arXiv subjects

Sanath K. Devalapurkar

Publications and source records attributed to Sanath K. Devalapurkar.

15 recordsLinked to original sources

arXiv2026

THH(Z) and the image of J

Let $p$ be an odd prime number and $\mathrm{j}_p$ the $p$-complete connective image of J spectrum. We establish an equivalence of cyclotomic $\mathbb{E}_\infty$-rings $\mathrm{THH}(\mathbb{Z})^{\wedge}_p \simeq \mathrm{sh}(\mathrm{j}_p^{\mathrm{triv}})$ and an equivalence of $\mathbb{E}_\infty$-rings $\mathrm{TP}(\mathbb{Z})^{\wedge}_p \simeq \mathrm{j}_p^{\mathrm{t}\mathrm{S}^1}$. We also record a few applications of this: a new perspective, with some new information, on the description of $\mathrm{TC}(\mathbb{Z})^{\wedge}_p$ as a spectrum; height $1$ analogues of the fiber squares of Antieau-Mathew-Morrow-Nikolaus, resulting in new calculations in $\mathrm{K}(1)$-localized algebraic K-theory; and a proof of a slight refinement of the noncommutative crystalline-de Rham comparison result of Petrov-Vologodsky.

math.AT↗

arXiv2026

The Smith Fiber Sequence and Invertible Field Theories

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in a companion paper.

math.AT↗

arXiv2025

$p$-typical curves on $p$-adic Tate twists and de Rham-Witt forms

We show that de Rham--Witt forms are naturally isomorphic to $p$-typical curves on $p$-adic Tate twists, which answers a question of Artin--Mazur from 1977 pursued in the earlier work of Bloch and Kato. We show this by more generally equipping a related result of Hesselholt on topological cyclic homology with the motivic filtrations introduced by Bhatt--Morrow--Scholze.

math.KT↗

arXiv2024

Chromatic aberrations of geometric Satake over the regular locus

Let $G$ be a connected, simply-laced, almost simple algebraic group over $\mathbf{C}$, let $G_c$ be a maximal compact subgroup of $G(\mathbf{C})$, and let $T_c$ be a maximal torus therein. Let $\mathrm{Gr}_G$ denote the affine Grassmannian of $G$, and let $\check{G}$ denote the Langlands dual group to $G$ with Lie algebra $\check{\mathfrak{g}}$. The derived geometric Satake equivalence of Bezrukavnikov-Finkelberg gives an equivalence between the $\infty$-category $\mathrm{Loc}_{G_c}(\mathrm{Gr}_G; \mathbf{C})$ of $G_c$-equivariant local systems of $\mathbf{C}$-vector spaces on $\mathrm{Gr}_G$ and the $\infty$-category of quasicoherent sheaves on a large open substack of $\check{\mathfrak{g}}^\ast[2]/\check{G}$. In this article, we study the analogous story when $\mathrm{Loc}_{G_c}(\mathrm{Gr}_G; \mathbf{C})$ is replaced by the $\infty$-category of $T_c$-equivariant local systems of $k$-modules over $\mathrm{Gr}_G(\mathbf{C})$, where $k$ is ($2$-periodic) rational cohomology, (complex) K-theory, or elliptic cohomology. Crucial to our work is the genuine equivariant refinement of these cohomology theories. We show that, although there may not be an equivalence as in derived geometric Satake, the $\infty$-category $\mathrm{Loc}_{T_c}(\mathrm{Gr}_G; k)$ admits a 1-parameter degeneration to an $\infty$-category of quasicoherent sheaves built out of the geometry of various Langlands-dual stacks associated to $k$ and the $1$-dimensional group scheme computing $S^1$-equivariant $k$-cohomology. For example, when $k$ is an elliptic cohomology theory with elliptic curve $E$, the $\infty$-category $\mathrm{Loc}_{T_c}(\mathrm{Gr}_G; k)$ degenerates to the $\infty$-category of quasicoherent sheaves on a large open locus in the moduli stack of $\check{B}$-bundles of degree $0$ on $E$. We also study several applications of these equivalences.

math.RT↗

arXiv2024

Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$

The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper $\mathbf{F}_p$-schemes $X$ with $\dim(X)<p^n$ as soon as its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum $\mathrm{BP}\langle n-1\rangle$, and the Hochschild-Kostant-Rosenberg spectral sequence for $X$ degenerates at the $E_2$-page. This is obtained from a noncommutative version, whose proof is essentially the same as Mathew's argument in arXiv:1710.09045.

math.AT↗

arXiv2024

A Long Exact Sequence in Symmetry Breaking: order parameter constraints, defect anomaly-matching, and higher Berry phases

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is studied in a companion paper. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases.

hep-th↗

arXiv2024

ku-theoretic spectral decompositions for spheres and projective spaces

Ben-Zvi--Sakellaridis--Venkatesh described a conjectural extension of the geometric Satake equivalence to spherical varieties, whose spectral decomposition is described by Hamiltonian varieties. The goal of this article is to study their conjecture, especially in the case of spherical varieties of relative rank 1, using tools from homotopy theory. Our discussion relates their conjecture to classical topics in homotopy theory such as the EHP sequence and Hopf fibrations, as well as more modern topics such as Hochschild (co)homology. We will also study an analogue of the derived geometric Satake equivalence and of the Ben-Zvi--Sakellaridis--Venkatesh conjecture with coefficients in connective complex K-theory. In this generalized setting, the dual group (a la Langlands, Gaitsgory--Nadler, Sakellaridis--Venkatesh, Knop--Schalke) remains unchanged, but the specific dual "representation" of the dual group changes. On the spectral/Langlands dual side, we expect that the appropriate replacement of Hamiltonian varieties are given by what we term "ku-Hamiltonian varieties"; this is a notion interpolating between Hamiltonian and quasi-Hamiltonian varieties (a la Alekseev--Malkin--Meinrenken). Finally, we suggest possible generalizations to more exotic cohomology theories such as complex cobordism.

math.AT↗

arXiv2024

Derived geometric Satake for $\mathrm{PGL}_2^{\times 3}/\mathrm{PGL}_2^\mathrm{diag}$

In this note, we study the local relative geometric Langlands conjecture of Ben-Zvi--Sakellaridis--Venkatesh for the spherical subgroup $\mathrm{PGL}_2^\mathrm{diag}$ of the triple product $\mathrm{PGL}_2^{\times 3}$ (and also for the spherical subgroup $\mathrm{G}_2$ of $\mathrm{SO}_8/μ_2$), whose corresponding Langlands dual $\mathrm{SL}_2^{\times 3}$-variety can be identified with the symplectic vector space $(\mathbf{A}^2)^{\otimes 3} \cong \mathbf{A}^8$ of $2\times 2 \times 2$-cubes. Our analysis relies on a construction of Bhargava relating $2 \times 2 \times 2$-cubes to Gauss composition on quadratic forms, arising here as the moment map for the Hamiltonian $\mathrm{SL}_2^{\times 3}$-action on $(\mathbf{A}^2)^{\otimes 3}$, and the Cayley hyperdeterminant as studied by Gelfand-Kapranov-Zelevinsky.

math.RT↗

arXiv2023

Generalized $n$-series and de Rham complexes

The goal of this article is to study some basic algebraic and combinatorial properties of "generalized $n$-series" over a commutative ring $R$, which are functions $s: \mathbf{Z}_{\geq 0} \to R$ satisfying a mild condition. A special example of generalized $n$-series is given by the $q$-integers $\frac{q^n-1}{q-1} \in \mathbf{Z}[\![q-1]\!]$. Given a generalized $n$-series $s$, one can define $s$-analogues of factorials (via $n!_s = \prod_{i=1}^n s(n)$) and binomial coefficients. We prove that Pascal's identity, the binomial identity, Lucas' theorem, and the Vandermonde identity admit $s$-analogues; each of these specialize to their appropriate $q$-analogue in the case of the $q$-integer generalized $n$-series. We also study the growth rates of generalized $n$-series defined over the integers. Finally, we define an $s$-analogue of the ($q$-)derivative, and prove $s$-analogues of the Poincaré lemma and the Cartier isomorphism for the affine line, as well as a pullback square due to Bhatt-Lurie.

math.CO↗

arXiv2021

On the James and Hilton-Milnor Splittings, & the metastable EHP sequence

This note provides modern proofs of some classical results in algebraic topology, such as the James Splitting, the Hilton-Milnor Splitting, and the metastable EHP sequence. We prove fundamental splitting results \begin{equation*} ΣΩΣX \simeq ΣX \vee (X\wedge ΣΩΣX) \quad \text{and} \quad Ω(X \vee Y) \simeq ΩX\times ΩY\times ΩΣ(ΩX \wedge ΩY) \end{equation*} in the maximal generality of an $\infty$-category with finite limits and pushouts in which pushouts squares remain pushouts after basechange along an arbitrary morphism (i.e., Mather's Second Cube Lemma holds). For connected objects, these imply the classical James and Hilton-Milnor Splittings. Moreover, working in this generality shows that the James and Hilton-Milnor splittings hold in many new contexts, for example in: elementary $\infty$-topoi, profinite spaces, and motivic spaces over arbitrary base schemes. The splitting result in this last context extend Wickelgren and Williams' splitting result for motivic spaces over a perfect field. We also give two proofs of the metastable EHP sequence in the setting of $\infty$-topoi: the first is a new, non-computational proof that only utilizes basic connectedness estimates involving the James filtration and the Blakers-Massey Theorem, while the second reduces to the classical computational proof.

math.AT↗

arXiv2020

The Ando-Hopkins-Rezk orientation is surjective

We show that the map $π_\ast \mathrm{MString}\to π_\ast \mathrm{tmf}$ induced by the Ando-Hopkins-Rezk orientation is surjective. This proves an unpublished claim of Hopkins and Mahowald. We do so by constructing an $\mathbf{E}_1$-ring $B$ and a map $B \to \mathrm{MString}$ such that the composite $B \to \mathrm{MString} \to \mathrm{tmf}$ is surjective on homotopy. Applications to differential topology, and in particular to Hirzebruch's prize question, are discussed.

math.AT↗

arXiv2019

Hodge theory for elliptic curves and the Hopf element $ν$

We show that the vector bundle on the moduli stack $M_\mathrm{ell}$ of elliptic curves associated to the $2$-cell complex $Cν$ is isomorphic to the de Rham cohomology sheaf $\mathrm{H}^1_\mathrm{dR}(\mathcal{E}/M_\mathrm{ell})$ of the universal elliptic curve $\mathcal{E}\to M_\mathrm{ell}$. We use this to calculate the homotopy groups of the $\mathbf{E}_{1}$-quotient $\mathrm{tmf} /\!\!/ ν$ of $\mathrm{tmf}$ by $ν$, called the spectrum of "topological quasimodular forms", by relating its Adams-Novikov spectral sequence to the cohomology of the moduli stack of cubic curves with a chosen splitting of the Hodge-de Rham filtration.

math.AT↗

arXiv2019

Roots of unity in $K(n)$-local rings

The goal of this paper is to address the following question: if $A$ is an $\mathbf{E}_{k}$-ring for some $k\geq 1$ and $f\colonπ_0 A \to B$ is a map of commutative rings, when can we find an $\mathbf{E}_{k}$-ring $R$ with an $\mathbf{E}_{k}$-ring map $g\colon A \to R$ such that $π_0 g = f$? A classical result in the theory of realizing $\mathbf{E}_\infty$-rings, due to Goerss--Hopkins, gives an affirmative answer to this question if $f$ is etale. The goal of this paper is to provide answers to this question when $f$ is ramified. We prove a non-realizability result in the $K(n)$-local setting for every $n\geq 1$ for $H_\infty$-rings containing primitive $p$th roots of unity. As an application, we give a proof of the folk result that the Lubin--Tate tower from arithmetic geometry does not lift to a tower of $H_\infty$-rings over Morava $E$-theory.

math.AT↗

arXiv2018

The Lubin-Tate stack and Gross-Hopkins duality

Morava $E$-theory $E$ is an $E_\infty$-ring with an action of the Morava stabilizer group $Γ$. We study the derived stack $\operatorname{Spf} E/Γ$. Descent-theoretic techniques allow us to deduce a theorem of Hopkins-Mahowald-Sadofsky on the $K(n)$-local Picard group, as well as a recent result of Barthel-Beaudry-Stojanoska on the Anderson duals of higher real $K$-theories.

math.AT↗

arXiv2018

Galois representations and ordinary reduction

We provide conditions on the p-adic Galois representation of a smooth proper variety over a complete nonarchimedean extension of Q_p to have (potentially) good ordinary reduction.

math.AG↗