GLSM's for gerbes (and other toric stacks)

In this paper we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) ${\bf C}^{\times}$ quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks, and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also check in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFT's to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using Morrison-Plesser-Hori-Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev's mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFT's, involving fields valued in roots of unity.Comment: 43 pages, LaTeX, 3 figures; v2: typos fixe

Decomposition and the Gross-Taylor string theory

It was recently argued by Nguyen-Tanizaki-Unsal that two-dimensional pure Yang-Mills theory is equivalent to (decomposes into) a disjoint union of (invertible) quantum field theories, known as universes. In this paper we compare this decomposition to the Gross-Taylor expansion of two-dimensional pure SU(N) Yang-Mills theory in the large N limit as the string field theory of a sigma model. Specifically, we study the Gross-Taylor expansion of individual Nguyen-Tanizaki-Unsal universes. These differ from the Gross-Taylor expansion of the full Yang-Mills theory in two ways: a restriction to single instanton degrees, and some additional contributions not present in the expansion of the full Yang-Mills theory. We propose to interpret the restriction to single instanton degree as implying a constraint, namely that the Gross-Taylor string has a global (higher-form) symmetry with Noether current related to the worldsheet instanton number. We compare two-dimensional pure Maxwell theory as a prototype obeying such a constraint, and also discuss in that case an analogue of the Witten effect arising under two-dimensional theta angle rotation. We also propose a geometric interpretation of the additional terms, in the special case of Yang-Mills theories on two-spheres. In addition, also for the case of theories on two-spheres, we propose a reinterpretation of the terms in the Gross-Taylor expansion of the Nguyen-Tanizaki-Unsal universes, replacing sigma models on branched covers by counting disjoint unions of stacky copies of the target Riemann surface, that makes the Nguyen-Tanizaki-Unsal decomposition into invertible field theories more nearly manifest. As the Gross-Taylor string is a sigma model coupled to worldsheet gravity, we also briefly outline the tangentially-related topic of decomposition in two-dimensional theories coupled to gravity.Comment: 95 pages, LaTe

Cluster decomposition, T-duality, and gerby CFT's

In this paper we study CFT's associated to gerbes. These theories suffer from a lack of cluster decomposition, but this problem can be resolved: the CFT's are the same as CFT's for disconnected targets. Such theories also lack cluster decomposition, but in that form, the lack is manifestly not very problematic. In particular, we shall see that this matching of CFT's, this duality between noneffective gaugings and sigma models on disconnected targets, is a worldsheet duality related to T-duality. We perform a wide variety of tests of this claim, ranging from checking partition functions at arbitrary genus to D-branes to mirror symmetry. We also discuss a number of applications of these results, including predictions for quantum cohomology and Gromov-Witten theory and additional physical understanding of the geometric Langlands program.Comment: 61 pages, LaTeX; v2,3: typos fixed; v4: writing improved in several sections; v5: typos fixe

Non-birational twisted derived equivalences in abelian GLSMs

In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between non-birational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with non-birational Kahler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kahler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov's `homological projective duality.' Along the way, we shall see how `noncommutative spaces' (in Kontsevich's sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.Comment: 54 pages, LaTeX; v2: typo fixe

Quantization of Fayet-Iliopoulos Parameters in Supergravity

In this short note we discuss quantization of the Fayet-Iliopoulos parameter in supergravity theories. We argue that in supergravity, the Fayet-Iliopoulos parameter determines a lift of the group action to a line bundle, and such lifts are quantized. Just as D-terms in rigid N=1 supersymmetry are interpreted in terms of moment maps and symplectic reductions, we argue that in supergravity the quantization of the Fayet-Iliopoulos parameter has a natural understanding in terms of linearizations in geometric invariant theory (GIT) quotients, the algebro-geometric version of symplectic quotients.Comment: 21 pages, utarticle class; v2: typos and tex issue fixe

Effects of musical training and event probabilities on encoding of complex tone patterns

Background: The human auditory cortex automatically encodes acoustic input from the environment and differentiates regular sound patterns from deviant ones in order to identify important, irregular events. The Mismatch Negativity (MMN) response is a neuronal marker for the detection of sounds that are unexpected, based on the encoded regularities. It is also elicited by violations of more complex regularities and musical expertise has been shown to have an effect on the processing of complex regularities. Using magnetoencephalography (MEG), we investigated the MMN response to salient or less salient deviants by varying the standard probability (70%, 50% and 35%) of a pattern oddball paradigm. To study the effects of musical expertise in the encoding of the patterns, we compared the responses of a group of non-musicians to those of musicians. Results: We observed significant MMN in all conditions, including the least salient condition (35% standards), in response to violations of the predominant tone pattern for both groups. The amplitude of MMN from the right hemisphere was influenced by the standard probability. This effect was modulated by long-term musical training: standard probability changes influenced MMN amplitude in the group of non-musicians only. Conclusion: This study indicates that pattern violations are detected automatically, even if they are of very low salience, both in non-musicians and musicians, with salience having a stronger impact on processing in the right hemisphere of non-musicians. Long-term musical training influences this encoding, in that non-musicians benefit to a greater extent from a good signal-to-noise ratio (i.e. high probability of the standard pattern), while musicians are less dependent on the salience of an acoustic environment.<br

Playing and Listening to Tailor-Made Notched Music: Cortical Plasticity Induced by Unimodal and Multimodal Training in Tinnitus Patients

Background. The generation and maintenance of tinnitus are assumed to be based on maladaptive functional cortical reorganization. Listening to modified music, which contains no energy in the range of the individual tinnitus frequency, can inhibit the corresponding neuronal activity in the auditory cortex. Music making has been shown to be a powerful stimulator for brain plasticity, inducing changes in multiple sensory systems. Using magnetoencephalographic (MEG) and behavioral measurements we evaluated the cortical plasticity effects of two months of (a) active listening to (unisensory) versus (b) learning to play (multisensory) tailor-made notched music in nonmusician tinnitus patients. Taking into account the fact that uni- and multisensory trainings induce different patterns of cortical plasticity we hypothesized that these two protocols will have different affects. Results. Only the active listening (unisensory) group showed significant reduction of tinnitus related activity of the middle temporal cortex and an increase in the activity of a tinnitus-coping related posterior parietal area. Conclusions. These findings indicate that active listening to tailor-made notched music induces greater neuroplastic changes in the maladaptively reorganized cortical network of tinnitus patients while additional integration of other sensory modalities during training reduces these neuroplastic effects