Knots, Three-Manifolds and Instantons
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A Space-Time Orbifold: a Toy Model for a Cosmological Singularity
hep-th/0202187 UPR-981-T, HIP-2002-07/TH A Space-Time Orbifold: A Toy Model for a Cosmological Singularity Vijay Balasubramanian,1 ∗ S. F. Hassan,2† Esko Keski-Vakkuri,2‡ and Asad Naqvi1§ 1David Rittenhouse Laboratories, University of Pennsylvania Philadelphia, PA 19104, U.S.A. 2Helsinki Institute of Physics, P.O.Box 64, FIN-00014 University of Helsinki, Finland Abstract We explore bosonic strings and Type II superstrings in the simplest time dependent backgrounds, namely orbifolds of Minkowski space by time reversal and some spatial reflections. We show that there are no negative norm physi- cal excitations. However, the contributions of negative norm virtual states to quantum loops do not cancel, showing that a ghost-free gauge cannot be chosen. The spectrum includes a twisted sector, with strings confined to a “conical” sin- gularity which is localized in time. Since these localized strings are not visible to asymptotic observers, interesting issues arise regarding unitarity of the S- matrix for scattering of propagating states. The partition function of our model is modular invariant, and for the superstring, the zero momentum dilaton tad- pole vanishes. Many of the issues we study will be generic to time-dependent arXiv:hep-th/0202187v2 19 Apr 2002 cosmological backgrounds with singularities localized in time, and we derive some general lessons about quantizing strings on such spaces. 1 Introduction Time-dependent space-times are difficult to study, both classically and quantum me- chanically. For example, non-static solutions are harder to find in General Relativity, while the notion of a particle is difficult to define clearly in field theory on time- dependent backgrounds. -
D-Instanton Perturbation Theory
D-instanton Perturbation Theory Ashoke Sen Harish-Chandra Research Institute, Allahabad, India Sao Paolo, June 2020 1 Plan: 1. Overview of the problem and the solution 2. Review of basic aspects of world-sheet string theory and string field theory 3. Some explicit computations A.S., arXiv:1908.02782, 2002.04043, work in progress 2 The problem 3 String theory began with Veneziano amplitude – tree level scattering amplitude of four tachyons in open string theory World-sheet expression for the amplitude (in α0 = 1 unit) Z 1 dy y2p1:p2 (1 − y)2p2:p3 0 – diverges for 2p1:p2 ≤ −1 or 2p2:p3 ≤ −1. Our convention: a:b ≡ −a0b0 + ~a:~b Conventional viewpoint: Define the amplitude for 2p1:p2 > −1, 2p2:p3 > −1 and then analytically continue to the other kinematic regions. 4 However, analytic continuation does not always work It may not be possible to move away from the singularity by changing the external momenta Examples: Mass renormalization, Vacuum shift – discussed earlier In these lectures we shall discuss another situation where analytic continuation fails – D-instanton contribution to string amplitudes 5 D-instanton: A D-brane with Dirichlet boundary condition on all non-compact directions including (euclidean) time. D-instantons give non-perturbative contribution to string amplitudes that are important in many situations Example: KKLT moduli stabilization uses non-perturbative contribution from D-instanton (euclidean D3-brane) Systematic computation of string amplitudes in such backgrounds will require us to compute amplitudes in the presence of D-instantons Problem: Open strings living on the D-instanton do not carry any continuous momenta ) we cannot move away from the singularities by varying the external momenta 6 Some examples: Let X be the (euclidean) time direction Since the D-instanton is localized at some given euclidean time, it has a zero mode that translates it along time direction 4-point function of these zero modes: Z 1 A = dy y−2 + (y − 1)−2 + 1 0 Derivation of this expression will be discussed later. -
Deformations of G2-Structures, String Dualities and Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected]
University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons, and the Quantum Physics Commons Recommended Citation Barbosa, Rodrigo De Menezes, "Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles" (2019). Publicly Accessible Penn Dissertations. 3279. https://repository.upenn.edu/edissertations/3279 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3279 For more information, please contact [email protected]. Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Abstract We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3-manifold Q. The afl tness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we obtain an explicit description of the moduli space for the dual type IIA string compactification. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of "flat Higgs bundles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes. -
Jhep05(2019)105
Published for SISSA by Springer Received: March 21, 2019 Accepted: May 7, 2019 Published: May 20, 2019 Modular symmetries and the swampland conjectures JHEP05(2019)105 E. Gonzalo,a;b L.E. Ib´a~neza;b and A.M. Urangaa aInstituto de F´ısica Te´orica IFT-UAM/CSIC, C/ Nicol´as Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain bDepartamento de F´ısica Te´orica, Facultad de Ciencias, Universidad Aut´onomade Madrid, 28049 Madrid, Spain E-mail: [email protected], [email protected], [email protected] Abstract: Recent string theory tests of swampland ideas like the distance or the dS conjectures have been performed at weak coupling. Testing these ideas beyond the weak coupling regime remains challenging. We propose to exploit the modular symmetries of the moduli effective action to check swampland constraints beyond perturbation theory. As an example we study the case of heterotic 4d N = 1 compactifications, whose non-perturbative effective action is known to be invariant under modular symmetries acting on the K¨ahler and complex structure moduli, in particular SL(2; Z) T-dualities (or subgroups thereof) for 4d heterotic or orbifold compactifications. Remarkably, in models with non-perturbative superpotentials, the corresponding duality invariant potentials diverge at points at infinite distance in moduli space. The divergence relates to towers of states becoming light, in agreement with the distance conjecture. We discuss specific examples of this behavior based on gaugino condensation in heterotic orbifolds. We show that these examples are dual to compactifications of type I' or Horava-Witten theory, in which the SL(2; Z) acts on the complex structure of an underlying 2-torus, and the tower of light states correspond to D0-branes or M-theory KK modes. -
A Knot-Vice's Guide to Untangling Knot Theory, Undergraduate
A Knot-vice’s Guide to Untangling Knot Theory Rebecca Hardenbrook Department of Mathematics University of Utah Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 1 / 26 What is Not a Knot? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 2 / 26 What is a Knot? 2 A knot is an embedding of the circle in the Euclidean plane (R ). 3 Also defined as a closed, non-self-intersecting curve in R . 2 Represented by knot projections in R . Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 3 / 26 Why Knots? Late nineteenth century chemists and physicists believed that a substance known as aether existed throughout all of space. Could knots represent the elements? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 4 / 26 Why Knots? Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 5 / 26 Why Knots? Unfortunately, no. Nevertheless, mathematicians continued to study knots! Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 6 / 26 Current Applications Natural knotting in DNA molecules (1980s). Credit: K. Kimura et al. (1999) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 7 / 26 Current Applications Chemical synthesis of knotted molecules – Dietrich-Buchecker and Sauvage (1988). Credit: J. Guo et al. (2010) Rebecca Hardenbrook A Knot-vice’s Guide to Untangling Knot Theory 8 / 26 Current Applications Use of lattice models, e.g. the Ising model (1925), and planar projection of knots to find a knot invariant via statistical mechanics. Credit: D. Chicherin, V.P. -
Holography Principle and Topology Change in String Theory 1 Abstract
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server SNUTP 98-089 hep-th/9807241 Holography Principle and Topology Change in String Theory 1 Soo-Jong Rey Physics Department, Seoul National University, Seoul 151-742 KOREA abstract D-instantons of Type IIB string theory are Ramond-Ramond counterpart of Giddings-Strominger wormholes connecting two asymptotic regions of spacetime. Such wormholes, according to Coleman, might lead to spacetime topology change, third-quantized baby universes and prob- abilistic determination of fundamental coupling parameters. Utilizing correspondence between AdS5 × M5 Type IIB supergravity and d = 4 super Yang-Mills theory, we point out that topol- ogy change and sum over topologies not only take place in string theory but also are required for consistency with holography. Nevertheless, we argue that the effects of D-instanton wormholes remain completely deterministic, in sharp contrast to Coleman’s scenario. 1Work supported in part by KOSEF SRC-Program, Ministry of Education Grant BSRI 98-2418, SNU Re- search Fund and Korea Foundation for Advanced Studies Faculty Fellowship. One of the most vexing problems in quantum gravity has been the issue of spacetime topol- ogy change: whether topology of spacetime can fluctuate and, if so, spacetime of different topologies should be summed over. Any positive answer to this question would bear profound implications to the fundamental interactions of Nature. For example, in Coleman’s scenario [1] of baby universes [2], fluctuation of spacetime topology induce effective lon-local interactions and third quantization of the universe thereof 2. As a result, the cosmological constant is no longer a calculable, deterministic parameter but is turned into a quantity of probabilistic dis- tribution sharply peaked around zero. -
Arxiv:Hep-Th/9404101V3 21 Nov 1995
hep-th/9404101 April 1994 Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions ⋆ Petr Horavaˇ ∗ Enrico Fermi Institute University of Chicago 5640 South Ellis Avenue Chicago IL 60637, USA ABSTRACT Chern-Simons gauge theory is formulated on three dimensional Z2 orbifolds. The locus of singular points on a given orbifold is equivalent to a link of Wilson lines. This allows one to reduce any correlation function on orbifolds to a sum of more complicated correlation functions in the simpler theory on manifolds. Chern-Simons theory on manifolds is known arXiv:hep-th/9404101v3 21 Nov 1995 to be related to 2D CFT on closed string surfaces; here I show that the theory on orbifolds is related to 2D CFT of unoriented closed and open string models, i.e. to worldsheet orb- ifold models. In particular, the boundary components of the worldsheet correspond to the components of the singular locus in the 3D orbifold. This correspondence leads to a simple identification of the open string spectra, including their Chan-Paton degeneration, in terms of fusing Wilson lines in the corresponding Chern-Simons theory. The correspondence is studied in detail, and some exactly solvable examples are presented. Some of these examples indicate that it is natural to think of the orbifold group Z2 as a part of the gauge group of the Chern-Simons theory, thus generalizing the standard definition of gauge theories. E-mail address: [email protected] ⋆∗ Address after September 1, 1994: Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544. 1. Introduction Since the first appearance of the notion of “orbifolds” in Thurston’s 1977 lectures on three dimensional topology [1], orbifolds have become very appealing objects for physicists. -
An Introduction to Orbifolds
An introduction to orbifolds Joan Porti UAB Subdivide and Tile: Triangulating spaces for understanding the world Lorentz Center November 2009 An introduction to orbifolds – p.1/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. An introduction to orbifolds – p.2/20 Motivation • Γ < Isom(Rn) or Hn discrete and acts properly discontinuously (e.g. a group of symmetries of a tessellation). • If Γ has no fixed points ⇒ Γ\Rn is a manifold. • If Γ has fixed points ⇒ Γ\Rn is an orbifold. ··· (there are other notions of orbifold in algebraic geometry, string theory or using grupoids) An introduction to orbifolds – p.2/20 Examples: tessellations of Euclidean plane Γ= h(x, y) → (x + 1, y), (x, y) → (x, y + 1)i =∼ Z2 Γ\R2 =∼ T 2 = S1 × S1 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 An introduction to orbifolds – p.3/20 Examples: tessellations of Euclidean plane Rotations of angle π around red points (order 2) 2 2 2 2 2 2 2 2 2 2 2 2 An introduction to orbifolds – p.3/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) An introduction to orbifolds – p.4/20 Example: tessellations of hyperbolic plane Rotations of angle π, π/2 and π/3 around vertices (order 2, 4, and 6) 2 4 2 6 An introduction to orbifolds – p.4/20 Definition Informal Definition • An orbifold O is a metrizable topological space equipped with an atlas modelled on Rn/Γ, Γ < O(n) finite, with some compatibility condition. -
Gravity, Scale Invariance and the Hierarchy Problem
Motivation Setup Warm-up: Dilaton+Gravity One example: Higgs+Gravity Further Examples: Higgs+Dilaton+Gravity Discussion and Outlook Backups Gravity, Scale Invariance and the Hierarchy Problem M. Shaposhnikov, A. Shkerin EPFL & INR RAS Based on arXiv:1803.08907, arXiv:1804.06376 QUARKS-2018 Valday, May 31, 2018 1 / 39 Motivation Setup Warm-up: Dilaton+Gravity One example: Higgs+Gravity Further Examples: Higgs+Dilaton+Gravity Discussion and Outlook Backups Overview 1 Motivation 2 Setup 3 Warm-up: Dilaton+Gravity 4 One example: Higgs+Gravity 5 Further Examples: Higgs+Dilaton+Gravity 6 Discussion and Outlook 2 / 39 Motivation Setup Warm-up: Dilaton+Gravity One example: Higgs+Gravity Further Examples: Higgs+Dilaton+Gravity Discussion and Outlook Backups Drowning by numbers The fact is that G 2 F ~ ∼ 1033, where G | Fermi constant, G | Newton constant 2 F N GN c Two aspects of the hierarchy problem (G. F. Giudice'08): \classical" \quantum": Let MX be some heavy mass scale. Then one expects 2 2 δmH;X ∼ MX : Even if one assumes that there are no heavy thresholds beyond the EW scale, then, naively, 2 2 δmH;grav: ∼ MP : 3 / 39 Motivation Setup Warm-up: Dilaton+Gravity One example: Higgs+Gravity Further Examples: Higgs+Dilaton+Gravity Discussion and Outlook Backups EFT approach and beyond A common approach to the hierarchy problem lies within the effective field theory framework: Low energy description of Nature, provided by the SM, can be affected by an unknown UV physics only though a finite set of parameters This \naturalness principle" is questioned now in light of the absence of signatures of new physics at the TeV scale. -
Department of Mathematics
Department of Mathematics The Department of Mathematics seeks to maintain its top ranking in mathematics research and education in the United States. The department is a key part of MIT’s educational mission at both the undergraduate and graduate levels and produces the most sought-after young researchers. Key to the department’s success is recruitment of the best faculty members, postdoctoral associates, and graduate students in an ever more competitive environment. The department strives to be diverse at all levels in terms of race, gender, and ethnicity. It continues to serve the varied needs of its graduate students, undergraduate students majoring in mathematics, and the broader MIT community. Awards and Honors The faculty received numerous distinctions this year. Professor Victor Kac received the Leroy P. Steele Prize for Lifetime Achievement, for “groundbreaking contributions to Lie Theory and its applications to Mathematics and Mathematical Physics.” Larry Guth (together with Netz Katz of the California Institute of Technology) won the Clay Mathematics Institute Research Award. Tomasz Mrowka was elected as a member of the National Academy of Sciences and William Minicozzi was elected as a fellow of the American Academy of Arts and Sciences. Alexei Borodin received the 2015 Loève Prize in Probability, given by the University of California, Berkeley, in recognition of outstanding research in mathematical probability by a young researcher. Alan Edelman received the 2015 Babbage Award, given at the IEEE International Parallel and Distributed Processing Symposium, for exceptional contributions to the field of parallel processing. Bonnie Berger was elected vice president of the International Society for Computational Biology. -
Matrix Integrals and Knot Theory 1
Matrix Integrals and Knot Theory 1 Matrix Integrals and Knot Theory Paul Zinn-Justin et Jean-Bernard Zuber math-ph/9904019 (Discr. Math. 2001) math-ph/0002020 (J.Kn.Th.Ramif. 2000) P. Z.-J. (math-ph/9910010, math-ph/0106005) J. L. Jacobsen & P. Z.-J. (math-ph/0102015, math-ph/0104009). math-ph/0303049 (J.Kn.Th.Ramif. 2004) G. Schaeffer and P. Z.-J., math-ph/0304034 P. Z.-J. and J.-B. Z., review article in preparation Carghjese, March 25, 2009 Matrix Integrals and Knot Theory 2 Courtesy of “U Rasaghiu” Matrix Integrals and Knot Theory 3 Main idea : Use combinatorial tools of Quantum Field Theory in Knot Theory Plan I Knot Theory : a few definitions II Matrix integrals and Link diagrams 1 2 4 dMeNtr( 2 M + gM ) N N matrices, N ∞ − × → RemovalsR of redundancies reproduces recent results of Sundberg & Thistlethwaite (1998) ⇒ based on Tutte (1963) III Virtual knots and links : counting and invariants Matrix Integrals and Knot Theory 4 Basics of Knot Theory a knot , links and tangles Equivalence up to isotopy Problem : Count topologically inequivalent knots, links and tangles Represent knots etc by their planar projection with minimal number of over/under-crossings Theorem Two projections represent the same knot/link iff they may be transformed into one another by a sequence of Reidemeister moves : ; ; Matrix Integrals and Knot Theory 5 Avoid redundancies by keeping only prime links (i.e. which cannot be factored) Consider the subclass of alternating knots, links and tangles, in which one meets alternatingly over- and under-crossings. For n 8 (resp. -
Supersymmetry and the Multi-Instanton Measure
View metadata, citation and similar papers at core.ac.uk brought to you by CORE hep-th/9708036provided by CERN Document Server Supersymmetry and the Multi-Instanton Measure Nicholas Dorey Physics Department, University of Wales Swansea Swansea SA2 8PP UK [email protected] Valentin V. Khoze Department of Physics, Centre for Particle Theory, University of Durham Durham DH1 3LE UK [email protected] and Michael P. Mattis Theoretical Division T-8, Los Alamos National Laboratory Los Alamos, NM 87545 USA [email protected] We propose explicit formulae for the integration measure on the moduli space of charge-n ADHM multi-instantons in N = 1 and N =2super- symmetric gauge theories. The form of this measure is fixed by its (su- per)symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n ≤ 2. Knowledge of the measure for all n al- lows us to revisit, and strengthen, earlier N = 2 results, chiefly: (1) For any number of flavors NF , we provide a closed formula for Fn, the n-instanton contribution to the Seiberg-Witten prepotential, as a finite-dimensional col- lective coordinate integral. This amounts to a solution, in quadratures, of the Seiberg-Witten models, without appeal to electric-magnetic duality. (2) In the conformal case NF =4,this means reducing to quadratures the previously unknown finite renormalization that relates the microscopic and effective coupling constants, τmicro and τeff. (3) Similar expressions are given for the 4-derivative/8-fermion term in the gradient expansion of N = 2 supersymmetric QCD.