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

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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;
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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

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.

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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

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.