum and the observed spectrum to obtain what is essentially a high signal-to-noise ratio representation of the mean line profile. From the correlation intensity as a function of time, we calculated periodograms for every point across the correlation function. In this way, we are able to determine accurate periods for each mode. By phasing the correlation function with the period belonging to a given mode, we can isolate the line profile variations for the mode and deduce the approximate value of the azimuthal order, m. Two of the modes are retrograde, the remainder are prograde and most of them lie in the range 4

Using Mikhailovʼs map from holomorphic functions to supersymmetric D3-brane solutions, we show how to construct giant gravitons in AdS5×S5 with toroidal topologies. In the 1/4-BPS sector we show that these are always of the form ♯K(S2×S1), and in the limit in which this becomes a set of m+n perpendicular spherical giants re-connected near to their intersections, we find K in terms of m, n. In the 1/8-BPS sector we find a similar class of solutions.

We extend the proposal for a detailed map between wrapped D-branes in Anti-de Sitter space and baryon-like operators in the associated dual conformal field theory provided in hep-th/0202150 to the recently formulated AdS4xCP3/ABJM correspondence. In this example, the role of the dibaryon operator of the 3-dimensional CFT is played by a D4-brane wrapping a CP2xCP3. This topologically stable D-brane in the AdS4xCP3 is nothing but one-half of the maximal giant graviton on CP3.

In this article we extend the construction of giant gravitons from holomorphic surfaces [arXiv:hep-th/0010206] to the ABJM correspondence. We construct a new class of 1/6-BPS M5-branes wrapping 5-manifolds in S7/Zk and supported by a large angular momentum in the orbifold space. These orbifold giant gravitons undergo a supersymmetry enhancement to 1/3-BPS and 1/2-BPS configurations in special cases. The compactification of M-theory on AdS4xS7/Zk to type IIA superstring theory on AdS4xCP3 then gives rise to another new class of 1/6-BPS D4 and NS5-branes wrapping 4 and 5-manifolds in CP3. The D4-branes carry a combination of D0-brane charge and angular momentum in the complex projective space, while the NS5-branes are supported only by D0-brane charge. Finally, we present a detailed analysis of a one-parameter family of 1/2-BPS M5-brane orbifold giant gravitons, and their D4 and NS5-brane CP3 descendants.

### Additional publications

##### Theses and dissertations

- A. Prinsloo, “
*Applications of the gauge theory/gravity correspondence*” (PhD thesis), Department of Mathematics and Applied Mathematics (Supervisor: Dr Jeffrey Murugan), University of Cape Town (2010). - A. Prinsloo, “
*AdS/CFT correspondence in a nonsupersymmetric γ**i**-deformed background*” (MSc dissertation), Department of Physics and Centre for Theoretical Physics (Supervisor: Prof Robert de Mello Koch), University of the Witwatersrand (2007).

##### Invited conference talks

- "
*Giant gravitons on AdS**3**x S**3**x S**3**x S**1*" (PDF), Permutations and Gauge-String Duality 2014 Workshop, Queen Mary College, University of London, 21-25 July 2014. - "
*The topology of giant gravitons*" (PDF), 1st South-East Mathematical Physics Seminars Meeting, University of Kent, 2 April 2014.

##### Conference posters

- "
*Integrable open spin-chains in AdS**3**/CFT**2*" (PDF), Young Researchers Integrability School, Durham University, 6-10 July 2015; and IGST 2015 Conference, King's College, University of London, 13-17 July 2015.