My research area is theoretical physics: quantum field theory in particle physics; field theoretical techniques in condensed matter; general relativity.
Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in f... more Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in four space-time dimensions can be used to analyse transport properties in two space dimensions. A non-linear renormalisation group equation for the conductivity in two dimensions is derived in this model and, as an example of its application, both the Ohmic and Hall DC and AC conductivities are studied in the presence of a magnetic field, using a bulk dyonic solution of the Einstein-Maxwell equations in asymptotically AdS4 space-time. The $$ \mathcal{Q} $$ Q -factor of the cyclotron resonance is shown to decrease as the temperature is increased and increase as the charge density is increased in a fixed magnetic field. Likewise the dissipative Ohmic conductivity at resonance increases as the temperature is decreased and as the charge density is increased. The analysis also involves a discussion of the piezoelectric effect in the context of the AdS/CMT framework.
Geometrically the phase space of a mechanical system involves the cotangent bundle of the configu... more Geometrically the phase space of a mechanical system involves the cotangent bundle of the configuration space. The phase space of a relativistic field theory is infinite dimensional and can be endowed with a symplectic structure defined in a perfectly co-variant manner that is very useful for discussing symmetries and conserved quantities of the system. In general relativity the symplectic structure takes Darboux form and it is shown in this work that the presence of a cosmological constant does not change this conclusion. For space-times that admit time-like Killing vectors the formalism can be used to define mass in general relativity and it is known that, for asymptotically flat black holes, this mass is identical the the usual Arnowitt-Desner-Misner mass while for asymptotically anti-de Sitter Kerr metrics it is the same as the Henneaux-Teitelboim mass. We show that the same formalism can also be used to derive the Brown-York mass and the Bondi mass for stationary space times, in particular the Brown-York mass has a natural interpretation in terms of differential forms on the space of solutions of the theory.
Journal of Physics A: Mathematical and Theoretical
The quantum Hall effect is studied in a spherical geometry using the Dirac operator for nonintera... more The quantum Hall effect is studied in a spherical geometry using the Dirac operator for noninteracting fermions in a background magnetic field, which is supplied by a Wu-Yang magnetic monopole at the centre of the sphere. Wave functions are cross-section of a non-trivial U (1) bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah-Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions ν = 1 2k+1 are recovered.
For a field theory that is invariant under diffeomorphisms there is a subtle interplay between sy... more For a field theory that is invariant under diffeomorphisms there is a subtle interplay between symmetries, conservation laws and the phase space of the theory. The natural language for describing these ideas is that of differential forms and both differential forms on space-time and differential forms on the infinite dimensional space of solutions of the equations of motion of the field theory play an important role. There are exterior derivatives on both spaces and together they weave a double differential complex which captures the cohomology of the theory. This is important in the definition of invariants in general relativity, such as mass and angular momentum and is also relevant to the study of quantum anomalies in gauge theories. We derive the structure of this double complex and show how it relates to conserved quantities in gravitational theories. One consequence of the construction is that conserved quantities can be calculated exactly at finite distance-for example it is not necessary to go to asymptotic regimes to calculate the mass or angular momentum of a stationary solution of Einstein's equations, they can be obtained exactly by an integration over any sphere outside the mass even at finite radius. 1 If Σ and Σ ′ have boundaries the fields are assumed to vanish sufficiently fast there that there is no contribution from any time-like component of ∂M.
An invariant definition of mass in asymptotically de-Sitter space-times is given that relies on t... more An invariant definition of mass in asymptotically de-Sitter space-times is given that relies on the existence of a time-like Killing vector on a sphere surrounding the mass but does not require going to an asymptotic region. In particular the mass can be calculated exactly on a sphere inside the cosmological horizon. The formalism requires varying the background metric solution by a perturbation that satisfies the linearized equations of motion but need not share the Killing symmetry of the solution and is therefore ideally suited to calculating masses in stationary spacetimes perturbed by a gravitational wave without going beyond the cosmological horizon.
The equation of state associated with N = 4 supersymmetric Yang-Mills in 4 dimensions, for SU(N) ... more The equation of state associated with N = 4 supersymmetric Yang-Mills in 4 dimensions, for SU(N) in the large N limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang-Mills theory on the boundary in which the cosmological constant is equivalent to a volume. The thermodynamic variable conjugate to the cosmological constant is a pressure and the P − V diagram is studied. It is known that there is a critical point where the heat capacity diverges and this is reflected in the isothermal compressibility. Critical exponents are derived and found to be mean field in the large N limit. The same analysis applied to 3 and 6 dimensional conformal field theories again yields mean field exponents associated with the compressibility at the critical point. 1 DIAS-STP-16-05 1 Feature invited paper for special edition "Black hole thermodynamics II" in Entropy.
The renormalisation group equation for N-point correlation functions can be interpreted in a geom... more The renormalisation group equation for N-point correlation functions can be interpreted in a geometrical manner as an equation for Lie transport of amplitudes in the space of couplings. The vector field generating the diffeomorphism has components given by the βfunctions of the theory. It is argued that this simple picture requires modification whenever any one of the points at which the amplitude is evaluated becomes close to any other. This modification requires the introduction of a connection on the space of couplings and new terms appear in the renormalisation group equation involving co-variant derivatives of the β-function and the curvature associated with the connection. It is shown how the connection is related to the operator expansion coefficients , but there remains an arbitrariness in its definition.
We explore the consequences of introducing a complex conductivity into the quantum Hall effect. T... more We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule $|p_1q_2 - p_2q_1|=1$ for quantum Hall transitions between filling factors $\nu_1=p_1/q_1$ and $\nu_2=p_2/q_2$; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus.
Skip to main content. Help us improve the CERN Document Server for you: take part! CERN Logo CERN... more Skip to main content. Help us improve the CERN Document Server for you: take part! CERN Logo CERN Document Server. Related links. CDS; Indico; Library; Bulletin; EDMS. Main navigation links: Search; Submit; Help; Your CDS: Your alerts; Your baskets; Your searches. login. Home > Articles & Preprints > Published Articles > Non-symmetric coset spaces with torsion. Information; Discussion; Files. Article. Title, Non-symmetric coset spaces with torsion : alternative compactifications of 10-d superstrings. ...
Thermodynamic potentials relevant to the microcanonical, the canonical and the grand canonical en... more Thermodynamic potentials relevant to the microcanonical, the canonical and the grand canonical ensembles, associated with rotating black holes in D-dimensions, are analysed and compared. Such black holes are known to be thermodynamically unstable, but the instability is a consequence of a subtle interplay between specific heats and the moments of inertia and it manifests itself differently in the different ensembles. A simple relation between the product of the specific heat and the determinant of the moment of inertia in both the canonical and the grand canonical ensembles is derived. Myers-Perry black holes in arbitrary dimension are studied in detail. All temperature extrema in the microcanonical ensemble are determined and classified. The specific heat and the moment of inertia tensor are evaluated in both the canonical and the grand canonical ensembles in any dimension. All zeros and poles of the specific heats, as a function of the angular momenta, are determined and the eigenvalues of the isentropic moment of inertia tensor are studied and classified. It is further shown that many of the thermodynamic properties of a Myers-Perry black hole in D − 2 dimensions can be obtained from those of a black hole in D dimensions by sending one of the angular momenta to infinity.
A new 501ut10n 0f the 6050n1c 5ect0r 0fD = 11 5uper9rav1ty 15 pre5ented 1n wh1ch the 1nterna1 5pa... more A new 501ut10n 0f the 6050n1c 5ect0r 0fD = 11 5uper9rav1ty 15 pre5ented 1n wh1ch the 1nterna1 5pace ha5 the t0p0109y CP 2 × 52 × 51 and the 1nterna1 150metry 9r0up 15 5U(3) × 5U(2) X U(1).
For any K(= 2k +1)th-order Lovelock gravity with fine-tuned Lovelock couplings, we demonstrate th... more For any K(= 2k +1)th-order Lovelock gravity with fine-tuned Lovelock couplings, we demonstrate the existence of a special isolated critical point characterized by non-standard critical exponents in the phase diagram of hyperbolic vacuum black holes. In the Gibbs free energy this corresponds to a place wherefrom two swallowtails emerge, giving rise to two first-order phase transitions between small and large black holes. We believe that this is a first example of a critical point with nonstandard critical exponents obtained in a geometric theory of gravity.
Conditions for thermodynamic stability of asymptotically anti-de Sitter rotating black holes in D... more Conditions for thermodynamic stability of asymptotically anti-de Sitter rotating black holes in D-dimensions are determined. Local thermodynamic stability requires not only positivity conditions on the specific heat and the moment of inertia tensor but it is also necessary that the adiabatic compressibility be positive. It is shown that, in the absence of a cosmological constant, neither rotation nor charge is sufficient to ensure full local thermodynamic stability of a black hole. Thermodynamic stability properties of anti-de Sitter Myers-Perry black holes are investigated for both singly spinning and multi-spinning black holes. Simple expressions are obtained for the specific heat and moment of inertia tensor in any dimension. An analytic expression is obtained for the boundary of the region of parameter space in which such space-times are thermodynamically stable.
When the cosmological constant is considered to be a thermodynamical variable in black hole therm... more When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary SU(N) gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of AdS 5 × S 5 , for N = 4 SUSY Yang-Mills at large N, and the variable thermodynamically conjugate to N, a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.
It is shown how Pauli couplings can be eliminated from Kaluza-Klein theories by introducing torsi... more It is shown how Pauli couplings can be eliminated from Kaluza-Klein theories by introducing torsion. Freeing the connection from the metric also allows time-like internal dimensions to be considered. If the torsion arises purely from Dirac fermions, an expression is derived for the Yang-Mills field strength in terms of fermion bilinears.
We show that, for the transition between any two quantum Hall states, the semicircle law and the ... more We show that, for the transition between any two quantum Hall states, the semicircle law and the existence of a duality symmetry follow solely from the consistency of the law of corresponding states with the two-dimensional scaling flow. This puts these two effects on a sound theoretical footing, implying that both should hold exactly at zero temperature, independently of the details of the microscopic electron dynamics. This derivation also shows how the experimental evidence favours taking the two-dimensional flow seriously for the whole transition, and not just near the critical points.
We review some recent developments in black hole thermodynamics for asymptotically anti-de Sitter... more We review some recent developments in black hole thermodynamics for asymptotically anti-de Sitter black holes. Viewing the cosmological constant as a thermodynamic pressure allows a thermodynamic volume to be defined, the thermodynamically conjugate variable to the pressure. The properties of the equation of state can then be studied and a compressibility and a speed of sound is defined for such black holes.
It is shown that the quarks and leptons of the standard model, including a right-handed neutrino,... more It is shown that the quarks and leptons of the standard model, including a right-handed neutrino, can be obtained by gauging the holonomy groups of complex projective spaces of complex dimensions two and three. The spectrum emerges as chiral zero modes of the Dirac operator coupled to gauge fields and the demonstration involves an index theorem analysis on a general complex projective space in the presence of topologically non-trivial SU(n) × U(1) gauge fields. The construction may have applications in type IIA string theory and non-commutative geometry.
Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in f... more Within the framework of the AdS/CMT correspondence asymptotically anti-de Sitter black holes in four space-time dimensions can be used to analyse transport properties in two space dimensions. A non-linear renormalisation group equation for the conductivity in two dimensions is derived in this model and, as an example of its application, both the Ohmic and Hall DC and AC conductivities are studied in the presence of a magnetic field, using a bulk dyonic solution of the Einstein-Maxwell equations in asymptotically AdS4 space-time. The $$ \mathcal{Q} $$ Q -factor of the cyclotron resonance is shown to decrease as the temperature is increased and increase as the charge density is increased in a fixed magnetic field. Likewise the dissipative Ohmic conductivity at resonance increases as the temperature is decreased and as the charge density is increased. The analysis also involves a discussion of the piezoelectric effect in the context of the AdS/CMT framework.
Geometrically the phase space of a mechanical system involves the cotangent bundle of the configu... more Geometrically the phase space of a mechanical system involves the cotangent bundle of the configuration space. The phase space of a relativistic field theory is infinite dimensional and can be endowed with a symplectic structure defined in a perfectly co-variant manner that is very useful for discussing symmetries and conserved quantities of the system. In general relativity the symplectic structure takes Darboux form and it is shown in this work that the presence of a cosmological constant does not change this conclusion. For space-times that admit time-like Killing vectors the formalism can be used to define mass in general relativity and it is known that, for asymptotically flat black holes, this mass is identical the the usual Arnowitt-Desner-Misner mass while for asymptotically anti-de Sitter Kerr metrics it is the same as the Henneaux-Teitelboim mass. We show that the same formalism can also be used to derive the Brown-York mass and the Bondi mass for stationary space times, in particular the Brown-York mass has a natural interpretation in terms of differential forms on the space of solutions of the theory.
Journal of Physics A: Mathematical and Theoretical
The quantum Hall effect is studied in a spherical geometry using the Dirac operator for nonintera... more The quantum Hall effect is studied in a spherical geometry using the Dirac operator for noninteracting fermions in a background magnetic field, which is supplied by a Wu-Yang magnetic monopole at the centre of the sphere. Wave functions are cross-section of a non-trivial U (1) bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah-Singer index theorem constrains the degeneracy of the ground state. The fractional quantum Hall effect is also studied in the composite Fermion model. Vortices of the statistical gauge field are supplied by Dirac strings associated with the monopole field. A unique ground state is attained only if the vortices have an even number of flux units and act to counteract the background field, reducing the effective field seen by the composite fermions. There is a unique gapped ground state and, for large particle numbers, fractions ν = 1 2k+1 are recovered.
For a field theory that is invariant under diffeomorphisms there is a subtle interplay between sy... more For a field theory that is invariant under diffeomorphisms there is a subtle interplay between symmetries, conservation laws and the phase space of the theory. The natural language for describing these ideas is that of differential forms and both differential forms on space-time and differential forms on the infinite dimensional space of solutions of the equations of motion of the field theory play an important role. There are exterior derivatives on both spaces and together they weave a double differential complex which captures the cohomology of the theory. This is important in the definition of invariants in general relativity, such as mass and angular momentum and is also relevant to the study of quantum anomalies in gauge theories. We derive the structure of this double complex and show how it relates to conserved quantities in gravitational theories. One consequence of the construction is that conserved quantities can be calculated exactly at finite distance-for example it is not necessary to go to asymptotic regimes to calculate the mass or angular momentum of a stationary solution of Einstein's equations, they can be obtained exactly by an integration over any sphere outside the mass even at finite radius. 1 If Σ and Σ ′ have boundaries the fields are assumed to vanish sufficiently fast there that there is no contribution from any time-like component of ∂M.
An invariant definition of mass in asymptotically de-Sitter space-times is given that relies on t... more An invariant definition of mass in asymptotically de-Sitter space-times is given that relies on the existence of a time-like Killing vector on a sphere surrounding the mass but does not require going to an asymptotic region. In particular the mass can be calculated exactly on a sphere inside the cosmological horizon. The formalism requires varying the background metric solution by a perturbation that satisfies the linearized equations of motion but need not share the Killing symmetry of the solution and is therefore ideally suited to calculating masses in stationary spacetimes perturbed by a gravitational wave without going beyond the cosmological horizon.
The equation of state associated with N = 4 supersymmetric Yang-Mills in 4 dimensions, for SU(N) ... more The equation of state associated with N = 4 supersymmetric Yang-Mills in 4 dimensions, for SU(N) in the large N limit, is investigated using the AdS/CFT correspondence. An asymptotically AdS black-hole on the gravity side provides a thermal background for the Yang-Mills theory on the boundary in which the cosmological constant is equivalent to a volume. The thermodynamic variable conjugate to the cosmological constant is a pressure and the P − V diagram is studied. It is known that there is a critical point where the heat capacity diverges and this is reflected in the isothermal compressibility. Critical exponents are derived and found to be mean field in the large N limit. The same analysis applied to 3 and 6 dimensional conformal field theories again yields mean field exponents associated with the compressibility at the critical point. 1 DIAS-STP-16-05 1 Feature invited paper for special edition "Black hole thermodynamics II" in Entropy.
The renormalisation group equation for N-point correlation functions can be interpreted in a geom... more The renormalisation group equation for N-point correlation functions can be interpreted in a geometrical manner as an equation for Lie transport of amplitudes in the space of couplings. The vector field generating the diffeomorphism has components given by the βfunctions of the theory. It is argued that this simple picture requires modification whenever any one of the points at which the amplitude is evaluated becomes close to any other. This modification requires the introduction of a connection on the space of couplings and new terms appear in the renormalisation group equation involving co-variant derivatives of the β-function and the curvature associated with the connection. It is shown how the connection is related to the operator expansion coefficients , but there remains an arbitrariness in its definition.
We explore the consequences of introducing a complex conductivity into the quantum Hall effect. T... more We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects, including: universality; the selection rule $|p_1q_2 - p_2q_1|=1$ for quantum Hall transitions between filling factors $\nu_1=p_1/q_1$ and $\nu_2=p_2/q_2$; critical values for the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaus.
Skip to main content. Help us improve the CERN Document Server for you: take part! CERN Logo CERN... more Skip to main content. Help us improve the CERN Document Server for you: take part! CERN Logo CERN Document Server. Related links. CDS; Indico; Library; Bulletin; EDMS. Main navigation links: Search; Submit; Help; Your CDS: Your alerts; Your baskets; Your searches. login. Home > Articles & Preprints > Published Articles > Non-symmetric coset spaces with torsion. Information; Discussion; Files. Article. Title, Non-symmetric coset spaces with torsion : alternative compactifications of 10-d superstrings. ...
Thermodynamic potentials relevant to the microcanonical, the canonical and the grand canonical en... more Thermodynamic potentials relevant to the microcanonical, the canonical and the grand canonical ensembles, associated with rotating black holes in D-dimensions, are analysed and compared. Such black holes are known to be thermodynamically unstable, but the instability is a consequence of a subtle interplay between specific heats and the moments of inertia and it manifests itself differently in the different ensembles. A simple relation between the product of the specific heat and the determinant of the moment of inertia in both the canonical and the grand canonical ensembles is derived. Myers-Perry black holes in arbitrary dimension are studied in detail. All temperature extrema in the microcanonical ensemble are determined and classified. The specific heat and the moment of inertia tensor are evaluated in both the canonical and the grand canonical ensembles in any dimension. All zeros and poles of the specific heats, as a function of the angular momenta, are determined and the eigenvalues of the isentropic moment of inertia tensor are studied and classified. It is further shown that many of the thermodynamic properties of a Myers-Perry black hole in D − 2 dimensions can be obtained from those of a black hole in D dimensions by sending one of the angular momenta to infinity.
A new 501ut10n 0f the 6050n1c 5ect0r 0fD = 11 5uper9rav1ty 15 pre5ented 1n wh1ch the 1nterna1 5pa... more A new 501ut10n 0f the 6050n1c 5ect0r 0fD = 11 5uper9rav1ty 15 pre5ented 1n wh1ch the 1nterna1 5pace ha5 the t0p0109y CP 2 × 52 × 51 and the 1nterna1 150metry 9r0up 15 5U(3) × 5U(2) X U(1).
For any K(= 2k +1)th-order Lovelock gravity with fine-tuned Lovelock couplings, we demonstrate th... more For any K(= 2k +1)th-order Lovelock gravity with fine-tuned Lovelock couplings, we demonstrate the existence of a special isolated critical point characterized by non-standard critical exponents in the phase diagram of hyperbolic vacuum black holes. In the Gibbs free energy this corresponds to a place wherefrom two swallowtails emerge, giving rise to two first-order phase transitions between small and large black holes. We believe that this is a first example of a critical point with nonstandard critical exponents obtained in a geometric theory of gravity.
Conditions for thermodynamic stability of asymptotically anti-de Sitter rotating black holes in D... more Conditions for thermodynamic stability of asymptotically anti-de Sitter rotating black holes in D-dimensions are determined. Local thermodynamic stability requires not only positivity conditions on the specific heat and the moment of inertia tensor but it is also necessary that the adiabatic compressibility be positive. It is shown that, in the absence of a cosmological constant, neither rotation nor charge is sufficient to ensure full local thermodynamic stability of a black hole. Thermodynamic stability properties of anti-de Sitter Myers-Perry black holes are investigated for both singly spinning and multi-spinning black holes. Simple expressions are obtained for the specific heat and moment of inertia tensor in any dimension. An analytic expression is obtained for the boundary of the region of parameter space in which such space-times are thermodynamically stable.
When the cosmological constant is considered to be a thermodynamical variable in black hole therm... more When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary SU(N) gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of AdS 5 × S 5 , for N = 4 SUSY Yang-Mills at large N, and the variable thermodynamically conjugate to N, a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.
It is shown how Pauli couplings can be eliminated from Kaluza-Klein theories by introducing torsi... more It is shown how Pauli couplings can be eliminated from Kaluza-Klein theories by introducing torsion. Freeing the connection from the metric also allows time-like internal dimensions to be considered. If the torsion arises purely from Dirac fermions, an expression is derived for the Yang-Mills field strength in terms of fermion bilinears.
We show that, for the transition between any two quantum Hall states, the semicircle law and the ... more We show that, for the transition between any two quantum Hall states, the semicircle law and the existence of a duality symmetry follow solely from the consistency of the law of corresponding states with the two-dimensional scaling flow. This puts these two effects on a sound theoretical footing, implying that both should hold exactly at zero temperature, independently of the details of the microscopic electron dynamics. This derivation also shows how the experimental evidence favours taking the two-dimensional flow seriously for the whole transition, and not just near the critical points.
We review some recent developments in black hole thermodynamics for asymptotically anti-de Sitter... more We review some recent developments in black hole thermodynamics for asymptotically anti-de Sitter black holes. Viewing the cosmological constant as a thermodynamic pressure allows a thermodynamic volume to be defined, the thermodynamically conjugate variable to the pressure. The properties of the equation of state can then be studied and a compressibility and a speed of sound is defined for such black holes.
It is shown that the quarks and leptons of the standard model, including a right-handed neutrino,... more It is shown that the quarks and leptons of the standard model, including a right-handed neutrino, can be obtained by gauging the holonomy groups of complex projective spaces of complex dimensions two and three. The spectrum emerges as chiral zero modes of the Dirac operator coupled to gauge fields and the demonstration involves an index theorem analysis on a general complex projective space in the presence of topologically non-trivial SU(n) × U(1) gauge fields. The construction may have applications in type IIA string theory and non-commutative geometry.
Graduate level talk on the Einstein/Cartan formulation of General Relativity using differential f... more Graduate level talk on the Einstein/Cartan formulation of General Relativity using differential forms. An example is given of how to calculate connection one-forms and curvature two-forms explicitly.
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Papers by Brian Dolan