3 years ago

# Geometric classification of 4d $\mathcal{N}=2$ SCFTs.

Matteo Caorsi, Sergio Cecotti

The classification of 4d $\mathcal{N}=2$ SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected $\mathbb{Q}$-factorial log-Fano variety with Hodge numbers $h^{p,q}=\delta_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring $\mathscr{R}$ is a graded polynomial ring generated by global holomorphic functions $u_i$ of dimension $\Delta_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension $k$-tuples $\{\Delta_1,\Delta_2,\cdots,\Delta_k\}$ which are realized as Coulomb branch dimensions of some rank-$k$ CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible $\{\Delta_1,\cdots,\Delta_k\} s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory. The number$\boldsymbol{N}(k)$of dimensions allowed in rank$k$is given by a certain sum of the Erd\"os-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large$k \boldsymbol{N}(k)=\frac{2\,\zeta(2)\,\zeta(3)}{\zeta(6)}\,k^2+o(k^2). $In the special case$k=2$our dimension formula reproduces a recent result by Argyres et al. Class Field Theory implies a subtlety: certain dimension$k$-tuples$\{\Delta_1,\cdots,\Delta_k\}$are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted. We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few$k

s.

Publisher URL: http://arxiv.org/abs/1801.04542

DOI: arXiv:1801.04542v3

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s. For Lagrangian SCFTs the Universal\nFormula reduces to the fundamental theorem of Springer Theory.\n

\n

The number $\\boldsymbol{N}(k)$ of dimensions allowed in rank $k$ is given by\na certain sum of the Erd\\\"os-Bateman Number-Theoretic function (sequence\nA070243 in OEIS) so that for large $k$ $\n\\boldsymbol{N}(k)=\\frac{2\\,\\zeta(2)\\,\\zeta(3)}{\\zeta(6)}\\,k^2+o(k^2).$ In the\nspecial case $k=2$ our dimension formula reproduces a recent result by Argyres\net al.\n

\n

Class Field Theory implies a subtlety: certain dimension $k$-tuples\n$\\{\\Delta_1,\\cdots,\\Delta_k\\}$ are consistent only if supplemented by\nadditional selection rules on the electro-magnetic charges, that is, for a SCFT\nwith these Coulomb dimensions not all charges/fluxes consistent with Dirac\nquantization are permitted.\n

\n

We illustrate the various aspects with several examples and perform a number\nof explicit checks. We include tables of dimensions for the first few $k; window.__REDUX_STATE__ = {"feed":{"scrollPos":0,"openAccess":false,"performRefetch":{}},"history":{"historyChanges":0},"onboarding":{"stepsList":[{"stepId":"type","stepName":"What kind of researcher are you?","stepDesc":"","options":[]},{"stepId":"Role","stepName":"What role describes you the best?","stepDesc":"","options":[]},{"stepId":"Org","stepName":"Where do you work or study?","stepDesc":""},{"stepId":"ra","stepName":"Research Areas","stepDesc":"Select the research areas you are interested in","options":[]},{"stepId":"topics","stepName":"Topics","stepDesc":"Select the topics you are interested in","options":[]},{"stepId":"publications","stepName":"Publications","stepDesc":"We have selected some popular publications for you to follow","options":[]},{"stepId":"feeds","stepName":"Feeds","stepDesc":"We have created this feed based on your interests, you can edit and add more from the side menu","options":[]}],"step":1,"loading":false,"loadingText":"Loading...","selections":[{"name":"type","selection":null,"type":"single","mandatory":true},{"name":"role","selection":null,"type":"single","mandatory":true},{"name":"work_study","selection":null,"type":"single","mandatory":false},{"name":"ra","selection":[],"type":"multiple","mandatory":true},{"name":"topics","selection":[],"type":"multiple","mandatory":true},{"name":"publications","selection":[],"type":"multiple","mandatory":false},{"name":"feeds","selection":[],"type":"multiple","mandatory":false}],"topicsNextCursor":null,"topicsFetchingNext":false}}; 3 years ago # Geometric classification of 4d$\mathcal{N}=2$SCFTs. Matteo Caorsi, Sergio Cecotti The classification of 4d$\mathcal{N}=2$SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected$\mathbb{Q}$-factorial log-Fano variety with Hodge numbers$h^{p,q}=\delta_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring$\mathscr{R}$is a graded polynomial ring generated by global holomorphic functions$u_i$of dimension$\Delta_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension$k$-tuples$\{\Delta_1,\Delta_2,\cdots,\Delta_k\}$which are realized as Coulomb branch dimensions of some rank-$k$CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible$\{\Delta_1,\cdots,\Delta_k\}

s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory.

The number $\boldsymbol{N}(k)$ of dimensions allowed in rank $k$ is given by a certain sum of the Erd\"os-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large $k$ $\boldsymbol{N}(k)=\frac{2\,\zeta(2)\,\zeta(3)}{\zeta(6)}\,k^2+o(k^2).$ In the special case $k=2$ our dimension formula reproduces a recent result by Argyres et al.

Class Field Theory implies a subtlety: certain dimension $k$-tuples $\{\Delta_1,\cdots,\Delta_k\}$ are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted.

We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few $k s. Publisher URL: http://arxiv.org/abs/1801.04542 DOI: arXiv:1801.04542v3 You might also like Discover & Discuss Important Research Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free. Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article. s.\n ","htmlAbstract":null,"createdDate":"2019-10-03T09:42:45.000Z","media":null,"settings":null,"pdfUrl":"https://arxiv.org/pdf/1801.04542","openAccessUrl":null,"authors":{"type":"json","json":["Matteo Caorsi","Sergio Cecotti"]},"contentType":null,"actionButton":null,"journal":{"type":"id","generated":false,"id":"Journal:1221","typename":"Journal"},"headlineImage":null,"__typename":"V4Paper","doi":"arXiv:1801.04542v3","paperUrl":"http://arxiv.org/abs/1801.04542","metrics":{"type":"id","generated":true,"id":"$V4Paper:295094.metrics","typename":"Metrics"}},"Journal:1221":{"id":"1221","name":"arXiv High Energy Physics - Theory","cover":{"type":"id","generated":true,"id":"$Journal:1221.cover","typename":"Image"},"__typename":"Journal"},"$Journal:1221.cover":{"baseURL":"https://s3-eu-west-1.amazonaws.com/stackademic/assets/journal_cover_arxviv.png","__typename":"Image"},"$V4Paper:295094.metrics":{"selectedCount":null,"viewedCount":null,"bookmarkedCount":null,"__typename":"Metrics"}}; window.__REDUX_STATE__ = {"feed":{"scrollPos":0,"openAccess":false,"performRefetch":{}},"history":{"historyChanges":0},"onboarding":{"stepsList":[{"stepId":"type","stepName":"What kind of researcher are you?","stepDesc":"","options":[]},{"stepId":"Role","stepName":"What role describes you the best?","stepDesc":"","options":[]},{"stepId":"Org","stepName":"Where do you work or study?","stepDesc":""},{"stepId":"ra","stepName":"Research Areas","stepDesc":"Select the research areas you are interested in","options":[]},{"stepId":"topics","stepName":"Topics","stepDesc":"Select the topics you are interested in","options":[]},{"stepId":"publications","stepName":"Publications","stepDesc":"We have selected some popular publications for you to follow","options":[]},{"stepId":"feeds","stepName":"Feeds","stepDesc":"We have created this feed based on your interests, you can edit and add more from the side menu","options":[]}],"step":1,"loading":false,"loadingText":"Loading...","selections":[{"name":"type","selection":null,"type":"single","mandatory":true},{"name":"role","selection":null,"type":"single","mandatory":true},{"name":"work_study","selection":null,"type":"single","mandatory":false},{"name":"ra","selection":[],"type":"multiple","mandatory":true},{"name":"topics","selection":[],"type":"multiple","mandatory":true},{"name":"publications","selection":[],"type":"multiple","mandatory":false},{"name":"feeds","selection":[],"type":"multiple","mandatory":false}],"topicsNextCursor":null,"topicsFetchingNext":false}}; 3 years ago # Geometric classification of 4d$\mathcal{N}=2$SCFTs. Matteo Caorsi, Sergio Cecotti The classification of 4d$\mathcal{N}=2$SCFTs boils down to the classification of conical special geometries with closed Reeb orbits (CSG). Under mild assumptions, one shows that the underlying complex space of a CSG is (birational to) an affine cone over a simply-connected$\mathbb{Q}$-factorial log-Fano variety with Hodge numbers$h^{p,q}=\delta_{p,q}$. With some plausible restrictions, this means that the Coulomb branch chiral ring$\mathscr{R}$is a graded polynomial ring generated by global holomorphic functions$u_i$of dimension$\Delta_i$. The coarse-grained classification of the CSG consists in listing the (finitely many) dimension$k$-tuples$\{\Delta_1,\Delta_2,\cdots,\Delta_k\}$which are realized as Coulomb branch dimensions of some rank-$k$CSG: this is the problem we address in this paper. Our sheaf-theoretical analysis leads to an Universal Dimension Formula for the possible$\{\Delta_1,\cdots,\Delta_k\}

s. For Lagrangian SCFTs the Universal Formula reduces to the fundamental theorem of Springer Theory.

The number $\boldsymbol{N}(k)$ of dimensions allowed in rank $k$ is given by a certain sum of the Erd\"os-Bateman Number-Theoretic function (sequence A070243 in OEIS) so that for large $k$ $\boldsymbol{N}(k)=\frac{2\,\zeta(2)\,\zeta(3)}{\zeta(6)}\,k^2+o(k^2).$ In the special case $k=2$ our dimension formula reproduces a recent result by Argyres et al.

Class Field Theory implies a subtlety: certain dimension $k$-tuples $\{\Delta_1,\cdots,\Delta_k\}$ are consistent only if supplemented by additional selection rules on the electro-magnetic charges, that is, for a SCFT with these Coulomb dimensions not all charges/fluxes consistent with Dirac quantization are permitted.

We illustrate the various aspects with several examples and perform a number of explicit checks. We include tables of dimensions for the first few \$k

s.

Publisher URL: http://arxiv.org/abs/1801.04542

DOI: arXiv:1801.04542v3

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Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.