We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of th... more We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of the next-to-lowest nontrival primary operator in unitary, modularinvariant two-dimensional conformal field theories without chiral primary operators, with total central charge c tot > 2. The bound we find is of the same form as found by Hellerman for ∆ 1 : ∆ 2 ≤ ctot 12 + O(1). We obtain a similar bound on the conformal dimension ∆ 3 , and present a method for deriving bounds on ∆ n for any n, under slightly modified assumptions. For asymptotically large c tot and n exp(πc/12), we show that ∆ n ≤ ctot 12 + O(1). This implies an asymptotic lower bound of order exp(πc tot /12) on the number of primary operators of dimension ≤ c tot /12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G N .
in the spring term of 2015. Although the course was offered primarily for graduate students, thes... more in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students.
in the spring term of 2015. Although the course was offered primarily for graduate students, thes... more in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students.
We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d co... more We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d conformal field theories with discrete spectra using modular invariance, including CFTs with chiral currents. We derive a bound on the conformal dimensions ∆ 1 and ∆ 2 going as c tot /12+O(1). The bound is of the same form found for CFTs without chiral currents in [1] and [2]. We then prove the inequality ∆ n ≤ c tot /12+ O(1) for large c tot and with appropriate restrictions on n. Using the AdS 3 /CFT 2 correspondence, our bounds correspond to upper bounds on the masses of the lightest few states and a lower bound on the number of states. We conclude by checking our results against several candidate conformal field theories.
In this note, we extend the striking connections between quantum integrable systems and conformal... more In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in [1] in several directions. First, we explicitly demonstrate that the action of quartic conformal Casimir operator on general d-dimensional scalar conformal blocks, can be expressed in terms of certain combinations of commuting integrals of motions of two particle hyperbolic BC 2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in SCFTs with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system, this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with arbitrary space-time spins in four dimensional CFTs can also be linearly expanded in terms of Calogero-Sutherland eigenfunctions.
We prove using invariance under the modular S− and ST −transformations that every unitary two-dim... more We prove using invariance under the modular S− and ST −transformations that every unitary two-dimensional conformal field theory (CFT) having only evenspin primary operators (with no extended chiral algebra and with right-and leftcentral charges c,c > 1) contains a primary operator with dimension ∆ 1 satisfying 0 < ∆ 1 < c+c 24 + 0.09280.... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS 3 /CFT 2 dictionary, the bound on ∆ 1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/8G N + O(√ −Λ); the bounds on the number of operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole. 2 Constraints from ST-invariance 3 2.1 Modular invariance 3 2.2 Intermediate Temperature Expansion 4 2.3 Polynomial Constraint 6 3 Proof of a bound for even spin 8 3.1 Proof by contradiction 8 3.2 Analytic and numerical bounds on ∆ 1 9 4 Gravitational interpretation of bounds 11 A Properties of G 0 (Ê 0) 13
Right click to open a feedback form in a new tab to let us know how this document benefits you. R... more Right click to open a feedback form in a new tab to let us know how this document benefits you. Right click to open a feedback form in a new tab to let us know how this document benefits you. Recommended Citation Recommended Citation Qualls, Joshua D., "UNIVERSAL CONSTRAINTS ON 2D CFTS AND 3D GRAVITY" (2014). Theses and Dissertations-Physics and Astronomy. 21.
We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of th... more We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of the next-to-lowest nontrival primary operator in unitary, modular-invariant two-dimensional conformal field theories without chiral primary operators, with total central charge c tot > 2. The bound we find is of the same form as found by Hellerman for ∆ 1 : ∆ 2 ≤ ctot 12 + O(1). We obtain a similar bound on the conformal dimension ∆ 3 , and present a method for deriving bounds on ∆ n for any n, under slightly modified assumptions. For asymptotically large c tot and n exp(πc/12), we show that ∆ n ≤ ctot 12 + O(1). This implies an asymptotic lower bound of order exp(πc tot /12) on the number of primary operators of dimension ≤ c tot /12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G N. Contents 1 Introduction 1 2 Review of the bound on ∆ 1 2 3 Bounds on ∆ 2 , ∆ 3 5 4 Bounds on ∆ n 8 5 Gravitational Interpretation 10 A Behavior of F 2 , F 3 11 B The O(1) term in ∆ + 2 12 C Condition on n, c tot 13 D Derivation for ∆ n 14
Interdisciplinary courses like PHYS481/MATH481 “Mathematics for Scientists and Engineers” play an... more Interdisciplinary courses like PHYS481/MATH481 “Mathematics for Scientists and Engineers” play an important role in undergraduate curriculum: physics students get exposure to advanced mathematical ideas that they might otherwise miss in their undergraduate studies, and both mathematics and physics students get experience directly applying mathematical ideas and techniques to real-world problems. Several in-state and peer institutions offer an additional course in mathematical physics. Although the advantages of a second course are plentiful, Morehead State University currently does not. In this talk we report on the progress of creating the cross-listed course “Advanced Mathematics for Physics” and mathematical techniques currently missing from coursework that would be most beneficial to graduating physics majors. As an example, we focus specifically on Schwarz-Christoffel (SC) transformations from complex analysis. SC transformations are functions that map points in the extended up...
There are several difficulties with introducing students to quantum mechanics and quantum computi... more There are several difficulties with introducing students to quantum mechanics and quantum computing in most settings. For many students, their first exposure is only after years of basic and advanced mathematics and multiple introductory physics courses in physics. A pilot study was designed to explore the possibility of introducing fundamental concepts of quantum computing and quantum mechanics in a concise standalone module using educational games. The research subjects (N=50) completed identical pretest and posttest separated by introductory quantum computing curriculum. This curriculum is introduced and reinforced using Tic Tac Toe, a Semi-classical Tic Tac Toe and Quantum Tic Tac Toe. We report on the results of this pilot study and the planned modifications before launching our upcoming research study to investigate the educational effectiveness of this tripartite lesson plan
Quantum computing (QC) is a radically new approach to computation. Classical computing is based o... more Quantum computing (QC) is a radically new approach to computation. Classical computing is based on the idea of bits: strings of 1\u27s and 0\u27s on which we do arithmetic operations. QC, however, is based on quantum bits, qubits, and new physical operations. Despite the complexity of QC, we contend that it is an appropriate and useful topic for undergraduate physics students. In this talk, we present our results from three small QC projects relating to flipping coins. In the first, we investigate the results of a secret coin flip and use QC to guess whether the coin landed on heads or tails. In the second, we determine whether a given coin is fair or unfair using only a single qubit ( instead of multiple bits). In the third, we explore how quantum communication can alleviate dishonest player biases in coin flipping cryptographic primitives
Quantum tic-tac-toe (QTTT) is an educational tool illustrating novel features of quantum systems.... more Quantum tic-tac-toe (QTTT) is an educational tool illustrating novel features of quantum systems. Most versions include states, superposition, collapse, and entanglement. While QTTT can provide a foundation for understanding quantum principles, QTTT can also be overwhelming for students new to quantum mechanics. In this talk, we introduce a simplified version of QTTT that does not include entanglement: semiclassical tic-tac-toe (STTT). As a first step toward a systematic development of interactive quantum computing demonstrations and curriculum, we have designed materials for an STTT lesson plan and pilot study. Materials include a pre- and post-test over basic quantum mechanical ideas, an instructor's guide to STTT and QTTT, and a classroom-ready Python-based program featuring classical tic-tac-toe, STTT, and QTTT. We are implementing a pilot study to explore their effectivness in communicating scientific content and the extent to which the inclusion of STTT might enhance stude...
We prove using invariance under the modular S− and ST −transformations that every unitary two-dim... more We prove using invariance under the modular S− and ST −transformations that every unitary two-dimensional conformal field theory (CFT) having only even-spin primary operators (with no extended chiral algebra and with right-and left-central charges c,c > 1) contains a primary operator with dimension ∆ 1 satisfying 0 < ∆ 1 < c+c 24 + 0.09280. . .. After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS 3 /CFT 2 dictionary, the bound on ∆ 1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/8G N + O(√ −Λ); the bounds on the number of operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
In this note, we extend the striking connections between quantum integrable systems and conformal... more In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in http://arxiv.org/abs/1602.01858 in several directions. First, we explicitly demonstrate that the action of quartic conformal Casimir operator on general d-dimensional scalar conformal blocks, can be expressed in terms of certain combinations of commuting integrals of motions of the two particle hyperbolic BC2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in SCFTs with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system, this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with ar...
We prove using invariance under the modular S- and ST-transformations that every unitary two-dime... more We prove using invariance under the modular S- and ST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c̃>1) contains a primary operator with dimension Δ_1 satisfying 0 < Δ_1 < (c+c̃)/24 + 0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS_3/CFT_2 dictionary, the bound on Δ_1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/(8G_N)+O(√(-Λ)); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d co... more We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d conformal field theories with discrete spectra using modular invariance, including CFTs with chiral currents. We derive a bound on the conformal dimensions Δ_1 and Δ_2 going as c_ tot/12 + O(1). The bound is of the same form found for CFTs without chiral currents in arXiv:0902.2790 and arXiv:1312.0038. We then prove the inequality Δ_n ≤ c_ tot/12 + O(1) for large c_ tot and with appropriate restrictions on n. Using the AdS_3/CFT_2 correspondence, our bounds correspond to upper bounds on the masses of the lightest few states and a lower bound on the number of states. We conclude by checking our results against several candidate conformal field theories.
These lectures notes are based on courses given at National Taiwan University, National Chiao-Tun... more These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions, with applications related to topics of particular interest: topics include the conformal bootstrap program, boundary conformal field theory, and applications related to the AdS/CFT correspondence. We assume the reader to be familiar with quantum mechanics at the graduate level and to have some basic knowledge of quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help.
We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of th... more We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of the next-to-lowest nontrival primary operator in unitary, modular-invariant two-dimensional conformal field theories without chiral primary operators, with total central charge c tot > 2. The bound we find is of the same form as found by Hellerman for ∆ 1 : ∆ 2 ≤ ctot 12 + O(1). We obtain a similar bound on the conformal dimension ∆ 3 , and present a method for deriving bounds on ∆ n for any n, under slightly modified assumptions. For asymptotically large c tot and n exp(πc/12), we show that ∆ n ≤ ctot 12 + O(1). This implies an asymptotic lower bound of order exp(πc tot /12) on the number of primary operators of dimension ≤ c tot /12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G N. Contents 1 Introduction 1 2 Review of the bound on ∆ 1 2 3 Bounds on ∆ 2 , ∆ 3 5 4 Bounds on ∆ n 8 5 Gravitational Interpretation 10 A Behavior of F 2 , F 3 11 B The O(1) term in ∆ + 2 12 C Condition on n, c tot 13 D Derivation for ∆ n 14
in the spring term of 2015. Although the course was offered primarily for graduate students, thes... more in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students.
We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of th... more We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of the next-to-lowest nontrival primary operator in unitary, modularinvariant two-dimensional conformal field theories without chiral primary operators, with total central charge c tot > 2. The bound we find is of the same form as found by Hellerman for ∆ 1 : ∆ 2 ≤ ctot 12 + O(1). We obtain a similar bound on the conformal dimension ∆ 3 , and present a method for deriving bounds on ∆ n for any n, under slightly modified assumptions. For asymptotically large c tot and n exp(πc/12), we show that ∆ n ≤ ctot 12 + O(1). This implies an asymptotic lower bound of order exp(πc tot /12) on the number of primary operators of dimension ≤ c tot /12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G N .
in the spring term of 2015. Although the course was offered primarily for graduate students, thes... more in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students.
in the spring term of 2015. Although the course was offered primarily for graduate students, thes... more in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students.
We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d co... more We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d conformal field theories with discrete spectra using modular invariance, including CFTs with chiral currents. We derive a bound on the conformal dimensions ∆ 1 and ∆ 2 going as c tot /12+O(1). The bound is of the same form found for CFTs without chiral currents in [1] and [2]. We then prove the inequality ∆ n ≤ c tot /12+ O(1) for large c tot and with appropriate restrictions on n. Using the AdS 3 /CFT 2 correspondence, our bounds correspond to upper bounds on the masses of the lightest few states and a lower bound on the number of states. We conclude by checking our results against several candidate conformal field theories.
In this note, we extend the striking connections between quantum integrable systems and conformal... more In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in [1] in several directions. First, we explicitly demonstrate that the action of quartic conformal Casimir operator on general d-dimensional scalar conformal blocks, can be expressed in terms of certain combinations of commuting integrals of motions of two particle hyperbolic BC 2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in SCFTs with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system, this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with arbitrary space-time spins in four dimensional CFTs can also be linearly expanded in terms of Calogero-Sutherland eigenfunctions.
We prove using invariance under the modular S− and ST −transformations that every unitary two-dim... more We prove using invariance under the modular S− and ST −transformations that every unitary two-dimensional conformal field theory (CFT) having only evenspin primary operators (with no extended chiral algebra and with right-and leftcentral charges c,c > 1) contains a primary operator with dimension ∆ 1 satisfying 0 < ∆ 1 < c+c 24 + 0.09280.... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS 3 /CFT 2 dictionary, the bound on ∆ 1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/8G N + O(√ −Λ); the bounds on the number of operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole. 2 Constraints from ST-invariance 3 2.1 Modular invariance 3 2.2 Intermediate Temperature Expansion 4 2.3 Polynomial Constraint 6 3 Proof of a bound for even spin 8 3.1 Proof by contradiction 8 3.2 Analytic and numerical bounds on ∆ 1 9 4 Gravitational interpretation of bounds 11 A Properties of G 0 (Ê 0) 13
Right click to open a feedback form in a new tab to let us know how this document benefits you. R... more Right click to open a feedback form in a new tab to let us know how this document benefits you. Right click to open a feedback form in a new tab to let us know how this document benefits you. Recommended Citation Recommended Citation Qualls, Joshua D., "UNIVERSAL CONSTRAINTS ON 2D CFTS AND 3D GRAVITY" (2014). Theses and Dissertations-Physics and Astronomy. 21.
We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of th... more We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of the next-to-lowest nontrival primary operator in unitary, modular-invariant two-dimensional conformal field theories without chiral primary operators, with total central charge c tot > 2. The bound we find is of the same form as found by Hellerman for ∆ 1 : ∆ 2 ≤ ctot 12 + O(1). We obtain a similar bound on the conformal dimension ∆ 3 , and present a method for deriving bounds on ∆ n for any n, under slightly modified assumptions. For asymptotically large c tot and n exp(πc/12), we show that ∆ n ≤ ctot 12 + O(1). This implies an asymptotic lower bound of order exp(πc tot /12) on the number of primary operators of dimension ≤ c tot /12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G N. Contents 1 Introduction 1 2 Review of the bound on ∆ 1 2 3 Bounds on ∆ 2 , ∆ 3 5 4 Bounds on ∆ n 8 5 Gravitational Interpretation 10 A Behavior of F 2 , F 3 11 B The O(1) term in ∆ + 2 12 C Condition on n, c tot 13 D Derivation for ∆ n 14
Interdisciplinary courses like PHYS481/MATH481 “Mathematics for Scientists and Engineers” play an... more Interdisciplinary courses like PHYS481/MATH481 “Mathematics for Scientists and Engineers” play an important role in undergraduate curriculum: physics students get exposure to advanced mathematical ideas that they might otherwise miss in their undergraduate studies, and both mathematics and physics students get experience directly applying mathematical ideas and techniques to real-world problems. Several in-state and peer institutions offer an additional course in mathematical physics. Although the advantages of a second course are plentiful, Morehead State University currently does not. In this talk we report on the progress of creating the cross-listed course “Advanced Mathematics for Physics” and mathematical techniques currently missing from coursework that would be most beneficial to graduating physics majors. As an example, we focus specifically on Schwarz-Christoffel (SC) transformations from complex analysis. SC transformations are functions that map points in the extended up...
There are several difficulties with introducing students to quantum mechanics and quantum computi... more There are several difficulties with introducing students to quantum mechanics and quantum computing in most settings. For many students, their first exposure is only after years of basic and advanced mathematics and multiple introductory physics courses in physics. A pilot study was designed to explore the possibility of introducing fundamental concepts of quantum computing and quantum mechanics in a concise standalone module using educational games. The research subjects (N=50) completed identical pretest and posttest separated by introductory quantum computing curriculum. This curriculum is introduced and reinforced using Tic Tac Toe, a Semi-classical Tic Tac Toe and Quantum Tic Tac Toe. We report on the results of this pilot study and the planned modifications before launching our upcoming research study to investigate the educational effectiveness of this tripartite lesson plan
Quantum computing (QC) is a radically new approach to computation. Classical computing is based o... more Quantum computing (QC) is a radically new approach to computation. Classical computing is based on the idea of bits: strings of 1\u27s and 0\u27s on which we do arithmetic operations. QC, however, is based on quantum bits, qubits, and new physical operations. Despite the complexity of QC, we contend that it is an appropriate and useful topic for undergraduate physics students. In this talk, we present our results from three small QC projects relating to flipping coins. In the first, we investigate the results of a secret coin flip and use QC to guess whether the coin landed on heads or tails. In the second, we determine whether a given coin is fair or unfair using only a single qubit ( instead of multiple bits). In the third, we explore how quantum communication can alleviate dishonest player biases in coin flipping cryptographic primitives
Quantum tic-tac-toe (QTTT) is an educational tool illustrating novel features of quantum systems.... more Quantum tic-tac-toe (QTTT) is an educational tool illustrating novel features of quantum systems. Most versions include states, superposition, collapse, and entanglement. While QTTT can provide a foundation for understanding quantum principles, QTTT can also be overwhelming for students new to quantum mechanics. In this talk, we introduce a simplified version of QTTT that does not include entanglement: semiclassical tic-tac-toe (STTT). As a first step toward a systematic development of interactive quantum computing demonstrations and curriculum, we have designed materials for an STTT lesson plan and pilot study. Materials include a pre- and post-test over basic quantum mechanical ideas, an instructor's guide to STTT and QTTT, and a classroom-ready Python-based program featuring classical tic-tac-toe, STTT, and QTTT. We are implementing a pilot study to explore their effectivness in communicating scientific content and the extent to which the inclusion of STTT might enhance stude...
We prove using invariance under the modular S− and ST −transformations that every unitary two-dim... more We prove using invariance under the modular S− and ST −transformations that every unitary two-dimensional conformal field theory (CFT) having only even-spin primary operators (with no extended chiral algebra and with right-and left-central charges c,c > 1) contains a primary operator with dimension ∆ 1 satisfying 0 < ∆ 1 < c+c 24 + 0.09280. . .. After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS 3 /CFT 2 dictionary, the bound on ∆ 1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/8G N + O(√ −Λ); the bounds on the number of operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
In this note, we extend the striking connections between quantum integrable systems and conformal... more In this note, we extend the striking connections between quantum integrable systems and conformal blocks recently found in http://arxiv.org/abs/1602.01858 in several directions. First, we explicitly demonstrate that the action of quartic conformal Casimir operator on general d-dimensional scalar conformal blocks, can be expressed in terms of certain combinations of commuting integrals of motions of the two particle hyperbolic BC2 Calogero-Sutherland system. The permutation and reflection properties of the underlying Dunkl operators play crucial roles in establishing such a connection. Next, we show that the scalar superconformal blocks in SCFTs with four and eight supercharges and suitable chirality constraints can also be identified with the eigenfunctions of the same Calogero-Sutherland system, this demonstrates the universality of such a connection. Finally, we observe that the so-called "seed" conformal blocks for constructing four point functions for operators with ar...
We prove using invariance under the modular S- and ST-transformations that every unitary two-dime... more We prove using invariance under the modular S- and ST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c̃>1) contains a primary operator with dimension Δ_1 satisfying 0 < Δ_1 < (c+c̃)/24 + 0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS_3/CFT_2 dictionary, the bound on Δ_1 proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ < 0 can be no heavier than 1/(8G_N)+O(√(-Λ)); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d co... more We derive a bound on the conformal dimensions of the lightest few states in general unitary 2d conformal field theories with discrete spectra using modular invariance, including CFTs with chiral currents. We derive a bound on the conformal dimensions Δ_1 and Δ_2 going as c_ tot/12 + O(1). The bound is of the same form found for CFTs without chiral currents in arXiv:0902.2790 and arXiv:1312.0038. We then prove the inequality Δ_n ≤ c_ tot/12 + O(1) for large c_ tot and with appropriate restrictions on n. Using the AdS_3/CFT_2 correspondence, our bounds correspond to upper bounds on the masses of the lightest few states and a lower bound on the number of states. We conclude by checking our results against several candidate conformal field theories.
These lectures notes are based on courses given at National Taiwan University, National Chiao-Tun... more These lectures notes are based on courses given at National Taiwan University, National Chiao-Tung University, and National Tsing Hua University in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions, with applications related to topics of particular interest: topics include the conformal bootstrap program, boundary conformal field theory, and applications related to the AdS/CFT correspondence. We assume the reader to be familiar with quantum mechanics at the graduate level and to have some basic knowledge of quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help.
We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of th... more We extend the work of Hellerman [1] to derive an upper bound on the conformal dimension ∆ 2 of the next-to-lowest nontrival primary operator in unitary, modular-invariant two-dimensional conformal field theories without chiral primary operators, with total central charge c tot > 2. The bound we find is of the same form as found by Hellerman for ∆ 1 : ∆ 2 ≤ ctot 12 + O(1). We obtain a similar bound on the conformal dimension ∆ 3 , and present a method for deriving bounds on ∆ n for any n, under slightly modified assumptions. For asymptotically large c tot and n exp(πc/12), we show that ∆ n ≤ ctot 12 + O(1). This implies an asymptotic lower bound of order exp(πc tot /12) on the number of primary operators of dimension ≤ c tot /12 + O(1), in the large-c limit. In dual gravitational theories, this corresponds to a lower bound in the flat-space limit on the number of gravitational states without boundary excitations, of mass less than or equal to 1/4G N. Contents 1 Introduction 1 2 Review of the bound on ∆ 1 2 3 Bounds on ∆ 2 , ∆ 3 5 4 Bounds on ∆ n 8 5 Gravitational Interpretation 10 A Behavior of F 2 , F 3 11 B The O(1) term in ∆ + 2 12 C Condition on n, c tot 13 D Derivation for ∆ n 14
in the spring term of 2015. Although the course was offered primarily for graduate students, thes... more in the spring term of 2015. Although the course was offered primarily for graduate students, these lecture notes have been prepared for a more general audience. They are intended as an introduction to conformal field theories in various dimensions working toward current research topics in conformal field theory. We assume the reader to be familiar with quantum field theory. Familiarity with string theory is not a prerequisite for this lectures, although it can only help. These notes include over 80 homework problems and over 45 longer exercises for students.
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