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Functorial String Diagrams for Reverse-Mode Automatic Differentiation [article]

Mario Alvarez-Picallo, Dan R. Ghica, David Sprunger, Fabio Zanasi
2021 arXiv   pre-print
Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness  ...  We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure.  ...  ACKNOWLEDGMENTS This material is based upon work supported by the Engineering and Physical Sciences Research Council (UK) under Grant EP/V001612/1 (Nominal String Diagrams).  ... 
arXiv:2107.13433v1 fatcat:2bnsqea5nvgxhhsakuwlhwq6oi

Membrane sigma-models and quantization of non-geometric flux backgrounds

Dionysios Mylonas, Peter Schupp, Richard J. Szabo
2012 Journal of High Energy Physics  
We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M.  ...  We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space.  ...  In the following we will take the topological limit of (2.32) where g ≪ R; this essentially decouples the open string modes from the closed string modes.  ... 
doi:10.1007/jhep09(2012)012 fatcat:jaslw2ndsnawzjd6rdaog6iqjm

From Loop Space Mechanics to Nonabelian Strings [article]

Urs Schreiber
2005 arXiv   pre-print
In particular, there is a 2-group related to the String-group which should be the right structure 2-group for the global description of spinning strings.  ...  We show how these relate to nonabelian gerbes and go beyond by obtaining global nonabelian surface holonomy, thus providing a class of action functionals for nonabelian strings.  ...  (The 0-mode of the worldsheet supercharge is a generalized Dirac(-Ramond) operator on loop space (for the closed string).)  ... 
arXiv:hep-th/0509163v1 fatcat:xt5ae2mtkjahlbrzaiam5r3inq

Belief propagation in monoidal categories

Jason Morton
2014 Electronic Proceedings in Theoretical Computer Science  
Reverse-mode automatic differentiation is categorical belief propagation in the category of vectors and matrices over D.  ...  Compact closed, spidered, and dungeon categories A compact closed category is a monoidal category with duals for objects and a compatible symmetric braiding, and string diagrams define the free compact  ...  An interpretation [27] assigns values to each of the variables in the tensor scheme in a functorial way.  ... 
doi:10.4204/eptcs.172.18 fatcat:fa37qdcblbfwpndjwedbzziowi

Tensor constructions of open string theories (I) foundations

Matthias R. Gaberdiel, Barton Zwiebach
1997 Nuclear Physics B  
The resulting string theory is shown to be physically equivalent to a string theory with a familiar open string gauge group.  ...  The possible tensor constructions of open string theories are analyzed from first principles.  ...  Stasheff for many helpful discussions, for explanations on homotopy equivalence, and for criticisms on a draft version of this paper. We are also grateful to S. Halperin, A.  ... 
doi:10.1016/s0550-3213(97)00580-4 fatcat:643mpzxqlzgizhasx47kgqnj2m

Quantization via Linear homotopy types [article]

Urs Schreiber
2014 arXiv   pre-print
For the reader inclined to foundations we provide a refinement of the proposal by Lawvere for a formal foundation of physics, lifted from classical continuum mechanics to local Lagrangian quantum gauge  ...  For the reader inclined to the interpretation of quantum mechanics we exhibit quantum superposition and interference as existential quantification in linear homotopy-type theory.  ...  Thanks to Adeel Khan Yusufzai for discussion of aspects of geometric motives and for pointing out Tabuada's proof of the relation to noncommutative motives; and thanks to Thomas Nikolaus for pointing out  ... 
arXiv:1402.7041v1 fatcat:yl2efwi2bfgzhdrxenirukvplu

The spectral gluing theorem revisited [article]

Dario Beraldo
2020 arXiv   pre-print
For instance, the category of G-Fourier modes I(G, G) coincides with QCoh(LS G ).  ...  where String(Par) is the 1-category of strings of parabolics defined in [AG18, Section 7.1.2], and β is the natural map that retains only the first and last element of a string.  ... 
arXiv:1804.04861v4 fatcat:5gcqyte6ojgadntz2l623k3hwy

The spectral gluing theorem revisited

Dario Beraldo
2020 Épijournal de Géométrie Algébrique  
For instance, the category of G-Fourier modes I(G, G) coincides with QCoh(LS G ).  ...  where String(Par) is the 1-category of strings of parabolics defined in [AG18, Section 7.1.2], and β is the natural map that retains only the first and last element of a string.  ... 
doi:10.46298/epiga.2020.volume4.5940 fatcat:woze3v3vaff33bbjxpgklyjp34

Extended Functorial Field Theories and Anomalies in Quantum Field Theories [article]

Lukas Müller
2020 arXiv   pre-print
We develop a general framework for the description of anomalies using extended functorial field theories extending previous work by Freed and Monnier.  ...  We provide precise mathematical definitions for all concepts involved using the language of symmetric monoidal bicategories.  ...  (Co)limits in Grpd are in general not compatible with equivalences of categories: consider the equivalence of diagrams in is a string of morphisms d 0 → d 1 → · · · → d n in D.  ... 
arXiv:2003.08217v1 fatcat:kalhwhz5wvgxzdnf6pilkdzif4

Azumaya-type noncommutative spaces and morphisms therefrom: Polchinski's D-branes in string theory from Grothendieck's viewpoint [article]

Chien-Hao Liu, Shing-Tung Yau
2007 arXiv   pre-print
Several originally open-string induced properties of D-branes can be reproduced solely by this intrinsic definition.  ...  We thank Andrew Strominger and Cumrun Vafa for lectures/discussions that influence the project; Duiliu-Emanuel Diaconescu, Gang Liu, Kefeng Liu, Pan Peng for discussions on open/closed string duality;  ...  [Ox]; participants of GSS, organized by Monica Maria Guica, for discussions on themes in string theory; Nima Arkani-Hamed, John Duncan, Dennis Gaitsgory, Katrin Wehrheim for topic courses; Rev.  ... 
arXiv:0709.1515v1 fatcat:ts33tdmmqbeshbmvf2tigxyxju

Categories of Physical Processes [article]

Stanisław Szawiel
2017 arXiv   pre-print
Both functors and families of objects and morphisms can be differentiated, and these two modes of differentiation lead to the traditional differential equations of quantum theory (Heisenberg and Schrödinger  ...  The functorial nature of GNS makes this essentially trivial. We show how to deal with time reversal and inhomogeneous, irreversible time evolution.  ...  A On The Notion of Gauge Theory The ideas presented here are not really new, but deserve being intensely stressed, for they deeply challenge any claim to understanding the general notion of gauge theory  ... 
arXiv:1709.09096v2 fatcat:3zlwsidjevednbdavqbpaxmbwa

Knowledge Representation in Bicategories of Relations [article]

Evan Patterson
2017 arXiv   pre-print
On a practical level, we show by example that relational ologs have a friendly and intuitive--yet fully precise--graphical syntax, derived from the string diagrams of monoidal categories.  ...  It is inspired by Spivak and Kent's olog, a recent categorical framework for knowledge representation.  ...  Finally, I thank the many contributors to the nLab for creating an invaluable online resource for aspiring and professional category theorists.  ... 
arXiv:1706.00526v2 fatcat:iyr2xxdteveqzlzjy6l6gq73oe

STRINGS, GAUGE FIELDS AND MEMBRANES [chapter]

RICHARD J. SZABO
2005 From Fields to Strings: Circumnavigating Theoretical Physics  
We present an overview of the intimate relationship between string and D-brane dynamics, and the dynamics of gauge and gravitational fields in three spacetime dimensions.  ...  The successes, prospects and open problems in describing both perturbative and nonperturbative aspects of string theory in terms of three-dimensional quantum field theory are highlighted.  ...  Polyakov for interesting discussions and correspondence. This work was supported in part by an Advanced Fellowship from the Particle Physics and Astronomy Research Council (U.K.).  ... 
doi:10.1142/9789812775344_0047 fatcat:dmj6i32onnakziiss6m6r57y5u

On the Existence of Heterotic-String and Type-II-Superstring Field Theory Vertices [article]

Seyed Faroogh Moosavian, Yehao Zhou
2019 arXiv   pre-print
Using this, we prove that heterotic-string and type-II-superstring field theory vertices exist.  ...  We consider two problems associated to the space of Riemann surfaces that are closely related to string theory.  ...  We also thank Ivo Sachs for useful discussions. SFM is grateful to Davide Gaiotto for his support during his stay at Perimeter Institute.  ... 
arXiv:1911.04343v1 fatcat:hioi6zojhngt5huhc5q5ylerky

T-Duality: Topology Change from H-Flux

Peter Bouwknegt, Jarah Evslin, Varghese Mathai
2004 Communications in Mathematical Physics  
When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G_4 fails.  ...  Acknowledgements We would like to thank Nick Halmagyi, Michael Schulz and Eric Sharpe for help and crucial references.  ...  JE would like to thank the University of Adelaide for hospitality during this project. PB and VM are financially supported by the Australian Research Council and JE by the INFN Sezione di Pisa.  ... 
doi:10.1007/s00220-004-1115-6 fatcat:v7bpqo6mqfajrktlddxog6kxia
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